关于Spherical Harmonic
EGM2008
EGM2008 - WGS 84 VersionIntroductionThe official Earth Gravitational Model EGM2008 has been publicly released by the U.S. National Geospatial-Intelligence Agency (NGA) EGM Development Team. This gravitational model is complete to spherical harmonic degree and order 2159, and contains additional coefficients extending to degree 2190 and order 2159. Full access to the model's coefficients and other descriptive files with additional details about EGM2008 are provided herein.Those wishing to use EGM2008 to compute geoid undulation values with respect to WGS 84,may do so using the self-contained suite of coefficient files, FORTRAN software, and pre-computed geoid grids provided on this web page. For other applications, the previous release of the full 'Geoscience' package for EGM2008 can be accessed through the link at the bottom of this web page.The WGS 84 constants used to define the reference ellipsoid, and the associated normal gravity field, to which the geoid undulations are referenced are:•a=6378137.00 m (semi-major axis of WGS 84 ellipsoid)•f=1/298.257223563 (flattening of WGS 84 ellipsoid)•GM=3.986004418 x 1014 m3s-2 (Product of the Earth's mass and the Gravitational Constant)•Ï‰=7292115 x 10-11 radians/sec (Earth's angular velocity)All synthesis software, coefficients, and pre-computed geoid grids listed below assume a Tide Free system, as far as permanent tide is concerned.Note that the harmonic synthesis software provided below applies a constant,zero-degree term of -41 cm to all geoid undulations computed using EGM2008 with the height_anomaly-to-geoid_undulation correction model (also provided). Similarly, all pre-computed geoid undulations incorporate this constant zero-degree term. This term converts geoid undulations that are intrinsically referenced to an ideal mean-earth ellipsoid into undulations that are referenced to WGS 84. The value of -41 cm derives from a mean-earth ellipsoid for which the estimated parameters in the Tide Free system are: a=6378136.58 m and 1/f=298.257686.Description of Software and DataTo compute point geoid undulations from spherical harmonic synthesis of the EGM2008 Tide Free Spherical Harmonic Coefficients and its associated Correction Model, at any WGS 84 latitude/longitude coordinate pair listed in a coordinate input file (such as INPUT.DAT), use the FORTRAN harmonic synthesis program, hsynth_WGS84.f.At present, we are also providing two grids of pre-computed geoid undulations: one at 1 x 1-minute resolution and one at 2.5 x 2.5-minute resolution. To interpolate geoid undulations from the 1 x 1-Minute Geoid Undulation Grid file, for any WGS 84 latitude/longitude coordinate pair listed in a coordinate input file (such as INPUT.DAT), use the FORTRAN interpolation program interp_1min.f. Similarly, the FORTRAN interpolation program interp_2p5min.f, will interpolate geoid undulations from the 2.5 x 2.5-Minute Geoid Undulation Grid file.Filenames containing a ".gz" in the suffix have been compressed using the Unix "gzip" command.All files that are SMALL ENDIAN format are highlighted in green. All files that are BIG ENDIAN format are highlighted in purple.Please carefully review the README_WGS84 file for a complete description of coefficient and data files for computing EGM2008 geoid undulations with respect to WGS 84. Also, before using any of the data files on this web page, please read the disclaimer.For GIS data formats, please visit the EGM2008 GIS Data Page.Software and Coefficients for WGS 84 Geoid Undulation Computations by Harmonic Synthesis•EGM2008 Harmonic Synthesis Program(hsynth_WGS84.f -184 KB) - Use this FORTRAN program to generate WGS 84 geoid undulations by spherical harmonic synthesis of EGM2008 and its associated height_anomaly-to-geoid_undulation correction model. This program requires that the coefficients for both EGM2008 and the correction model, and an INPUT.DAT file, all be located in the same directory as the hsynth_WGS84.f program.•Harmonic Synthesis Executable for Windows XP(hsynth_WGS84.exe - 696 KB) - Windows executable version of EGM2008 Harmonic Synthesis Program. Use as described above.•Spherical Harmonic Coefficients for Earth's Gravitational Potential (Tide Free System)-(EGM2008_to2190_TideFree.gz - 72 MB) - EGM2008 coefficients required by the harmonic synthesis program.•Correction Coefficients(Zeta-to-N_to2160_egm2008.gz - 50 MB) -Height_anomaly-to-geoid_undulation coefficients required by the harmonic synthesis program.•INPUT.DAT(4 KB) - sample input file of WGS 84 latitude/longitude coordinate pairs for testing the harmonic synthesis program.•OUTPUT.DAT(4 KB)- sample output file of geoid undulation values, generated by reading INPUT.DAT into the harmonic synthesis program (for test verification).*Users - Please verify results by comparing to OUTPUT.DAT file immediately above. Software and Grids for WGS 84 Geoid Undulation Computation by Interpolation*Note: The 1 x 1 minute interpolation program below requires a large PC RAM capacity.•Interpolation Program for 1 x 1-Minute Geoid Grid(interp_1min.f - 28 KB) - Use this FORTRAN program to interpolate geoid undulations from the 1 x 1-Minute Geoid Undulation Grid file, for any WGS 84 latitude/longitude coordinate pair listed in a coordinate input file (such as INPUT.DAT). When applied to the 1 x 1-Minute Geoid Undulation Grid file, this interp_1min.f program will generate geoid undulation values that match the corresponding values generated by harmonic synthesis (hsynth_WGS84.f or hsynth_WGS84.exe above) to within 1 mm. The interp_1min.f program requires that the 1 x 1-Minute Geoid Undulation Grid file and the INPUT.DAT are located in the same directory as interp_1min.f.• 1 x 1-Minute Grid Interpolation Executable for Windows XP(interp_1min.exe - 434 KB) - Windows executable version of the Interpolation Program for 1 x 1-Minute Geoid Grid. Use as described above. To be used with the SMALL ENDIAN Geoid Undulation Grid below.• 1 x 1-Minute Geoid Undulation Grid in WGS 84 - SMALL ENDIAN (Und_min1x1_egm2008_isw=82_WGS84_TideFree_SE.gz - 825 MB) - 1 x 1-minute global grid of pre-computed geoid undulations. This file has a SMALL ENDIAN internal binary representation.• 1 x 1-Minute Geoid Undulation Grid in WGS 84 - BIG ENDIAN(Und_min1x1_egm2008_isw=82_WGS84_TideFree.gz - 828 MB) - 1 x 1-minute global grid of pre-computed geoid undulations. This file has a BIG ENDIAN internal binary representation.•INPUT.DAT(4 KB) - sample input file of WGS 84 latitude/longitude coordinate pairs for testing the Interpolation Program for 1 x 1-Minute Geoid Grid.•OUTPUT.DAT(4 KB)- sample output file of geoid undulation values, generated by reading INPUT.DAT into the Interpolation Program for 1 x 1-Minute Geoid Grid above (for test verification).*Users - Please verify results by comparing to OUTPUT.DAT file immediately above.•Interpolation Program for 2.5 x 2.5 Minute Geoid Grid(interp_2p5min.f - 28 KB) - Use this FORTRAN program to interpolate geoid undulations from the 2.5 x2.5-Minute Geoid Undulation Grid file, for any WGS 84 latitude/longitude coordinatepair listed in a coordinate input file (such as INPUT.DAT). When applied to the 2.5 x2.5-Minute Geoid Undulation Grid file, this interp_2p5min.f program will generategeoid undulation values that match the corresponding values generated by harmonic synthesis (hsynth_WGS84.f or hsynth_WGS84.exe above) to within 1 cm. The interp_2p5min.f program requires that the 2.5 x 2.5-Minute Geoid Undulation Grid file and the INPUT.DAT are located in the same directory as interp_2p5min.f.• 2.5 x 2.5-Minute Interpolation Executable for Windows XP(interp_2p5min.exe - 434 KB) - Windows executable version of the Interpolation Program for 2.5 x 2.5 Minute Geoid Grid. Use as described above. To be used with the SMALL ENDIAN Geoid Undulation Grid below.• 2.5 x 2.5-Minute Geoid Undulation Grid in WGS 84 - SMALL ENDIAN (Und_min2.5x2.5_egm2008_isw=82_WGS84_TideFree_SE.gz - 135 MB) - 2.5 x2.5-minute global grid of pre-computed geoid undulations. This file has a SMALLENDIAN internal binary representation.• 2.5 x 2.5-Minute Geoid Undulation Grid in WGS 84 - BIG ENDIAN(Und_min2.5x2.5_egm2008_isw=82_WGS84_TideFree.gz - 135 MB) - 2.5 x2.5-minute global grid of pre-computed geoid undulations. This file has a BIG ENDIANinternal binary representation.•INPUT.DAT(4 KB) - sample input file of WGS 84 latitude/longitude coordinate pairs for testing the Interpolation Program for 2.5 x 2.5-Minute Geoid Grid.•OUTPUT.DAT(4 KB)- sample output file of geoid undulation values, generated by reading INPUT.DAT into the Interpolation Program for 2.5 x 2.5-Minute Geoid Grid above (for test verification).*Users - Please verify results by comparing to OUTPUT.DAT file immediately above. Software to Extract WGS 84 Geoid Undulations from Grid Files (No Interpolation)•Extract 1 x 1-Minute Grid Program(gridget_1min.f -20 KB) - Use this FORTRAN program to extract a user-defined sub-rectangle of geoid undulations from the 1 x 1-Minute Geoid Undulation Grid file. This program prompts the user for a rectangular area of interest and requires the 1 x 1-Minute Geoid Undulation Grid file to be located in the same directory as the gridget_1min.f program.•Extract 1 x 1-Minute Grid Executable for Windows XP(gridget_1min.exe - 401 KB) - Windows executable version of the Extract 1 x 1-Minute Grid Program. Use as described above. To be used with the SMALL ENDIAN Geoid Undulation Grid below.• 1 x 1-Minute Geoid Undulation Grid in WGS 84 - SMALL ENDIAN (Und_min1x1_egm2008_isw=82_WGS84_TideFree_SE.gz - 825 MB) - 1 x 1-minute global grid of pre-computed geoid undulations. This file has a SMALL ENDIAN internal binary representation.• 1 x 1-Minute Geoid Undulation Grid in WGS 84 - BIG ENDIAN(Und_min1x1_egm2008_isw=82_WGS84_TideFree.gz - 828 MB) - 1 x 1-minute global grid of pre-computed geoid undulations. This file has a BIG ENDIAN internal binary representation.•Extract 2.5 x 2.5-Minute Grid Program(gridget_2p5min.f - 20 KB) - Use thisFORTRAN program to extract a user-defined sub-rectangle of geoid undulations from the 2.5 x 2.5-Minute Geoid Undulation Grid file. This program prompts the user for a rectangular area of interest and requires the 2.5 x 2.5-Minute Geoid Undulation Grid file to be located in the same directory as the gridget_2p5min.f program.•Extract 2.5 x 2.5-Minute Grid Executable for Windows XP(gridget_2p5min.exe - 397 KB) - Windows executable version of the Extract 2.5 x 2.5-Minute Grid Program.Use as described above. To be used with the SMALL ENDIAN Geoid Undulation Grid below.• 2.5 x 2.5-Minute Geoid Undulation Grid in WGS 84 - SMALL ENDIAN (Und_min2.5x2.5_egm2008_isw=82_WGS84_TideFree_SE.gz - 135 MB) - 2.5 x2.5-minute global grid of pre-computed geoid undulations. This file has a SMALLENDIAN internal binary representation.• 2.5 x 2.5-Minute Geoid Undulation Grid in WGS 84 - BIG ENDIAN(Und_min2.5x2.5_egm2008_isw=82_WGS84_TideFree.gz - 135 MB) - 2.5 x2.5-minute global grid of pre-computed geoid undulations. This file has a BIG ENDIANinternal binary representation.Additional Information•Original Release of the EGM2008 Model Coefficients from EGU General Assembly, Vienna, Austria 2008: Geoscience Package•An Earth Gravitational Model to Degree 2160: EGM2008(NPavlis&al_EGU2008.ppt - 18.5 MB) - Presentation given at the 2008 European Geosciences Union General Assembly held in Vienna, Austria, April13-18, 2008.•Background Papers from Earlier Symposiums on the New EGM。
SphericalHarmonics球谐函数的理解与使用
SphericalHarmonics球谐函数的理解与使用球谐函数（Spherical Harmonics）是用于描述球对称性的函数。
它在数学、物理、计算机图形学等领域中具有广泛的应用。
本文将对球谐函数的理解与使用进行详细介绍。
首先，我们来了解球谐函数的定义。
给定单位球面上的点(x,y,z)，球谐函数Yₗⁿ(x,y,z)定义如下：Yₗⁿ(x, y, z) = (-1)^m * sqrt((2ℓ+1)/(4π)*(ℓ-，m，)!/(ℓ+，m，)!)*Pₗ，m，(cosθ)*e^(imφ)其中，Yₗⁿ表示度为ℓ，阶为，m，的球谐函数；ℓ是非负整数，表示球谐函数的度；，m，<=ℓ，m是整数，表示球谐函数的阶；Pₗ，m，(cosθ)是勒让德多项式；θ是点(x, y, z)相对于x轴的极角；φ是点(x, y, z)相对于x轴的方位角；e是自然对数的底。
球谐函数具有下述性质：1.球谐函数是单位球面上的正交基，即不同的球谐函数之间在单位球面上的内积等于0。
2.Yₗⁿ(x,y,z)关于极角θ是奇函数，关于方位角φ是偶函数。
3.在单位球面上，球谐函数Yₗⁿ(x,y,z)的绝对值平方是一个常数，即，Yₗⁿ(x,y,z)，²在球面上处处相等。
在物理学中，球谐函数被广泛应用于描述球对称的物理场。
例如，在量子力学中，球谐函数用于描述原子中的电子波函数；在电动力学中，球谐函数用于展开电磁场的球谐分量；在量子力学中，球谐函数用于描述自旋等。
在计算机图形学中，球谐函数也被广泛应用于实时渲染、全局光照以及球形图像处理等领域。
通过将光照场或图像投影到球谐函数系数上，可以实现基于球面光照的实时渲染。
球谐函数还可以用于创建全局光照环境贴图，用于增强场景的真实感。
此外，球谐函数还可以用于球形图像处理，例如球形全景图像的压缩和展开。
值得注意的是，球谐函数展开的精度和复杂度有一定的关系。
一般来说，较高阶的球谐函数能够更准确地近似光照场或图像，但计算复杂度也会增加。
亥姆霍兹方程helm_sp
Physics116CHelmholtz’s and Laplace’s Equations in Spherical Polar Coordinates:Spherical Harmonics and Spherical Bessel FunctionsPeter Young(Dated:October23,2009)I.HELMHOLTZ’S EQUATIONAs discussed in class,when we solve the diffusion equation or wave equation by separating out the time dependence,u( r,t)=F( r)T(t),(1) the part of the solution depending on spatial coordinates,F( r),satisfies Helmholtz’s equation∇2F+k2F=0,(2)where k2is a separation constant.In this handout we willfind the solution of this equation in spherical polar coordinates.The radial part of the solution of this equation is,unfortunately,not discussed in the book,which is the reason for this handout.Note,if k=0,Eq.(2)becomes Laplace’s equation∇2F=0.We shall discuss explicitly the solution for this(important)case.A.Reminder of the Solution in Circular PolarsRecall that the solution of Helmholtz’s equation in circular polars(two dimensions)is F(r,θ)= k∞ n=0J n(kr)(A kn cos nθ+B kn sin nθ)(2dimensions),(3) where J n(kr)is a Bessel function,and we have ignored the second solution of Bessel’s equation, the Neumann function1N n(kr),which diverges at the origin.For the special case of k=0(Laplace’s equation)you showed in the homework that the solution for the radial part isR(r)=C n r n+D n r−n,(4)1The Neumann function is often called the“Bessel function of the second kind”.(for n=0the solution is C0+D0ln r).The r n solution in Eq.(4)arises as the limit of the J n(kr) solution in Eq.(3)for k→0,while the r−n solution arises as the limit of the Neumann function N n(x)solution of Helmholtz’s equation(not displayed in Eq.(3)which only includes the solution regular at the origin).Since the solution of Helmholtz’s equation in circular polars(two dimensions)involves Bessel functions,you might expect that some sort of Bessel functions will also be involved here in spherical polars(three dimensions).This is correct and in fact we will see that the solution involves spherical Bessel functions.B.Separation of Variables in Spherical PolarsNow we set aboutfinding the solution of Helmholtz’s and Laplace’s equation in spherical polars. In this coordinate system,Helmholtz’s equation,Eq.(2),is1 r2∂∂r r2∂F∂r+1r2sinθ∂∂θ sinθ∂F∂θ +1r2sin2θ∂2F∂φ2+k2F=0.(5)To solve Eq.(5),we use the standard approach of separating the variables,i.e.we writeF(r,θ,φ)=R(r)Θ(θ)Φ(φ).(6) We then multiply by r2/(RΘΦ)which gives1 R ddr r2dR dr+k2r2+1Θsinθd dθ sinθdΘdθ +1Φsin2θd2Φdφ2=0.(7)C.Angular PartMultiplying Eq.(7)by sin2θ,the last term,Φ−1(d2Φ/dφ2),only involvesφ(whereas thefirst two terms only depend on r andθ),and so must be a constant which we call−m2,i.e.1Φd2Φdφ2=−m2.(8)The solution is clearlyΦ(φ)=e imφ,(9)with m an integer(in order that the solution is the same forφandφ+2π).Substituting into Eq.(7)gives1 R ddr r2dR dr+k2r2+1Θsinθd dθ sinθdΘdθ −m2sin2θ=0.(10)The third and fourth terms in Eq.(10)are only a function ofθ(whereas thefirst two only depend on r),and must therefore be a constant which,for reasons that will be clear later,we write as l(l+1),i.e.1Θsinθddθ sinθdΘdθ−m2sin2θ=−l(l+1),(11)which can be written as1 sinθddθ sinθdΘdθ+ l(l+1)−−m2sin2θΘ=0.(12)With the substitution x=cosθ,Eq.(12)becomesddx (1−x2)dΘ(x)dx + l(l+1)−m21−x2 Θ(x)=0.(13)Eq.(13)is the Associated Legendre equation,so the solution isΘ(x)=P m l(x)(x=cosθ),(14)where the P m l(cosθ)are Associated Legendre Polynomials,and,as shown in the book,we need l=0,1,2,···,and m runs over integer values from−l to l.If l is not an integer one can show that the solution of Eq.(12)diverges for cosθ=1or−1(θ=0orπ).Generally we require the solution to befinite in these limits,and this is the reason why we write the separation constant in Eq.(12)as l(l+1)with l an integer.The functionsΘandΦare often combined into a spherical harmonic,Y m l(θ,φ),whereY m l(θ,φ)=const.P m l(cosθ)e imφ,(15)where“const.”is a messy normalization constant,designed to get the right hand side of Eq.(16) below equal to unity when l=l′,m=m′.The spherical harmonics are orthogonal and normalized, i.e.2πdφ π0dθsinθY m l(θ,φ)⋆Y m′l′(θ,φ)=δl,l′δm,m′.(16) Note that since the spherical harmonics are complex we need to take the complex conjugate of one of them in this orthogonality-normalization relation.Thefirst few spherical harmonics areY00(θ,φ)= 14π,Y11(θ,φ)=− 38πsinθe iφ,Y11(θ,φ)= 34πcosθ,Y−11(θ,φ)= 38πsinθe−iφ.Spherical harmonics arise in many situations in physics in which there is spherical symmetry.An important example is the solution of the Schr¨o dinger equation in atomic physics.For the case of m=0,i.e.no dependence on the azimuthal angleφ,we haveΦ(φ)=1and also P m l(cosθ)=P l(cosθ),where the P l(x)are Legendre Polynomials.HenceY0l(θ,φ)=const.P l(cosθ).(17) You will recall from earlier classes that thefirst three Legendre polynomials areP0(x)=1,P1(x)=x,P2(x)=12(3x2−1).D.Radial PartWe now focus on the radial equation,which,from Eqs.(10)and(12),isd dr r2dR dr + k2r2−l(l+1) R=0,(18)or equivalentlyr2d2Rdr2+2rdRdr+ k2r2−l(l+1) R=0.(19)It turns out to be useful to define a function Z(r)byR(r)=Z(r)(kr)1/2.(20)Substituting this into Eq.(19)wefind that Z satisfiesr2d2Zdr2+rdZdr+ k2r2−(l+1/2)2 Z=0,(21)which is Bessel’s equation of order l+1/2.The solutions are J l+1/2(kr)and N l+1/2(kr)which, together with the factor(kr)−1/2in Eq.(20),means that the solutions for R(r)are the spherical Bessel and Neumann functions,j l(kr)and n l(kr)defined byj l(x)= π2x J l+1/2(x),n l(x)= π2x N l+1/2(x).(22) From now one we will assume that the solution isfinite at the origin,which rules out n l(kr), and soR(r)=j l(kr).(23) Hence,the general solution of Helmholtz’s equation which is regular at the origin isF(r,θ,φ)= k∞ l=0l m=−l a klm j l(kr)Y m l(θ,φ),(24) where the coefficients a klm would be determined by boundary conditions.Eq.(24)is the solution of Helmholtz’s equation in spherical polars(three dimensions)and is to be compared with the solution in circular polars(two dimensions)in Eq.(3).It turns out the spherical Bessel functions(i.e.Bessel functions of half-integer order,see Eq.(22)) are simpler than Bessel functions of integer order,because they are are related to trigonometric functions.For example,one hasj0(x)=sin xx,j1(x)=sin xx2−cos xx,n0(x)=−cos xx,n1(x)=−sin xx−cos xx2.(25)Hence Eq.(24)is not quite as formidable as it may seem.The only situations considered in detail in this course will be those in which there is no de-pendence on the azimuthal angleφ.In this case only the m=0terms contribute.For these, Y0l(θ,φ)=const.P l(cosθ),see Eq.(17),and so Eq.(24)simplifies toF(r,θ)= k∞ l=0a kl j l(kr)P l(cosθ)(azimuthal symmetry).(26)E.Example with azimuthal symmetry:a plane waveAs a special case of Eq.(26),consider a plane wave traveling in the z direction(the direction of the polar axis).We know that in cartesian coordinates the spatial part of the amplitude of the wave is just exp(ikz),which we can also write as exp(ikr cosθ).Since the amplitude of the wavesatisfies Helmholtz’s equation(and there is noφdependence),it must also be given by Eq.(26) (for the specified value of k),i.e.e ikr cosθ=∞l=0a l j l(kr)P l(cosθ),(27)for some choice of the a l.In fact one can show thata l=(2l+1)i l,(28) which givese ikr cosθ=∞l=0i l(2l+1)j l(kr)P l(cosθ),(29)a result which is very important in“scattering theory”in quantum mechanics.PLACE’S EQUATIONFinally we consider the special case of k=0,place’s equation∇2F=0.A.Separation of variablesSeparating the variables as above,the angular part of the solution is still a spherical harmonic Y m l(θ,φ).The difference between the solution of Helmholtz’s equation and Laplace’s equation lies in the radial equation,which becomesr2d2Rdr2+2rdRdr−l(l+1)R=0.As for the analogous case of circular polars,we can see by inspection that the solution just has a single power or r,i.e.R(r)∝rλfor some value ofλ.To determineλwe substitute rλin to the equation,which givesλ(λ+1)−l(l+1)=0.Factoring gives(λ−l)(λ+l+1)=0,so the two solutions of areλ=l andλ=−(l+1).If we specialize to the case of azimuthal symmetry for simplicity the general solution isF(r,θ)=∞l=0 A l r l+B l r l+1 P l(cosθ).(30)The r l term corresponds to the spherical Bessel function j l(kr)in Eq.(26)in the limit k→0, while the r−(l+1)term corresponds to the spherical Neumann function n l(kr)(not shown in Eq.(26) which only displays the solution of Helmholtz’s equation regular at the origin)in the same limit.Eq.(30)describes,for example,the electrostatic potential in regions of space where there is no charge.Coulomb’s law,F(r)∝1/r corresponds to the case of l=0(remember that P0(x)=1).If the solution is valid in the region where r→∞the A l vanish since the potential should go to zero far away from any charges,and soF(r,θ)=∞l=0B l r l+1P l(cosθ),(31)which is the multipole expansion that we discussed in the earlier part of the course.。
纳米固体的声子硬化和光子软化
Size-induced acoustic hardening and optic softening of phonons in CdS, InP, CeO2, SnO2, and Si nanostructuresChang Q SunSchl EEE, NTU, Singaporeecqsun@.sg; .sg/home/ecqsun/It has been puzzling that the Raman optical modes shift to lower frequency (or termed as optical mode softening) associated with creation of Raman acoustic modes that shift to higher energy (or called as acoustic hardening) upon nanosolid formation and size reduction. Understandings of the mechanism behind the size-induced acoustic hardening and optic softening have been quite controversial. On the basis of the recent bond order-length-strength (BOLS) correlation [Phys. Rev. B 69 045105 (2004)], we show that the optical softening arises from atomic cohesive energy weakening of surface atoms and the acoustic mode hardening is predominated by intergrain interaction. Agreement between predictions and observations has been reached for Si, CdS, InP, TiO2, CeO2, and SnO2 nanostructures with elucidation of vibration frequency of the corresponding isolated dimers. Findings further evidence the impact of bond order loss to low-dimensional systems and the essentiality of the BOLS correlation in describing the behavior of nanostructures.PACS: 61.46.+w; 78.30.-j; 78.67.-n; 63.22.+mKeywords: nanostructures; Raman shift; BOLS correlation; surface bond contraction; interparticle interaction- 1 -- 2 -I IntroductionAtomic vibration is of high interest because the behavior of phonons influence directly on the electrical and optical properties in solid materials and devices such as electron-phonon coupling in photoabsorption and photoemission, and phonon scattering in device transport dynamics.1 It has been long surprising that with structural miniaturization down to nanometer scale the transverse and the longitudinal optical (TO/LO) Raman modes shift towards lower frequency (or called as optical mode softening)2 accompanied with generation of low-frequency Raman (LFR) acoustic modes at wave numbers of a few or a few tens cm -1. The LFR peak shifts up (or called as acoustic mode hardening) towards higher frequency upon the solid size being reduced.3,4 Generally, the size dependent Raman shifts follow a scaling relation: 2,4κωωj f j K A K +∞=)()(where A f and κ are adjustable parameters for data fitting. K j , the dimensionless form of size, is the number of atoms with diameter d lined along the radius (R j ) of a spherical dot. For optical red shift, A f < 0. For Si example, ω(∞) = 520 cm -1 corresponds to wavelength of 2×104 nm and the index κ varies from 1.08 to = 1.44 or even 2.0, varying from source to source.5 For the LFR blue shift, A f > 0, κ = 1, and ω(∞) = 0. Therefore, the LFR disappears for large particles.The underlying mechanism behind the Raman shift is under debate with numerous theories. Theoretical studies of phonon frequency shift are often based on continuum dielectric mechanism.6,7 Sophisticated calculations have been carried out using models of correlation length,8 bulk phonon dispersion,9 lattice-dynamic matrix,10 associated with microscopic valence force field,4 phonon confinement,11 and bond polarization.2The mechanism of quadrupolar vibration taking the individual nanoparticle as a whole was assumed to be responsible for the LFR acoustic modes. The phonon energies are size dependent and vary with materials of the host matrix. The LFRscattering from silver nanoclusters embedded in porous alumina 12 andSiO 213 was suggested to arise from the quadrupolar vibration modes that are enhanced by the excitation of the surface plasmas of the encapsulated Ag particles. The selection of modes by LFR scattering is suggested to arise from the stronger plasmon-phonon coupling for these modes. For an Ag particle smaller than four nanometers, the size dependence of the peak frequency can be explained using Lamb’s theory 14 that gives vibrational frequencies of a homogeneous elastic body with a spherical form. On the other hand, lattice strain was suggested to be another possible mechanism for the LFR blue shift as size-dependent compressive strain has been observed from CdS x Se 1-x nanocrystals embedded in a borosilicate (B 2O 3-SiO 2) glass matrix.15 The lattice strain enhances the surface stress when the crystal size is reduced. Therefore, the observed blue shift of acoustic phonon energies was suggested to be consequence of thecompressive stress that overcomes the red shift caused by phonon confinement. Liang et al 16 presented a model for the Raman blue shift by relating the frequency shift to the bond length and bond strength that are functions of entropy latent heat of fusion and the critical temperature for solid-liquid transition.- 3 -The high-frequency Raman shift has ever been suggested to be activated by surface disorder,17 surface stress,18,19 and phonon quantum confinement,20,21 as well as surface chemical passivation.22 The phonon confinement model attributes the red shift of the Raman line to the relaxation of the wave-vector selection rule (∆q = 0) for the excitation of the Raman active phonons due to their localization. The relaxation of the selection rule arises from the finite crystalline size and the diameter distribution of nanosolid in the films. When the size is decreased the rule of momentum conservation will be relaxed and the Raman active modes will not be limited at the center of the Brillouin zone.18 A Gaussian-type phonon confinement model 21 indicates that strong phonon damping presents whereas calculations 23 using the correlation functions of the local dielectric constant ignores the role of phonon damping in the nanosolid. The large surface-to-volume ratio of a nanodot strongly affects the optical properties because of introducing surface polarization and surface states.24 Using a phenomenological Gaussian envelope function of phonon amplitudes, Tanaka et al.25 showed that the size dependence of optic red shift originated from the relaxation of the ∆q = 0 selection rule based on the phonon confinement argument with negative phonon dispersion. The phonon energies for all the glasses are reduced and the values of the phonon energies of CdSe nanodots are found to be quite different for different host glasses. A sophisticated analytical model of Hwang et al.5 indicates that the effect of lattice strain must be considered in explaining the optical red shift for CdSe nanodots embedded in different glass matrices. For a free surface, it has been derived that the red shift follows the relation:()()2−=∞∆jjBK K ωω (1)The value of B in eq (1) is a competition between the phonon negative dispersion and the size-dependent surface tension. Thus, a positive value of B indicates that the phonon negative dispersion exceeds the size-dependent surface tension and consequently causes the red shift of phonon frequency, and vice versa. In case of balance of the two effects, i.e. B = 0, the size dependence disappears. There are still some difficulties to use this equation, as remarked by Hwang et al.5It is noted that currently available models for the optical red shift are based on assumptions that the materials are homogeneous and isotropic which is valid only in the long-wavelength limit. When the size of the nanosolid is in the range of a few nanometers the continuum dielectric models exhibit limitations. Therefore, the discussed models could hardly reproduce with satisfactory the Raman frequency shifts at the lower end of the size limit. The objective of this work is to show that derivatives of the recent BOLS correlation mechanism 26,27,28 could reproduce the size induced Raman shifts leading to deeper and consistent insight into the mechanism behind with information about the vibration frequency of the corresponding dimers, which is beyond the scope of other sophisticated models.- 4 -II Principle2.1 Vibration modesRaman scattering is known to arise from the radiating dipole moment induced in a system by the electric field of incident electromagnetic radiation. The laws of momentum and energy conservation govern the interaction between a phonon and the incident photon. When we consider a solid containing numerous Bravais unit cells, and each cell contains n atoms, there will be 3n modes of vibrations. Among the 3n modes, there will be three acoustic modes, LA, TA 1, and TA 2, and 3(n-1) optical modes, LO and TOs. The acoustic modes represent the in-phase motion of the mass center of the unit cell or the entire solid as a whole. The long-range Coulomb interaction is responsible for the intercluster interaction. Therefore, the acoustic LFR should arise from the vibration of the entire nanosolid interacting with the host matrix or with other neighboring clusters. Therefore, it is expected that the LTR mode approaches zero if the particle size is infinitely large. The optical modes arise from the relative motion of the individual atoms in a complex unit cell. For elemental solids with a simple crystal structure such as the fcc of Ag, only acoustic modes present. Silicon or diamond is an interlock of two fcc unit cells that contain each cell two atoms in nonequivalent positions, there will be three acoustic modes and three optical modes.2.2 Optical phonon frequencyThe total energy E causing lattice vibration consists of the component of short-range interactions E S and the component of long-range Coulomb interaction E C 4.C S E E E +=(2)The long-range part corresponds to the LFR mode and represents the weak interaction between nanosolids. The short-range energy E S arising from nearest bonding atoms, which is composed of two parts. One is the lattice thermal vibration E V (T) and the other is interatomic binding energy at zero K, E b (r). The E S for a dimer can be expressed in a Taylor’s series, 26()()()()()()()()()()()()()T E d E d r k d r k d E d r dr r u d d r dr r u d d u d r dr n r u d T r E V b b d d n dr n n n S +=+−+−+=−+−++=−⎟⎟⎠⎞⎜⎜⎝⎛==∑ (6)'2!3!20!,32333222K(3)The term with index n = 0 corresponds to the minimal binding energy at T = 0 K, E b (d ) < 0. The term n = 1 is the force [()d r r u ∂∂= 0] at equilibrium and the terms n ≥ 2correspond to the thermal vibration energy, E V (T). By definition, the thermal vibration energy of a single bond is- 5 -()()()()()()222222!22d r d r dr n r u d d r k d r T E d n n n n v V −⎟⎟⎠⎞⎜⎜⎝⎛−=−=−=∑≥−µω(4)where r-d = x is the magnitude of lattice vibration. µ is the reduced mass of the dimer of concern. The k v = µω22/d E b ∝ is the force constant for lattice harmonic vibration with an angular frequency of ω. High-order terms correspond to nonlinear contribution that can be negligible in the first order approximation.For a single bond, the k v is strengthened because of the bond order loss induced bond contraction and bond strength gain.26-29 For a single atom, we have to count contribution from all the neighboring bonds. For a lower-coordinated atom the resultant k v could be lower because of the bond order loss. Considering the vibration amplitude x << d, it is convenient and reasonable to take the mean contribution from each coordinate to the force constant and to the magnitude of dislocation as the first order approximation:221ωµi z k k k ====L and z d r x x x z )(21−====L .Therefore the total energy of a certain atom with z coordinates is the sum over all the coordinates,()()()...!2...2,22222+−+=⎥⎥⎦⎤⎢⎢⎣⎡+⎟⎠⎞⎜⎝⎛−+=∑d r dr r u zd zE z d r E T d E d b z b S µω (5)This relation leads to the expression for phonon frequency as a function of bond energy and atomic CN, and bond length, ()d zE dr r u d z b d 212122∝⎥⎥⎦⎤⎢⎢⎣⎡×=µω(6)According to the BOLS correlation,26-29the bond order loss of a surface atom causes the remaining bonds of the lower-coordinated atoms to contract spontaneously (d i = c i d ) associated with bond strength gain (E i = c i –m E b ). The index m recognizes the nature of the bond involved. Such a BOLS correlation and its consequence modify not only the atomic cohesive energy (atomic CN multiplies the single bond energy) but also the Hamiltonain due to the densification of binding energy in the relaxed region. A physically detectable quantity that depends on the atomic cohesive energy or the Hamiltonian for a nanosolid can be expressed as Q (K j ) in a shell structure: ⎪⎪⎩⎪⎪⎨⎧∆=∞∞−−+=∑≤300)()()()()(i i j S S j q q Q Q K Q q q N Nq K Q γ- 6 -(7)where Q (∞) = Nq 0 is for a bulk solid. q 0 and q S correspond to the Q value per atomic volume inside the bulk and in the surface region, respectively. N S = ΣN i is the number of atoms in the surface atomic shells. Combining eqs. (6) and (7) gives the size-dependent optic red shift (where ()∞Q =())1(ωω−∞):()()()∑∑≤⎟⎠⎞⎜⎝⎛+−≤<∆=⎥⎥⎦⎤⎢⎢⎣⎡−=⎥⎦⎤⎢⎣⎡−=−∞∞−3123011)1(i p m i b i i i b i i c z z R γωωγωωωω where ())[]{}()⎪⎪⎪⎪⎩⎪⎪⎪⎪⎨⎧==−=−+=⎪⎩⎪⎨⎧≤≈==12;6)75.01(4812exp 12,3,1321z z spherical K z z z c else K c K V V N N j i i i j i j i i i τγ(8)ω(1) is the vibrational frequency of an isolated dimer which is the reference point for the optical red shift upon nanosolid and bulk formation. γi is the portion of atoms in the i th atomic layer over the total number of atoms of the entire solid of different shapes(τ = 1–3 correspond to a thin plate, a rod, and a spherical dot, respectively). The index i is counted up to three from the outmost atomic layer to the center of the solid as no atomic CN imperfection is justified at i > 3.III Results and discussion3.1 Optical modes and dimer vibrationIn experiment, one can only measure ()∞ω and ()j K ω in eq (8). However, with the known m value derived from measurement of other quantities such as the melting point or core level energy,26-29 one can determine )1(ω or the bulk shift ()∞ω-)1(ω by matching the measured data represented below to the predicted line in eq (8) without needing any other assumptions,()()()[]⎪⎩⎪⎨⎧−∞∆=−=∆)(,1)(,'Theory t Measuremen K A K R j j ωωωκ (9)Hence, the frequency shift from the dimer bond vibration to the bulk value,()()κωωj R K A ∆−≡−∞')1(, can be obtained. The matching of prediction with- 7 -measurement indicates that k ≡ 1, because 1−∝∆j R K .Figure 1 shows that the BOLS predictions match exceedingly well with the theoretically calculated or the experimentally measured optical red shift of a number of samples. Derived information about the corresponding dimer vibration is given in Table 1.3.2 A coustic modes and intercluster interactionFigure 2 shows the least-square-mean-root fitting of the size dependent LFR frequency for different nanosolids. The LFR frequency depends linearly on the inverse K j()()j j K A K '−=∞−ωω(10)The zero intercept at the vertical axis, ()∞ω= 0, indicates that when the K j approaches infinity the LFR peaks disappear, which implies that the LFR modes and their blue shifts originate from vibration of the individual nanoparticle as a whole. It seems not essential to involve the quadruple motion or the bond strain at the interface. However, the current derivative gives information about the strength of interparticle interaction, as summarized in Table 2.3.3 Surface atom vibration According to Einstein’s relation, it can be derived that T k z x c B =2)(2ωµ. At a given temperature, the vibrational amplitude and frequency of a given atom is correlated as: 121−∝ωz x , which is CN dependent. The frequency and magnitude of vibration for an surface atom at the surface (z = 4) or a metallic monatomic chain (MC with z = 2) can be derived as()()()()⎪⎪⎪⎩⎪⎪⎪⎨⎧========−−−+−1,2846.0670.01,404.0388.088.4,517.0388.0232344.3111m MC m Metal m Si c z m ib b ωω and()()()()()()⎪⎪⎩⎪⎪⎨⎧=×=×=×===+MC Metal Si c z z z z x x m b b b b43.170.0643.188.0309.188.0332344.31212111111ωω(11)The vibrational amplitude of an atom at the surface or a MC is indeed greater thanthat of a bulk atom while the frequency is lower. The magnitude and frequency are sensitive to the m value and varies insignificantly with the curvature of a spherical dot when K j > 3. This result verifies for the first time the assumption30,31 that the vibration amplitude of a surface atom is always greater than the bulk value and it keeps constant at all particle sizes.IV SummaryIn summary, a combination of the BOLS correlation and the scaling relation has enabled us to correlate the size-created and the size-hardened LFR acoustic phonons to the intergrain interaction and the optic phonon softening to the CN-imperfection reduced cohesive energy of atoms near the surface edge. The optic softening and acoustic hardening is realized in a K j-1 fashion. Decoding the measured size-dependence of Raman optical shift has derived vibrational information of Si, InP, CdS, CdSe, TiO2, CeO2, and SnO2 dimers and their bulk shifts, which is beyond the scope of direct measurement. As the approach proceeds in a way from bond-by-bond, atom-by-atom, shell-by-shell, no other constraints developed for the continuum medium are applied. One striking significance is that we are able to verify the correlation between the magnitude and the frequency of vibration of the lower-coordinated atoms. Consistency between the BOLS predictions and observations also verify the validity of other possible models that incorporate the size-induced Raman shift from different perspectives.Table and Figure captionsTable 1 Vibration frequencies of isolated dimers of various nanosolids and their red shift upon bulk formation derived from simulating the size dependent red shift of Raman optical modes as shown in Figure 1.Material d(nm)A′ω(∞)(cm-1) ω(1)(cm-1)ω(∞)-ω(1)(cm-1)CdS0.65Se0.35 0.286 23.9 203.4 158.8 44.60.28624.3 303 257.7 45.3CdSe 0.2947.76 210 195.2 14.8CeO20.22 20.89 464.5 415.1 49.4SnO20.20214.11 638 602.4 35.6InP 0.2947.06 347 333.5 13.5Si 0.26325.32520.0502.317.7Table 2 Linearization of the LFR acoustic modes of various nanosolids gives information about the strength of interparticle interaction for the specific solids.Sample A′Ag-a & Ag-b 23.6 ± 0.7Ag-c 18.2 ± 0.6TiO2-a TiO2-b 105.5 ± 0.1SnO2-a 93.5 ± 5.4- 8 -CdSe-1-a 146.1 ± 6.27CdSe-1-b 83.8 ± 2.8CdSe-1-c 46.7 ± 1.4CdSSe-a 129.4 ± 1.2CdSSe-b 58.4 ± 0.8Si-LA 97.77Si-TA1 45.57Si-TA2 33.78Figure 1 (link) Comparison of the BOLS predictions (lines for different shapes) with theoretical and experimental observations (scattered data) on the size-dependent optic phonon softening of nano-solid. (a) data labeled Si-1 was calculated using correlation length model,8 Si-3 (dot) and Si-4 (rod) were calculated using the bulk dispersion relation of phonons, 9 Si-5 was calculated from the lattice-dynamic matrix,4 Si-7 was calculated using phonon confinement model,11 and Si-8 (rod) and Si-9 (dot) were calculated using bond polarizability model.2 Data for Si-2,32 Si-6,33and Si-10, and Si-1118 are measured data. (b) CdS0.65Se0.35-1, CdS0.65Se0.35 (in glass)-LO2,CdS0.65Se0.35-2, CdS0.65Se0.35 (in glass)-LO1,34CdSe-1, CdSe(in B2O3SiO2)-LO, CdSe-2, CdSe(in SiO2)-LO, and CdSe-3 CdSe(in GeO2)-LO, CdSe-4, CdSe(inGeO2)-LO,25 (c) CeO2-1,35 SnO2-1,36 SnO2-2,17 and InP37 are all measurement. Figure 2 (link) Generation and blue shift of the LFR acoustic modes where the solid dotted and dashed lines are the corresponding results of the least squares fitting. (a) the Si-a, Si-b, and Si-c were calculated from the lattice-dynamic matrix by using a microscopic valence force field model,4 the Si-d and Si-e are the experimental results.3 (b) Ag-a (Ag in SiO2)38Ag-b (Ag in SiO2)13 Ag-c (Ag in Alumina).12 (c) TiO2-a39 TiO2-b39 SnO2-a.17 (d)CdSe-a (l = 0 n = 2) CdSe-b (l = 2 n = 1) and CdSe-c (l = 0 n = 1).40 (e) CdS0.65Se0.35-a [CdS0.65Se0.35 (in glass)-LF2] and CdS0.65Se0.35-b [CdS0.65Se0.35 (in glass)-LF1]34 are all measured data.- 9 -- 10 -R a m a n S h i f t (%)K j R a m a n S h i f t (%)R a m a n S h i f t (%)Fg-1RamanShift(cm-1)RamanShift(cm-1)1/R (nm-1)1/R (nm-1)RamanShift(cm-1)RamanShift(cm-1)RamanShift(cm-1)Fg-2- 11 -1 T. Takagahara, Phys. Rev. Lett.71, 3577 (1993).2 J. Zi, H. Büscher, C. Falter, W. Ludwig, K. M. Zhang, and X. D. Xie, Appl. Phys. Lett.69, 200(1996).3 M. Fujii, Y. Kanzawa, S. Hayashi, and K. Yamamoto, Phys. Rev. B 54, R8373 (1996).4 W. Cheng and S. F. Ren, Phys. Rev. B 65, 205305 (2002).5 Y.-N. Hwang, S. Shin, H. L. Park, S.-H. Park, U. Kim, H. S. Jeong, E.-J. Shin, and D. Kim, Phys.Rev. B 54, 15120 (1996).6 M. C. Klein, F. Hache, D. Ricard, and C. Flytzanis, Phys. Rev. B 42, 11123 (1990).7 C. Trallero-Giner, A. Debernardi, M. Cardona, E. Menendez-Proupi, and A. I. Ekimov, Phys. Rev. B57, 4664 (1998).8 G. Viera, S. Huet, and L. Boufendi, J. Appl. Phys. 90, 4175 (2001).9 P. M. Fauchet and I. H. Campell, Crit. Rev. Solid State Mater. Sci. 14, S79 (1988).10 X. H. Hu and J. Zi, J. Phys. Condens. Mat. 14,L671 (2002).11 A. K. Sood, K. Jayaram, D. Victor, and S. Muthu, J. Appl. Phys. 72, 4963 (1992).12 B. Palpant, H. Portales, L. Saviot, J. Lerme, B. Prevel, M. Pellarin, E. Duval, A. Perez, and M.Broyer, Phys. Rev. B 60, 17107 (1999).13 M. Fujii, T. Nagareda, S. Hayashi, and K. Yamamoto, Phys. Rev. B 44, 6243 (1991).14 H. Lamb, Proc. London Math. Soc. 13, 189 (1882).15 G. Scamarcio, M. Lugara, and D. Manno, Phys. Rev. B 45, 13792 (1992).16 L. H. Liang, C. M. Shen, X. P. Chen, W. M. Liu, and H. J. Gao, J. Phys. Condens. Matt. 16, 267(2004).17 A. Dieguez, A. Romano-Rodrýguez, A. Vila, and J. R. Morante, J. Appl. Phys. 90, 1550 (2001).18 Z. Iqbal and S. Vepřek, J. Phys. C15, 377 (1982).19 E. Anastassakis and E. Liarokapis, J. Appl. Phys. 62, 3346 (1987).20 H. Richter, Z. P. Wang, and L. Ley, Solid State Commun. 39, 625 (1981).21 I. H. Campbell and P. M. Fauchet, Solid State Commun.58, 739 (1986).22 C. Q. Sun, Prog. Mater. Sci. 48, 521 (2003).23 N. Ohtani and K. Kawamura, Solid State Commun. 75, 711 (1990).24 L. Banyai and S. W. Koch, Semiconductor Quantum Dots (World Scientific, Singapore, 1993).25 A. Tanaka, S. Onari, and T. Arai, Phys. Rev.B 45, 6587 (1992).26 L. K. Pan, C. Q. Sun, and C. M. Li, J. Phys. Chem. B 108, L3404 (2004).27C. Q. Sun, C. M. Li, S. Li, B. K. Tay, Phys. Rev. B 69, 245402 (2004).28 C. Q. Sun, Phys. Rev. B 69, 045105 (2004).29 C. Q. Sun, H. L. Bai, B. K. Tay, S. Li, and E. Y. Jiang, J. Phys. Chem. B 107, 7544 (2003).30 F. G. Shi, J. Mater. Res. 91, 307 (1994).31 Q. Jiang, Z. Zhang, and J. C. Li, Chem. Phys. Lett. 22, 549 (2003).32 C. Ossadnik, S. Vepřek, and I. Gregora, Thin Solid Films337, 148 (1999).33 G. X. Cheng, H. Xia, K. J. Chen, W. Zhang, and X. K. Zhang, Phys. Status. Solidi. A 118, K51(1990).34 P. Verma, L. Gupta, S. C. Abbi, and K. P. Jain, J. Appl. Phys. 88, 4109 (2000).35 J. E. Spanier, R. D. Robinson, F. Zhang, S. W. Chan, and I. P. Herman, Phys. Rev. B 64, 245407(2001).36 C. H. Shek, G. M. Lin, and J. K. L. Lai, Nanosructured Mater. 11, 831 (1999).37 M. J. Seong, O. I. Micic, A. J. Nozik, A. Mascarenhas, and H. M. Cheong, Appl. Phys. Lett.82,185 (2003).38 P. Gangopadhyay, R. Kesavamoorthy, K. G. M. Nair, and R. Dhandapani, J. Appl. Phys. 88, 4975(2000).39 M. Gotić, M. Ivanda, A. Sekulić, S. Musić, S. Popović, A. Turković, and K. Furić, Mater. Lett. 28,225 (1996).40 L. Saviot, B. champagnon, E. Duval, I. A. Kudriavtsev, and A. I. Ekimov, J. Non-Crystal Solids197, 238 (1996).- 12 -。
机械设计专业英语
机械设计专业英语机械设计专业英语圆柱螺旋扭转弹簧cylindroid helical-coil torsion spring圆柱螺旋压缩弹簧cylindroid helical-coil compression spring 圆柱凸轮cylindrical cam圆柱蜗杆cylindrical worm圆柱坐标操作器cylindrical coordinate manipulator圆锥螺旋扭转弹簧conoid helical-coil compression spring 圆锥滚子tapered roller圆锥滚子轴承tapered roller bearing圆锥齿轮机构bevel gears圆锥角cone angle原动件driving link约束constraint约束条件constraint condition约束反力constraining force跃度jerk跃度曲线jerk diagram运动倒置kinematic inversion运动方案设计kinematic precept design运动分析kinematic analysis运动副kinematic pair运动构件moving link运动简图kinematic sketch运动链kinematic chain运动失真undercutting运动设计kinematic design运动周期cycle of motion运动综合kinematic synthesis运转不均匀系数coefficient of velocity fluctuation运动粘度kenematic viscosity载荷load载荷—变形曲线load—deformation curve 载荷—变形图load—deformation diagram 窄V 带narrow V belt毡圈密封felt ring seal展成法generating张紧力tension 张紧轮tension pulley振动vibration振动力矩shaking couple振动频率frequency of vibration振幅amplitude of vibration正切机构tangent mechanism正向运动学direct (forward) kinematics正弦机构sine generator, scotch yoke织布机loom正应力、法向应力normal stress制动器brake直齿圆柱齿轮spur gear直齿锥齿轮straight bevel gear直角三角形right triangle直角坐标操作器Cartesian coordinate manipulator直径系数diametral quotient直径系列diameter series直廓环面蜗杆hindley worm直线运动linear motion直轴straight shaft质量mass质心center of mass执行构件executive link; working link质径积mass-radius product智能化设计intelligent design, ID中间平面mid-plane中心距center distance中心距变动center distance change中心轮central gear中径mean diameter终止啮合点final contact, end of contact周节pitch周期性速度波动periodic speed fluctuation 周转轮系epicyclic gear train肘形机构toggle mechanism轴shaft轴承盖bearing cup轴承合金bearing alloy轴承座bearing block轴承高度bearing height轴承宽度bearing width轴承内径bearing bore diameter轴承寿命bearing life轴承套圈bearing ring轴承外径bearing outside diameter轴颈journal轴瓦、轴承衬bearing bush轴端挡圈shaft end ring轴环shaft collar轴肩shaft shoulder轴角shaft angle轴向axial direction轴向齿廓axial tooth profile轴向当量动载荷dynamic equivalent axial load轴向当量静载荷static equivalent axial load 轴向基本额定动载荷basic dynamic axial load rating轴向基本额定静载荷basic static axial load rating轴向接触轴承axial contact bearing轴向平面axial plane轴向游隙axial internal clearance轴向载荷axial load轴向载荷系数axial load factor轴向分力axial thrust load主动件driving link主动齿轮driving gear主动带轮driving pulley转动导杆机构whitworth mechanism转动副revolute (turning) pair转速swiveling speed ; rotating speed转动关节revolute joint转轴revolving shaft转子rotor转子平衡balance of rotor装配条件assembly condition锥齿轮bevel gear锥顶common apex of cone锥距cone distance锥轮bevel pulley; bevel wheel锥齿轮的当量直齿轮equivalent spur gear of the bevel gear 锥面包络圆柱蜗杆milled helicoids worm准双曲面齿轮hypoid gear子程序subroutine 子机构sub-mechanism自动化automation自锁self-locking自锁条件condition of self-locking自由度degree of freedom, mobility总重合度total contact ratio总反力resultant force总效率combined efficiency; overall efficiency组成原理theory of constitution组合齿形composite tooth form组合安装stack mounting组合机构combined mechanism阻抗力resistance最大盈亏功maximum difference work between plus and minus work纵向重合度overlap contact ratio纵坐标ordinate组合机构combined mechanism最少齿数minimum teeth number最小向径minimum radius作用力applied force坐标系coordinate frame行星轮变速装置planetary speed changing devices行星轮系planetary gear train形封闭凸轮机构positive-drive (or form-closed) cam mechanism虚拟现实virtual reality虚拟现实技术virtual reality technology, VRT虚拟现实设计virtual reality design, VRD虚约束redundant (or passive) constraint许用不平衡量allowable amount of unbalance许用压力角allowable pressure angle许用应力allowable stress; permissible stress 悬臂结构cantilever structure悬臂梁cantilever beam循环功率流circulating power load旋转力矩running torque旋转式密封rotating seal旋转运动rotary motion选型type selection压力pressure压力中心center of pressure压缩机compressor压应力compressive stress压力角pressure angle牙嵌式联轴器jaw (teeth) positive-contact coupling雅可比矩阵Jacobi matrix摇杆rocker液力传动hydrodynamic drive液力耦合器hydraulic couplers液体弹簧liquid spring液压无级变速hydraulic stepless speed changes液压机构hydraulic mechanism一般化运动链generalized kinematic chain 移动从动件reciprocating follower移动副prismatic pair, sliding pair移动关节prismatic joint移动凸轮wedge cam盈亏功increment or decrement work应力幅stress amplitude应力集中stress concentration应力集中系数factor of stress concentration 应力图stressdiagram应力—应变图stress-strain diagram优化设计optimal design油杯oil bottle油壶oil can油沟密封oily ditch seal有害阻力useless resistance有益阻力useful resistance有效拉力effective tension有效圆周力effective circle force有害阻力detrimental resistance余弦加速度运动cosine acceleration (or simple harmonic) motion预紧力preload原动机primer mover圆带round belt圆带传动round belt drive圆弧齿厚circular thickness 圆弧圆柱蜗杆hollow flank worm 圆角半径fillet radius圆盘摩擦离合器disc friction clutch圆盘制动器disc brake原动机prime mover原始机构original mechanism圆形齿轮circular gear圆柱滚子cylindrical roller圆柱滚子轴承cylindrical roller bearing圆柱副cylindric pair圆柱式凸轮步进运动机构barrel (cylindric) cam圆柱螺旋拉伸弹簧cylindroid helical-coil extension spring凸轮cam凸轮倒置机构inverse cam mechanism凸轮机构cam , cam mechanism凸轮廓线cam profile凸轮廓线绘制layout of cam profile凸轮理论廓线pitch curve凸缘联轴器flange coupling图册、图谱atlas图解法graphical method推程rise推力球轴承thrust ball bearing推力轴承thrust bearing退刀槽tool withdrawal groove退火anneal陀螺仪gyroscopeV 带V belt外力external force外圈outer ring外形尺寸boundary dimension万向联轴器Hooks coupling ; universal coupling外齿轮external gear弯曲应力beading stress弯矩bending moment腕部wrist往复移动reciprocating motion往复式密封reciprocating seal网上设计on-net design, OND微动螺旋机构differential screw mechanism 位移displacement 位移曲线displacement diagram位姿pose , position and orientation稳定运转阶段steady motion period稳健设计robust design蜗杆worm蜗杆传动机构worm gearing蜗杆头数number of threads蜗杆直径系数diametral quotient蜗杆蜗轮机构worm and worm gear蜗杆形凸轮步进机构worm cam interval mechanism蜗杆旋向hands of worm蜗轮worm gear涡圈形盘簧power spring无级变速装置stepless speed changes devices无穷大infinite系杆crank arm, planet carrier现场平衡field balancing向心轴承radial bearing向心力centrifugal force相对速度relative velocity相对运动relative motion相对间隙relative gap象限quadrant橡皮泥plasticine细牙螺纹fine threads销pin消耗consumption小齿轮pinion小径minor diameter橡胶弹簧balata spring修正梯形加速度运动规律modified trapezoidal acceleration motion修正正弦加速度运动规律modified sine acceleration motion斜齿圆柱齿轮helical gear斜键、钩头楔键taper key泄漏leakage谐波齿轮harmonic gear谐波传动harmonic driving谐波发生器harmonic generator 斜齿轮的当量直齿轮equivalent spur gear of the helical gear心轴spindle行程速度变化系数coefficient of travel speed variation行程速比系数advance-to return-time ratio 行星齿轮装置planetary transmission行星轮planet gear平衡机balancing machine平衡品质balancing quality平衡平面correcting plane平衡质量balancing mass平衡重counterweight平衡转速balancing speed平面副planar pair, flat pair平面机构planar mechanism平面运动副planar kinematic pair平面连杆机构planar linkage平面凸轮planar cam平面凸轮机构planar cam mechanism平面轴斜齿轮parallel helical gears普通平键parallel key其他常用机构other mechanism in common use起动阶段starting period启动力矩starting torque气动机构pneumatic mechanism奇异位置singular position起始啮合点initial contact , beginning of contact气体轴承gas bearing千斤顶jack嵌入键sunk key强迫振动forced vibration切齿深度depth of cut曲柄crank曲柄存在条件Grashoff`s law曲柄导杆机构crank shaper (guide-bar) mechanism曲柄滑块机构slider-crank (or crank-slider) mechanism 曲柄摇杆机构crank-rocker mechanism曲齿锥齿轮spiral bevel gear曲率curvature曲率半径radius of curvature曲面从动件curved-shoe follower曲线拼接curve matching曲线运动curvilinear motion曲轴crank shaft驱动力driving force驱动力矩driving moment (torque)全齿高whole depth权重集weight sets球ball球面滚子convex roller球轴承ball bearing球面副spheric pair球面渐开线spherical involute球面运动spherical motion球销副sphere-pin pair球坐标操作器polar coordinate manipulator 燃点spontaneous ignition热平衡heat balance; thermal equilibrium人字齿轮herringbone gear冗余自由度redundant degree of freedom柔轮flexspline柔性冲击flexible impulse; soft shock柔性制造系统flexible manufacturing system; FMS柔性自动化flexible automation润滑油膜lubricant film润滑装置lubrication device润滑lubrication润滑剂lubricant三角形花键serration spline三角形螺纹V thread screw三维凸轮three-dimensional cam三心定理Kennedy`s theorem砂轮越程槽grinding wheel groove砂漏hour-glass少齿差行星传动planetary drive with small teeth difference设计方法学design methodology设计变量design variable设计约束design constraints深沟球轴承deep groove ball bearing生产阻力productive resistance 升程rise升距lift螺旋角helix angle螺旋线helix ,helical line绿色设计green design ; design for environment马耳他机构Geneva wheel ; Geneva gear马耳他十字Maltese cross脉动无级变速pulsating stepless speed changes脉动循环应力fluctuating circulating stress 脉动载荷fluctuating load铆钉rivet迷宫密封labyrinth seal密封seal密封带seal belt密封胶seal gum密封元件potted component密封装置sealing arrangement面对面安装face-to-face arrangement面向产品生命周期设计design for product`s life cycle, DPLC 名义应力、公称应力nominal stress模块化设计modular design, MD模块式传动系统modular system模幅箱morphology box模糊集fuzzy set模糊评价fuzzy evaluation模数module摩擦friction摩擦角friction angle摩擦力friction force摩擦学设计tribology design, TD摩擦阻力frictional resistance摩擦力矩friction moment摩擦系数coefficient of friction摩擦圆friction circle磨损abrasion ;wear; scratching末端执行器end-effector目标函数objective function耐腐蚀性corrosion resistance耐磨性wear resistance挠性机构mechanism with flexible elements挠性转子flexible rotor内齿轮internal gear内齿圈ring gear内力internal force内圈inner ring能量energy能量指示图viscosity逆时针counterclockwise (or anticlockwise) 啮出engaging-out 啮合engagement, mesh, gearing啮合点contact points啮合角working pressure angle啮合线line of action啮合线长度length of line of action啮入engaging-in牛头刨床shaper凝固点freezing point; solidifying point扭转应力torsion stress扭矩moment of torque扭簧helical torsion spring诺模图NomogramO 形密封圈密封O ring seal盘形凸轮disk cam盘形转子disk-like rotor抛物线运动parabolic motion疲劳极限fatigue limit疲劳强度fatigue strength偏置式offset偏( 心) 距offset distance偏心率eccentricity ratio偏心质量eccentric mass偏距圆offset circle偏心盘eccentric偏置滚子从动件offset roller follower偏置尖底从动件offset knife-edge follower 偏置曲柄滑块机构offset slider-crank mechanism拼接matching评价与决策evaluation and decision频率frequency平带flat belt平带传动flat belt driving平底从动件flat-face follower平底宽度face width 平分线bisector平均应力average stress平均中径mean screw diameter平均速度average velocity平衡balance可靠度degree of reliability可靠性reliability可靠性设计reliability design, RD空气弹簧air spring空间机构spatial mechanism空间连杆机构spatial linkage空间凸轮机构spatial cam空间运动副spatial kinematic pair空间运动链spatial kinematic chain空转idle宽度系列width series框图block diagram雷诺方程Reynolds‘s equation离心力centrifugal force离心应力centrifugal stress离合器clutch离心密封centrifugal seal理论廓线pitch curve理论啮合线theoretical line of action隶属度membership力force力多边形force polygon力封闭型凸轮机构force-drive (or force-closed) cam mechanism力矩moment力平衡equilibrium力偶couple力偶矩moment of couple连杆connecting rod, coupler连杆机构linkage连杆曲线coupler-curve连心线line of centers链chain链传动装置chain gearing链轮sprocket ; sprocket-wheel ; sprocket gear ; chain wheel 联组V 带tight-up V belt联轴器coupling ; shaft coupling两维凸轮two-dimensional cam临界转速critical speed六杆机构six-bar linkage龙门刨床double Haas planer轮坯blank轮系gear train螺杆screw螺距thread pitch螺母screw nut螺旋锥齿轮helical bevel gear螺钉screws螺栓bolts螺纹导程lead螺纹效率screw efficiency螺旋传动power screw螺旋密封spiral seal螺纹thread (of a screw)螺旋副helical pair螺旋机构screw mechanism基本额定寿命basic rating life基于实例设计case-based design,CBD 基圆base circle基圆半径radius of base circle基圆齿距base pitch基圆压力角pressure angle of base circle 基圆柱base cylinder基圆锥base cone急回机构quick-return mechanism急回特性quick-return characteristics急回系数advance-to return-time ratio 急回运动quick-return motion棘轮ratchet棘轮机构ratchet mechanism棘爪pawl极限位置extreme (or limiting) position极位夹角crank angle between extreme (or limiting) positions 计算机辅助设计computer aided design, CAD计算机辅助制造computer aided manufacturing, CAM计算机集成制造系统computer integrated manufacturing system, CIMS计算力矩factored moment; calculation moment 计算弯矩calculated bending moment加权系数weighting efficient加速度acceleration加速度分析acceleration analysis加速度曲线acceleration diagram尖点pointing; cusp尖底从动件knife-edge follower间隙backlash间歇运动机构intermittent motion mechanism减速比reduction ratio减速齿轮、减速装置reduction gear减速器speed reducer减摩性anti-friction quality渐开螺旋面involute helicoid渐开线involute渐开线齿廓involute profile渐开线齿轮involute gear渐开线发生线generating line of involute渐开线方程involute equation渐开线函数involute function渐开线蜗杆involute worm渐开线压力角pressure angle of involute渐开线花键involute spline简谐运动simple harmonic motion键key键槽keyway交变应力repeated stress交变载荷repeated fluctuating load交叉带传动cross-belt drive交错轴斜齿轮crossed helical gears胶合scoring角加速度angular acceleration角速度angular velocity角速比angular velocity ratio角接触球轴承angular contact ball bearing 角接触推力轴承angular contact thrust bearing角接触向心轴承angular contact radial bearing角接触轴承angular contact bearing铰链、枢纽hinge校正平面correcting plane接触应力contact stress接触式密封contact seal阶梯轴multi-diameter shaft结构structure结构设计structural design截面section节点pitch point节距circular pitch; pitch of teeth节线pitch line节圆pitch circle节圆齿厚thickness on pitch circle节圆直径pitch diameter节圆锥pitch cone节圆锥角pitch cone angle解析设计analytical design紧边tight-side紧固件fastener径节diametral pitch径向radial direction径向当量动载荷dynamic equivalent radial load径向当量静载荷static equivalent radial load 径向基本额定动载荷basic dynamic radial load rating径向基本额定静载荷basic static radial load tating径向接触轴承radial contact bearing径向平面radial plane径向游隙radial internal clearance径向载荷radial load径向载荷系数radial load factor径向间隙clearance静力static force静平衡static balance静载荷static load静密封static seal局部自由度passive degree of freedom矩阵matrix矩形螺纹square threaded form锯齿形螺纹buttress thread form矩形牙嵌式离合器square-jaw positive-contact clutch绝对尺寸系数absolute dimensional factor绝对运动absolute motion绝对速度absolute velocity 均衡装置load balancing mechanism抗压强度compression strength开口传动open-belt drive开式链open kinematic chain开链机构open chain mechanism高度系列height series功work工况系数application factor工艺设计technological design工作循环图working cycle diagram工作机构operation mechanism工作载荷external loads工作空间working space工作应力working stress工作阻力effective resistance工作阻力矩effective resistance moment公法线common normal line公共约束general constraint公制齿轮metric gears功率power功能分析设计function analyses design共轭齿廓conjugate profiles共轭凸轮conjugate cam构件link鼓风机blower固定构件fixed link; frame固体润滑剂solid lubricant关节型操作器jointed manipulator惯性力inertia force惯性力矩moment of inertia ,shaking moment 惯性力平衡balance of shaking force惯性力完全平衡full balance of shaking force惯性力部分平衡partial balance of shaking force惯性主矩resultant moment of inertia惯性主失resultant vector of inertia冠轮crown gear广义机构generation mechanism广义坐标generalized coordinate轨迹生成path generation轨迹发生器path generator滚刀hob滚道raceway滚动体rolling element滚动轴承rolling bearing滚动轴承代号rolling bearing identification code滚针needle roller滚针轴承needle roller bearing滚子roller滚子轴承roller bearing滚子半径radius of roller滚子从动件roller follower滚子链roller chain滚子链联轴器double roller chain coupling 滚珠丝杆ball screw 滚柱式单向超越离合器roller clutch过度切割undercutting函数发生器function generator函数生成function generation含油轴承oil bearing耗油量oil consumption耗油量系数oil consumption factor赫兹公式H. Hertz equation合成弯矩resultant bending moment合力resultant force合力矩resultant moment of force黑箱black box横坐标abscissa互换性齿轮interchangeable gears花键spline滑键、导键feather key滑动轴承sliding bearing滑动率sliding ratio滑块slider环面蜗杆toroid helicoids worm环形弹簧annular spring缓冲装置shocks; shock-absorber灰铸铁grey cast iron回程return回转体平衡balance of rotors混合轮系compound gear train积分integrate机电一体化系统设计mechanical-electrical integration system design机构mechanism 机构分析analysis of mechanism机构平衡balance of mechanism机构学mechanism机构运动设计kinematic design of mechanism机构运动简图kinematic sketch of mechanism机构综合synthesis of mechanism机构组成constitution of mechanism机架frame, fixed link机架变换kinematic inversion机器machine机器人robot机器人操作器manipulator机器人学robotics技术过程technique process技术经济评价technical and economic evaluation技术系统technique system机械machinery机械创新设计mechanical creation design, MCD机械系统设计mechanical system design, MSD机械动力分析dynamic analysis of machinery机械动力设计dynamic design of machinery 机械动力学dynamics of machinery机械的现代设计modern machine design机械系统mechanical system机械利益mechanical advantage机械平衡balance of machinery机械手manipulator机械设计machine design; mechanical design 机械特性mechanical behavior机械调速mechanical speed governors机械效率mechanical efficiency机械原理theory of machines and mechanisms机械运转不均匀系数coefficient of speed fluctuation机械无级变速mechanical stepless speed changes基础机构fundamental mechanism端面transverse plane端面参数transverse parameters端面齿距transverse circular pitch端面齿廓transverse tooth profile端面重合度transverse contact ratio端面模数transverse module端面压力角transverse pressure angle锻造forge对称循环应力symmetry circulating stress对心滚子从动件radial (or in-line ) roller follower对心直动从动件radial (or in-line ) translating follower对心移动从动件radial reciprocating follower对心曲柄滑块机构in-line slider-crank (or crank-slider) mechanism多列轴承multi-row bearing多楔带poly V-belt多项式运动规律polynomial motion多质量转子rotor with several masses惰轮idle gear额定寿命rating life额定载荷load ratingII 级杆组dyad发生线generating line发生面generating plane法面normal plane法面参数normal parameters法面齿距normal circular pitch法面模数normal module法面压力角normal pressure angle法向齿距normal pitch法向齿廓normal tooth profile法向直廓蜗杆straight sided normal worm法向力normal force反馈式组合feedback combining反向运动学inverse ( or backward) kinematics反转法kinematic inversion反正切Arctan范成法generating cutting仿形法form cutting方案设计、概念设计concept design, CD 防振装置shockproof device飞轮flywheel飞轮矩moment of flywheel非标准齿轮nonstandard gear非接触式密封non-contact seal非周期性速度波动aperiodic speed fluctuation非圆齿轮non-circular gear粉末合金powder metallurgy分度线reference line; standard pitch line分度圆reference circle; standard (cutting) pitch circle分度圆柱导程角lead angle at reference cylinder分度圆柱螺旋角helix angle at reference cylinder分母denominator分子numerator分度圆锥reference cone; standard pitch cone 分析法analytical method封闭差动轮系planetary differential复合铰链compound hinge复合式组合compound combining复合轮系compound (or combined) gear train 复合平带compound flat belt复合应力combined stress复式螺旋机构Compound screw mechanism 复杂机构complex mechanism杆组Assur group干涉interference刚度系数stiffness coefficient刚轮rigid circular spline钢丝软轴wire soft shaft刚体导引机构body guidance mechanism刚性冲击rigid impulse (shock)刚性转子rigid rotor刚性轴承rigid bearing刚性联轴器rigid coupling高度系列height series高速带high speed belt高副higher pair格拉晓夫定理Grashoff`s law根切undercutting公称直径nominal diameter阿基米德蜗杆Archimedes worm安全系数safety factor; factor of safety安全载荷safe load凹面、凹度concavity扳手wrench板簧flat leaf spring半圆键woodruff key变形deformation摆杆oscillating bar摆动从动件oscillating follower摆动从动件凸轮机构cam with oscillating follower 摆动导杆机构oscillating guide-bar mechanism 摆线齿轮cycloidal gear摆线齿形cycloidal tooth profile摆线运动规律cycloidal motion摆线针轮cycloidal-pin wheel包角angle of contact保持架cage背对背安装back-to-back arrangement背锥back cone ；normal cone背锥角back angle背锥距back cone distance比例尺scale比热容specific heat capacity闭式链closed kinematic chain闭链机构closed chain mechanism臂部arm变频器frequency converters变频调速frequency control of motor speed 变速speed change变速齿轮change gear ; change wheel变位齿轮modified gear变位系数modification coefficient标准齿轮standard gear标准直齿轮standard spur gear表面质量系数superficial mass factor表面传热系数surface coefficient of heat transfer表面粗糙度surface roughness并联式组合combination in parallel并联机构parallel mechanism并联组合机构parallel combined mechanism 并行工程concurrent engineering并行设计concurred design, CD不平衡相位phase angle of unbalance不平衡imbalance (or unbalance)不平衡量amount of unbalance不完全齿轮机构intermittent gearing波发生器wave generator波数number of waves补偿compensation参数化设计parameterization design, PD残余应力residual stress操纵及控制装置operation control device槽轮Geneva wheel槽轮机构Geneva mechanism ；Maltese cross槽数Geneva numerate槽凸轮groove cam侧隙backlash差动轮系differential gear train差动螺旋机构differential screw mechanism 差速器differential 常用机构conventional mechanism; mechanism in common use车床lathe承载量系数bearing capacity factor承载能力bearing capacity成对安装paired mounting尺寸系列dimension series齿槽tooth space齿槽宽spacewidth齿侧间隙backlash齿顶高addendum齿顶圆addendum circle齿根高dedendum齿根圆dedendum circle齿厚tooth thickness齿距circular pitch齿宽face width齿廓tooth profile齿廓曲线tooth curve齿轮gear齿轮变速箱speed-changing gear boxes齿轮齿条机构pinion and rack齿轮插刀pinion cutter; pinion-shaped shaper cutter齿轮滚刀hob ,hobbing cutter齿轮机构gear齿轮轮坯blank齿轮传动系pinion unit齿轮联轴器gear coupling齿条传动rack gear齿数tooth number齿数比gear ratio齿条rack齿条插刀rack cutter; rack-shaped shaper cutter 齿形链、无声链silent chain齿形系数form factor齿式棘轮机构tooth ratchet mechanism插齿机gear shaper重合点coincident points重合度contact ratio冲床punch传动比transmission ratio, speed ratio传动装置gearing; transmission gear传动系统driven system传动角transmission angle传动轴transmission shaft串联式组合combination in series串联式组合机构series combined mechanism串级调速cascade speed control创新innovation ; creation创新设计creation design垂直载荷、法向载荷normal load唇形橡胶密封lip rubber seal磁流体轴承magnetic fluid bearing从动带轮driven pulley从动件driven link, follower从动件平底宽度width of flat-face从动件停歇follower dwell从动件运动规律follower motion从动轮driven gear粗线bold line粗牙螺纹coarse thread大齿轮gear wheel打包机packer打滑slipping 带传动belt driving带轮belt pulley带式制动器band brake单列轴承single row bearing单向推力轴承single-direction thrust bearing 单万向联轴节single universal joint单位矢量unit vector当量齿轮equivalent spur gear; virtual gear 当量齿数equivalent teeth number; virtual number of teeth当量摩擦系数equivalent coefficient of friction当量载荷equivalent load刀具cutter导数derivative倒角chamfer导热性conduction of heat导程lead导程角lead angle等加等减速运动规律parabolic motion; constant acceleration and deceleration motion 等速运动规律uniform motion; constant velocity motion等径凸轮conjugate yoke radial cam等宽凸轮constant-breadth cam等效构件equivalent link等效力equivalent force等效力矩equivalent moment of force等效量equivalent等效质量equivalent mass等效转动惯量equivalent moment of inertia 等效动力学模型dynamically equivalent model底座chassis低副lower pair点划线chain dotted line（疲劳）点蚀pitting垫圈gasket垫片密封gasket seal碟形弹簧belleville spring顶隙bottom clearance定轴轮系ordinary gear train; gear train with fixed axes动力学dynamics动密封kinematical seal动能dynamic energy动力粘度dynamic viscosity动力润滑dynamic lubrication动平衡dynamic balance动平衡机dynamic balancing machine 动态特性dynamic characteristics动态分析设计dynamic analysis design 动压力dynamic reaction动载荷dynamic load。
Spherical Harmonics
• Use Microsoft DirectX SDK for spherical harmonics computations
Irradiance
• • The radiance of a purely diffuse surface is defined in terms of the surface’s irradiance Irradiance is an integral of the field-radiance function multiplied by the Lambertian cosine term over a hemisphere
[Greger98]
•
Every point in space has a radiance distribution function
– Radiance is a 5D function (3 spacial dimensions and 2 directional dimensions)
Radiance
– Approximates global illumination on the surface – Only for static scenes! – Does not address dynamic objects that move through the scene – Result in beautifully rendered, globally illuminated scenes that contain unrealistic, locally lit dynamic objects
•
Instead! Compress irradiance maps
– Represent each as a vector of spherical harmonic coefficients – Reduces both storage and bandwidth costs
【1】Lambertian Reflectance and Linear Subspaces
Lambertian Reflectance and Linear Subspaces Ronen Basri,Member,IEEE,and David W.Jacobs,Member,IEEE Abstract—We prove that the set of all Lambertian reflectance functions(the mapping from surface normals to intensities)obtained with arbitrary distant light sources lies close to a9D linear subspace.This implies that,in general,the set of images of a convex Lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low-dimensional linear subspace,explaining prior empirical results.We also provide a simple analytic characterization of this linear space.We obtain these results byrepresenting lighting using spherical harmonics and describing the effects of Lambertian materials as the analog of a convolution.These results allow us to construct algorithms for object recognition based on linear methods as well as algorithms that use convex optimization to enforce nonnegative lighting functions.We also show a simple way to enforce nonnegative lighting when the images of an object lie near a 4D linear space.We apply these algorithms to perform face recognition by finding the3D model that best matches a2D query image.Index Terms—Face recognition,illumination,Lambertian,linear subspaces,object recognition,specular,spherical harmonics.æ1I NTRODUCTIONV ARIABILITY in lighting has a large effect on the appearance of objects in images,as is illustrated in Fig.1.But we show in this paper that the set of images an object produces under different lighting conditions can,in some cases,be simply characterized as a nine dimensional subspace in the space of all possible images.This characterization can be used to construct efficient recogni-tion algorithms that handle lighting variations.Under normal conditions,light coming from all direc-tions illuminates an object.When the sources of light are distant from an object,we may describe the lighting conditions by specifying the intensity of light as a function of its direction.Light,then,can be thought of as a nonnegative function on the surface of a sphere.This allows us to represent scenes in which light comes from multiple sources,such as a room with a few lamps and, also,to represent light that comes from extended sources, such as light from the sky,or light reflected off a wall.Our analysis begins by representing these lighting func-tions using spherical harmonics.This is analogous to Fourier analysis,but on the surface of the sphere.With this representation,low-frequency light,for example,means light whose intensity varies slowly as a function of direction.To model the way diffuse surfaces turn light into an image,we look at the amount of light reflected as a function of the surface normal(assuming unit albedo),for each lighting condition.We show that these reflectance functions are produced through the analog of a convolution of the lighting function using a kernel that represents Lambert’s reflection. This kernel acts as a low-pass filter with99.2percent of its energy in the first nine components,the zero,first,and second order harmonics.(This part of our analysis was derived independently also by Ramamoorthi and Hanrahan[31].)We use this and the nonnegativity of light to prove that under any lighting conditions,a nine-dimensional linear subspace,for example,accounts for at least98percent of the variability in the reflectance function.This suggests that in general the set of images of a convex,Lambertian object can be approxi-mated accurately by a low-dimensional linear subspace.We further show how to analytically derive this subspace from a model of an object that includes3D structure and albedo.To provide some intuition about these results,consider the example shown in Fig.2.The figure shows a white sphere made of diffuse material,illuminated by three distant lights.The lighting function can be described in this case as the sum of three delta functions.The image of the sphere, however,is smoothly shaded.If we look at a cross-section of the reflectance function,describing how the sphere reflects light,we can see that it is a very smoothed version of three delta functions.The diffuse material acts as a filter,so that the reflected light varies much more slowly than the incoming light.Our results help to explain recent experimental work(e.g., Epstein et al.[10],Hallinan[15],Yuille et al.[40])that has indicated that the set of images produced by an object under a wide range of lighting conditions lies near a low dimensional linear subspace in the space of all possible images.Our results also allow us to better understand several existing recogni-tion methods.For example,previous work showed that,if we restrict every point on the surface of a diffuse object to face every light source(that is,ignoring attached shadows),then the set of images of the object lies in a3D linear space(e.g., Shashua[34]and Moses[26]).Our analysis shows that,in fact, this approach uses the linear space spanned by the three first order harmonics,but omits the significant zeroth order(DC) component.Koenderink and van Doorn[21]augmented this space in order to account for an additional,perfect diffuse component.The additional component in their method is the missing DC component.Our analysis also leads us to new methods of recognizing objects with unknown pose and lighting conditions.In particular,we discuss how the harmonic basis,which is derived analytically from a model of an object,can be used in a linear subspace-based object recognition algorithm,in place.R.Basri is with the Department of Computer Science,The WeizmannInstitute of Science,Rehovot,76100Israel.E-mail:ronen.basri@weizmann.ac.il.. D.W.Jacobs is with NEC Research Institute,4Independence Way,Princeton,NJ08540.E-mail:dwj@.Manuscript received19June2001;revised31Dec.2001;accepted30Apr.2002.Recommended for acceptance by P.Belhumeur.For information on obtaining reprints of this article,please send e-mail to:tpami@,and reference IEEECS Log Number114379.0162-8828/03/$17.00ß2003IEEE Published by the IEEE Computer Societyof a basis derived by performing SVD on large collections of rendered images.Furthermore,we show how we can enforce the constraint that light is nonnegative everywhere by projecting this constraint to the space spanned by the harmonic basis.With this constraint recognition is expressed as a nonnegative least-squares problem that can be solved using convex optimization.This leads to an algorithm for recognizing objects under varying pose and illumination that resembles Georghiades et al.[12],but works in a low-dimensional space that is derived analytically from a model.The use of the harmonic basis,in this case,allows us to rapidly produce a representation to the images of an object in poses determined at runtime.Finally,we discuss the case in which a first order approximation provides an adequate approxima-tion to the images of an object.The set of images then lies near a 4D linear subspace.In this case,we can express the nonnegative lighting constraint analytically.We use this expression to perform recognition in a particularly efficient way,without complex,iterative optimization techniques.The paper is divided as follows:Section 2briefly reviews the relevant studies.Section 3presents our analysis of Lambertian reflectance.Section 4uses this analysis to derive new algorithms for object recognition.Finally,Section 5discusses extensions to specular reflectance.2P AST A PPROACHESOur work is related to a number of recent approaches to object recognition that represent the set of images that an object can produce using low-dimensional linear subspaces of the space of all images.Ullman and Basri [38]analytically derive such a representation for sets of 3D points undergoing scaled orthographic projection.Shashua [34]and Moses [26](see also Nayar and Murase [28]and Zhao and Yang [41])derive a 3D linear representation of the set of images produced by a Lambertian object as lighting changes,but ignoring shadows.Hayakawa [16]uses factorization to build 3D models using this linear representation.Koenderink and van Doorn [21]extend this to a 4D space by allowing the light to include a diffuse component.Our work differs from these in that ourrepresentation accounts for attached shadows.These shadows occur when a surface faces away from a light source.We do not account for cast shadows,which occur when an intervening part of an object blocks the light from reaching a different part of the surface.For convex objects,only attached shadows occur.As is mentioned in Section 1,we show below that the 4D space used by Koenderink and van Doorn is in fact the space obtained by a first order harmonic approximation of the images of the object.The 3D space used by Shashua,Moses,and Hayakawa is the same space,but it lacks the significant DC component.Researchers have collected large sets of images and performed PCA to build representations that capture within class variations (e.g.,Kirby and Sirovich [19],Turk and Pentland [37],and Cootes et al.[7])and variations due to pose and lighting (Murase and Nayar [27],Hallinan [15],Belhumeur et al.[3],and Yuille et al.[40];see also Malzbender et al.[24]).This approach and its variations have been extremely popular in the last decade,particularly in applications to face recognition.Hallinan [15],Epstein et al.[10],and Yuille et al.[40]perform experiments that show that large numbers of images of real,Lambertian objects,taken with varied lighting conditions,do lie near a low-dimensional linear space,justifying this representation.Belhumeur and Kriegman [4]have shown that the set of images of an object under arbitrary illumination forms a convex cone in the space of all possible images.This analysis accounts for attached shadows.In addition,for convex,Lambertian objects,they have shown that this cone (called the illumination cone )may have unbounded dimension.They have further shown how to construct the cone from as few as three images.Georghiades et al.[11],[12]use this representa-tion for object recognition.To simplify the representation (an accurate representation of the illumination cone requires all the images that can be obtained with a single directional source),they further projected the images to low-dimen-sional subspaces obtained by rendering the objects and applying PCA to the rendered images.Our analysis allows us to further simplify this process by using instead the harmonic basis,which is derived analytically from a model of the object.This leads to a significant speed up of the recognition process (see Section 4).Spherical harmonics have been used in graphics to efficiently represent the bidirectional reflection distribution function (BRDF)of different materials by,e.g.,Cabral et al.[6]and Westin et al.[39].Koenderink and van Doorn [20]proposed replacing the spherical harmonics basis with a basis for functions on the half-sphere that is derived from the Zernike polynomials,since BRDFs are defined over a half sphere.Nimeroff et al.[29],Dobashi et al.[8],and Teo et al.Fig.1.The same face,under two different lighting conditions.Fig.2.On the left,a white sphere illuminated by three directional (distant point)sources of light.All the lights are parallel to the image plane,one source illuminates the sphere from above and the two others illuminate the sphere from diagonal directions.In the middle,a cross-section of the lighting function with three peaks corresponding to the three light sources.On the right,a cross-section indicating how the sphere reflects light.We will make precise the intuition that the material acts as a low-pass filtering,smoothing the light as it reflects it.[35]explore specific lighting configurations(e.g.,daylight) that can be represented efficiently as a linear combination of basis lightings.Dobashi et al.[8],in particular,use spherical harmonics to form such a basis.Miller and Hoffman[25]were first to describe the process of turning incoming light into reflection as a convolution. D’Zmura[9]describes this process in terms of spherical harmonics.With this representation,after truncating high order components,the reflection process can be written as a linear transformation and,so,the low-order components of the lighting can be recovered by inverting the transformation. He used this analysis to explore ambiguities in lighting.We extend this work by deriving subspace results for the reflectance function,providing analytic descriptions of the basis images,and constructing new recognition algorithms that use this analysis while enforcing nonnegative lighting.Independent of and contemporaneous with our work, Ramamoorthi and Hanrahan[31],[32],[33]have described the effect of Lambertian reflectance as a convolution and analyzed it in terms of spherical harmonics.Like D’Zmura, they use this analysis to explore the problem of recovering lighting from reflectances.Both the work of Ramamoorthi and Hanrahan and ours(first described in[1])show that Lambertian reflectance acts as a low-pass filter with most of the energy in the first nine components.In addition to this,we show that the space spanned by the first nine harmonics accurately approximates the reflectance function under any light configuration,even when the light is dominated by high frequencies.Furthermore,we show how to use this space for object recognition.Since the first introduction of our work,a number of related papers have further used and extended these ideas in a number of directions.Specifically,Ramamoorthi[30] analyzed the relationship between the principal components of the images produced by an object and the first nine harmonics.Lee et al.[23]constructed approximations to this space using physically realizable lighting.Basri and Jacobs[2] used the harmonic formulation to construct algorithms for photometric stereo under unknown,arbitrary lighting. Finally,Thornber and Jacobs[36]and Ramamoorthi and Hanrahan[32]further examined the effect of specularity and cast shadows.3M ODELING I MAGE F ORMATIONIn this section,we construct an analytically derived repre-sentation of the images produced by a convex,Lambertian object illuminated by distant light sources.We restrict ourselves to convex objects,so we can ignore the effect of shadows cast by one part of the object on another part of it.We assume that the surface of the object reflects light according to Lambert’s law[22],which states that materials absorb light and reflect it uniformly in all directions.The only parameter of this model is the albedo at each point on the object,which describes the fraction of the light reflected at that point.This relatively simple model applies to diffuse(nonshiny) materials.It has been analyzed and used effectively in a number of vision applications.By a“distant”light source we mean that it is valid to make the approximation that a light shines on each point in the scene from the same angle,and with the same intensity(this also rules out,for example,slide projectors).Lighting, however,may come from multiple sources,including diffuse sources such as the sky.We can therefore describe the intensity of the light as a single function of its direction that does not depend on the position in the scene.It is important to note that our analysis accounts for attached shadows,which occur when a point in the scene faces away from a light source.While we are interested in understanding the images created by an object,we simplify this problem by breaking it into two parts.We use an intermediate representation,the reflectance function(also called the reflectance map,see Horn [17,chapters10,11]).Given our assumptions,the amount of light reflected by a white surface patch(a patch with albedo of one)depends on the surface normal at that point,but not on its spatial position.For a specific lighting condition,the reflectance function describes how much light is reflected by each surface normal.In the first part of our analysis,we consider the set of possible reflectance functions produced under different illumination conditions.This analysis is independent of the structure of the particular object we are looking at;it depends only on lighting conditions and the properties of Lambertian reflectance.Then,we discuss the relationship between the reflectance function and the image. This depends on object structure and albedo,but not on lighting,except as it determines the reflectance function.We begin by discussing the relation of lighting and reflectance.Before we proceed,we would like to clarify the relation between the reflectance function and the bidirectional reflection distribution function(BRDF).The BRDF of a surface material is a function that describes the ratio of radiance,the amount of light reflected by the surface in every direction (measured in power per unit area per solid angle),to irradiance,the amount of light falling on the surface in every direction(measured in power per unit area).BRDF is commonly specified in a local coordinate frame,in which the surface normal is fixed at the north pole.The BRDF of a Lambertian surface is constant,since such a surface reflects light equally in all direction,and it is equal to1=%.In contrast, thereflectancefunctiondescribestheradianceofaunitsurface area given the entire distribution of light in the scene.The reflectance function is obtained by integrating the BRDF over all directions of incident light,weighting the intensity of the light by the foreshortening of the surface as seen from each lightsource.Inaddition,thereflectancefunctionisspecifiedin a global,viewer centered coordinate frame in which the viewing direction is fixed at the north pole.For example,if a scene is illuminated by a single directional source(a distant point source)of unit intensity,the reflectance function for every surface normal will contain the appropriate foreshortening of the surface with respect to the light source direction scaled by 1=%.(For surface normals that face away from the light source the reflectance function will vanish.)For simplicity,we omit below the extra factor of1=%that arises from the Lambertian BRDF since it only scales the intensities in the image by a constant factor.3.1Image Formation as the Analog of aConvolutionBoth lighting and reflectance can be described as functions on the surface of the sphere.We describe the intensity of light as a function of its direction.This formulation allows us to consider multiple light sources that illuminate an object simultaneously from many directions.We describe reflec-tance as a function of the direction of the surface normal.To begin,we introduce notation for describing such functions.Let S 2denote the surface of a unit sphere centered at the origin.We will use u;v to denote unit vectors.We denote their Cartesian coordinates as ðx;y;z Þ,with x 2þy 2þz 2¼1.When appropriate,we will denote such vectors by a pair of angles,ð ;0Þ,withu ¼ðx;y;z Þ¼ðcos 0sin ;sin 0sin ;cos Þ;ð1Þwhere 0 %and 0 0 2%.In this coordinate frame,the poles are set at ð0;0;Æ1Þ, denotes the angle between u and ð0;0;1Þ,and it varies with latitude,and 0varies with longitude.We will use ð l ;0l Þto denote a direction of light and ð r ;0r Þto denote a direction of reflectance,although we will drop this subscript when there is no ambiguity.Similarly,we may express the lighting or reflectance directions using unit vectors such as u l or v r .Since we assume that the sphere is illuminated by a distant set of lights all points are illuminated by identical lighting conditions.Consequently,the configuration of lights that illuminate the sphere can be expressed as a nonnegative function ‘ð l ;0l Þ,giving the intensity of the light reaching the sphere from each direction ð l ;0l Þ.We may also write this as ‘ðu l Þ,describing lighting direction with a unit vector.According to Lambert’s law,if a light ray of intensity l and coming from the direction u l reaches a surface point with albedo &and normal direction v r ,then the intensity,i ,reflected by the point due to this light is given byi ¼l ðu l Þ&max ðu l Áv r ;0Þ:ð2ÞIf we fix the lighting,and ignore &for now,then the reflected light is a function of the surface normal alone.We write this function as r ð r ;0r Þ,or r ðv r Þ.If light reaches a point from a multitude of directions,then the light reflected by the point would be the sum of (or in the continuous case the integral over)the contribution for each direction.If we denote k ðu Áv Þ¼max ðu Áv;0Þ,then we can write:r ðv r Þ¼ZS 2k ðu l Áv r Þ‘ðu l Þdu l ;ð3Þwhere RS 2denotes integration over the surface of the sphere.Below,we will occasionally abuse notation and write k ðu Þto denote the max of zero and the cosine of the angle between u and the north pole (that is,omitting v means that v is the north pole).We therefore call k the half-cosine function.We can also write k ð Þ,where is the latitude of u ,since k only depends on the component of u .For any fixed v ,as we vary u (as we do while integrating (3)),then k ðu Áv Þcomputes the half cosine function centered around v instead of the north pole.That is,since v r is fixed inside the integral,we can think of k as a function just of u ,which gives the max of zero and the cosine of the angle between u and v r .Thus,intuitively,(3)is analogous to a convolution,in which we center a kernel (the half-cosine function defined by k ),and integrate its product with a signal (‘).In fact,we will call this a convolution,and writer ðv r Þ¼k ѼdefZ S 2k ðu l Áv r Þ‘ðu l Þdu l :ð4ÞNote that there is some subtlety here since we cannot,ingeneral,speak of convolving a function on the surface of the sphere with an arbitrary kernel.This is because we have three degrees of freedom in how we position a convolution kernel on the surface of the sphere,but the output of theconvolution should be a function on the surface of the sphere,which has only two degrees of freedom.However,since k is rotationally symmetric this ambiguity disappears.In fact,we have been careful to only define convolution for rotationally symmetric k .3.2Spherical Harmonics and the Funk-Hecke TheoremJust as the Fourier basis is convenient for examining the results of convolutions in the plane,similar tools exist for understanding the results of the analog of convolutions on the sphere.We now introduce these tools,and use them to show that in producing reflectance,k acts as a low-pass filter.The surface spherical harmonics are a set of functions that form an orthonormal basis for the set of all functions on the surface of the sphere.We denote these functions by Y nm ,with n ¼0;1;2;...and Àn m n :Y nm ð ;0Þ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2n þ1Þ4%ðn Àj m jÞ!ðn þj m jÞ!s P n j m j ðcos Þe im0;ð5Þwhere P nm are the associated Legendre functions ,defined asP nm ðz Þ¼1Àz 2ðÞm=22n !d n þm dz z 2À1ÀÁn :ð6ÞWe say that Y nm is an n th order harmonic.In the course of this paper,it will sometimes be convenient to parameterize Y nm as a function of space coordinates ðx;y;z Þrather than angles.The spherical harmonics,written Y nm ðx;y;z Þ,then become polynomials of degree n in ðx;y;z Þ.The first nine harmonics then becomeY 00¼1ffiffiffiffi4%p Y 10¼ffiffiffiffi34%q z Y e 11¼ffiffiffiffi34%q x Y o11¼ffiffiffiffi34%q yY 20¼12ffiffiffiffi54%q ð3z 2À1ÞY e 21¼3ffiffiffiffiffiffi512%q xz Y o 21¼3ffiffiffiffiffiffi5q yz Y e 22¼3ffiffiffiffiffiffi5q x 2Ày 2ðÞY o 22¼3ffiffiffiffiffiffi512%q xy;ð7Þwhere the superscripts e and o denote the even and the odd components of the harmonics,respectively,(soY nm ¼Y e n j m j ÆiY on j m j ,according to the sign of m ;in fact the even and odd versions of the harmonics are more convenient to use in practice since the reflectance function is real).Because the spherical harmonics form an orthonormal basis,thismeansthatany piecewisecontinuousfunction,f ,on the surface of the sphere can be written as a linear combination of an infinite series of harmonics.Specifically,for any f ,f ðu Þ¼X 1n ¼0X n m ¼Ànf nm Y nm ðu Þ;ð8Þwhere f nm is a scalar value,computed as:f nm ¼ZS 2f ðu ÞY Ãnm ðu Þdu;ð9Þand Y Ãnmðu Þdenotes the complex conjugate of Y nm ðu Þ.If we rotate a function f,this acts as a phase shift.Define for every n the n th order amplitude of f asA n¼defffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12nþ1X nm¼Ànf2nms:ð10ÞThen,rotating f does not change the amplitude of a particular order.It may shuffle values of the coefficients, f nm,for a particular order,but it does not shift energy between harmonics of different orders.For example, consider a delta function.As in the case of the Fourier transform,the harmonic transform of a delta function has equal amplitude in every order.If the delta function is at the north pole,its transform is nonzero only for the zonal harmonics,in which m¼0.If the delta function is,in general,position,it has some energy in all harmonics.But in either case,the n th order amplitude is the same for all n.Both the lighting function,‘,and the Lambertian kernel, k,can be written as sums of spherical harmonics.Denote by‘¼X1n¼0X nm¼Ànl nm Y nm;ð11Þthe harmonic expansion of‘,and bykðuÞ¼X1n¼0k n Y n0:ð12ÞNote that,because kðuÞis circularly symmetric about the north pole,only the zonal harmonics participate in this expansion,andZ S2kðuÞYÃnmðuÞdu¼0;m¼0:ð13ÞSpherical harmonics are useful in understanding the effect of convolution by k because of the Funk-Hecke theorem, which is analogous to the convolution theorem.Loosely speaking,the theorem states that we can expand‘and k in terms of spherical harmonics and,then,convolving them is equivalent to multiplication of the coefficients of this expansion.We will state the Funk-Hecke theorem here in a form that is specialized to our specific concerns.Our treatment is based on Groemer[13],but Groemer presents a more general discussion in which,for example,the theorem is stated for spaces of arbitrary dimension.Theorem1(Funk-Hecke).Let kðuÁvÞbe a bounded,integrable function on[-1,1].Then:kÃY nm¼ n Y nmwithn¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi4%rk n:That is,the theorem states that the convolution of a (circularly symmetric)function k with a spherical harmonic Y mn(as defined in(4))results in the same harmonic,scaled by a scalar n. n depends on k and is tied directly to k n,the n th order coefficient of the harmonic expansion of k.Following the Funk-Hecke theorem,the harmonic ex-pansion of the reflectance function,r,can be written as:r¼kѼX1n¼0X nm¼Ànð n l nmÞY nm:ð14ÞThis is the chief implication of the Funk-Hecke theorem for our purposes.3.3Properties of the Convolution KernelThe Funk-Hecke theorem implies that in producing the reflectance function,r,the amplitude of the light,‘,at every order n is scaled by a factor n that depends only on the convolution kernel,k.We can use this to infer analytically what frequencies will dominate r.To achieve this,we treat‘as a signal and k as a filter,and ask how the amplitudes of ‘change as it passes through the filter.The harmonic expansion of the Lambertian kernel(12) can be derived(with some tedious manipulation detailed in Appendix A)yieldingk n¼ffiffi%p2n¼0ffiffi%3pn¼1ðÀ1Þn2þ1ffiffiffiffiffiffiffiffiffiffiffiffiffið2nþ1Þ%pnnn2n!2;even0n!2;odd:8>>>><>>>>:ð15ÞThe first few coefficients,for example,arek0¼ffiffi%p2%0:8862k1¼ffiffi%3p%1:0233k2¼ffiffiffiffi5%p8%0:4954k4¼Àffiffi%p16%À0:1108k6¼ffiffiffiffiffiffi13%p128%0:0499k8¼ffiffiffiffiffiffi17%p256%À0:0285:ð16Þ(k3¼k5¼k7¼0),j k n j approaches zero as OðnÀ2Þ.A graph representation of the coefficients is shown in Fig.3.The energy captured by every harmonic term is measured commonly by the square of its respective coefficient divided by the total squared energy of the transformed function.The total squared energy in the half cosine function is given byZ2%Z%k2ð Þsin d d0¼2%Z%2cos2 sin d ¼2%:ð17ÞFig.3.From left to right:A graph representation of the first11coefficients of the Lambertian kernel,the relative energy captured by each of the coefficients,and the cumulative energy.。
黑洞的准正模式(quasinormal modes)
Quasi-Normal Modes of Stars and Black HolesKostas D.KokkotasDepartment of Physics,Aristotle University of Thessaloniki,Thessaloniki54006,Greece.kokkotas@astro.auth.grhttp://www.astro.auth.gr/˜kokkotasandBernd G.SchmidtMax Planck Institute for Gravitational Physics,Albert Einstein Institute,D-14476Golm,Germany.bernd@aei-potsdam.mpg.dePublished16September1999/Articles/Volume2/1999-2kokkotasLiving Reviews in RelativityPublished by the Max Planck Institute for Gravitational PhysicsAlbert Einstein Institute,GermanyAbstractPerturbations of stars and black holes have been one of the main topics of relativistic astrophysics for the last few decades.They are of partic-ular importance today,because of their relevance to gravitational waveastronomy.In this review we present the theory of quasi-normal modes ofcompact objects from both the mathematical and astrophysical points ofview.The discussion includes perturbations of black holes(Schwarzschild,Reissner-Nordstr¨o m,Kerr and Kerr-Newman)and relativistic stars(non-rotating and slowly-rotating).The properties of the various families ofquasi-normal modes are described,and numerical techniques for calculat-ing quasi-normal modes reviewed.The successes,as well as the limits,of perturbation theory are presented,and its role in the emerging era ofnumerical relativity and supercomputers is discussed.c 1999Max-Planck-Gesellschaft and the authors.Further information on copyright is given at /Info/Copyright/.For permission to reproduce the article please contact livrev@aei-potsdam.mpg.de.Article AmendmentsOn author request a Living Reviews article can be amended to include errata and small additions to ensure that the most accurate and up-to-date infor-mation possible is provided.For detailed documentation of amendments, please go to the article’s online version at/Articles/Volume2/1999-2kokkotas/. Owing to the fact that a Living Reviews article can evolve over time,we recommend to cite the article as follows:Kokkotas,K.D.,and Schmidt,B.G.,“Quasi-Normal Modes of Stars and Black Holes”,Living Rev.Relativity,2,(1999),2.[Online Article]:cited on<date>, /Articles/Volume2/1999-2kokkotas/. The date in’cited on<date>’then uniquely identifies the version of the article you are referring to.3Quasi-Normal Modes of Stars and Black HolesContents1Introduction4 2Normal Modes–Quasi-Normal Modes–Resonances7 3Quasi-Normal Modes of Black Holes123.1Schwarzschild Black Holes (12)3.2Kerr Black Holes (17)3.3Stability and Completeness of Quasi-Normal Modes (20)4Quasi-Normal Modes of Relativistic Stars234.1Stellar Pulsations:The Theoretical Minimum (23)4.2Mode Analysis (26)4.2.1Families of Fluid Modes (26)4.2.2Families of Spacetime or w-Modes (30)4.3Stability (31)5Excitation and Detection of QNMs325.1Studies of Black Hole QNM Excitation (33)5.2Studies of Stellar QNM Excitation (34)5.3Detection of the QNM Ringing (37)5.4Parameter Estimation (39)6Numerical Techniques426.1Black Holes (42)6.1.1Evolving the Time Dependent Wave Equation (42)6.1.2Integration of the Time Independent Wave Equation (43)6.1.3WKB Methods (44)6.1.4The Method of Continued Fractions (44)6.2Relativistic Stars (45)7Where Are We Going?487.1Synergism Between Perturbation Theory and Numerical Relativity487.2Second Order Perturbations (48)7.3Mode Calculations (49)7.4The Detectors (49)8Acknowledgments50 9Appendix:Schr¨o dinger Equation Versus Wave Equation51Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt41IntroductionHelioseismology and asteroseismology are well known terms in classical astro-physics.From the beginning of the century the variability of Cepheids has been used for the accurate measurement of cosmic distances,while the variability of a number of stellar objects(RR Lyrae,Mira)has been associated with stel-lar oscillations.Observations of solar oscillations(with thousands of nonradial modes)have also revealed a wealth of information about the internal structure of the Sun[204].Practically every stellar object oscillates radially or nonradi-ally,and although there is great difficulty in observing such oscillations there are already results for various types of stars(O,B,...).All these types of pulsations of normal main sequence stars can be studied via Newtonian theory and they are of no importance for the forthcoming era of gravitational wave astronomy.The gravitational waves emitted by these stars are extremely weak and have very low frequencies(cf.for a discussion of the sun[70],and an im-portant new measurement of the sun’s quadrupole moment and its application in the measurement of the anomalous precession of Mercury’s perihelion[163]). This is not the case when we consider very compact stellar objects i.e.neutron stars and black holes.Their oscillations,produced mainly during the formation phase,can be strong enough to be detected by the gravitational wave detectors (LIGO,VIRGO,GEO600,SPHERE)which are under construction.In the framework of general relativity(GR)quasi-normal modes(QNM) arise,as perturbations(electromagnetic or gravitational)of stellar or black hole spacetimes.Due to the emission of gravitational waves there are no normal mode oscillations but instead the frequencies become“quasi-normal”(complex), with the real part representing the actual frequency of the oscillation and the imaginary part representing the damping.In this review we shall discuss the oscillations of neutron stars and black holes.The natural way to study these oscillations is by considering the linearized Einstein equations.Nevertheless,there has been recent work on nonlinear black hole perturbations[101,102,103,104,100]while,as yet nothing is known for nonlinear stellar oscillations in general relativity.The study of black hole perturbations was initiated by the pioneering work of Regge and Wheeler[173]in the late50s and was continued by Zerilli[212]. The perturbations of relativistic stars in GR werefirst studied in the late60s by Kip Thorne and his collaborators[202,198,199,200].The initial aim of Regge and Wheeler was to study the stability of a black hole to small perturbations and they did not try to connect these perturbations to astrophysics.In con-trast,for the case of relativistic stars,Thorne’s aim was to extend the known properties of Newtonian oscillation theory to general relativity,and to estimate the frequencies and the energy radiated as gravitational waves.QNMs werefirst pointed out by Vishveshwara[207]in calculations of the scattering of gravitational waves by a Schwarzschild black hole,while Press[164] coined the term quasi-normal frequencies.QNM oscillations have been found in perturbation calculations of particles falling into Schwarzschild[73]and Kerr black holes[76,80]and in the collapse of a star to form a black hole[66,67,68]. Living Reviews in Relativity(1999-2)5Quasi-Normal Modes of Stars and Black Holes Numerical investigations of the fully nonlinear equations of general relativity have provided results which agree with the results of perturbation calculations;in particular numerical studies of the head-on collision of two black holes [30,29](cf.Figure 1)and gravitational collapse to a Kerr hole [191].Recently,Price,Pullin and collaborators [170,31,101,28]have pushed forward the agreement between full nonlinear numerical results and results from perturbation theory for the collision of two black holes.This proves the power of the perturbation approach even in highly nonlinear problems while at the same time indicating its limits.In the concluding remarks of their pioneering paper on nonradial oscillations of neutron stars Thorne and Campollataro [202]described it as “just a modest introduction to a story which promises to be long,complicated and fascinating ”.The story has undoubtedly proved to be intriguing,and many authors have contributed to our present understanding of the pulsations of both black holes and neutron stars.Thirty years after these prophetic words by Thorne and Campollataro hundreds of papers have been written in an attempt to understand the stability,the characteristic frequencies and the mechanisms of excitation of these oscillations.Their relevance to the emission of gravitational waves was always the basic underlying reason of each study.An account of all this work will be attempted in the next sections hoping that the interested reader will find this review useful both as a guide to the literature and as an inspiration for future work on the open problems of the field.020406080100Time (M ADM )-0.3-0.2-0.10.00.10.20.3(l =2) Z e r i l l i F u n c t i o n Numerical solutionQNM fit Figure 1:QNM ringing after the head-on collision of two unequal mass black holes [29].The continuous line corresponds to the full nonlinear numerical calculation while the dotted line is a fit to the fundamental and first overtone QNM.In the next section we attempt to give a mathematical definition of QNMs.Living Reviews in Relativity (1999-2)K.D.Kokkotas and B.G.Schmidt6 The third and fourth section will be devoted to the study of the black hole and stellar QNMs.In thefifth section we discuss the excitation and observation of QNMs andfinally in the sixth section we will mention the more significant numerical techniques used in the study of QNMs.Living Reviews in Relativity(1999-2)7Quasi-Normal Modes of Stars and Black Holes 2Normal Modes–Quasi-Normal Modes–Res-onancesBefore discussing quasi-normal modes it is useful to remember what normal modes are!Compact classical linear oscillating systems such asfinite strings,mem-branes,or cavitiesfilled with electromagnetic radiation have preferred time harmonic states of motion(ωis real):χn(t,x)=e iωn tχn(x),n=1,2,3...,(1) if dissipation is neglected.(We assumeχto be some complex valuedfield.) There is generally an infinite collection of such periodic solutions,and the“gen-eral solution”can be expressed as a superposition,χ(t,x)=∞n=1a n e iωn tχn(x),(2)of such normal modes.The simplest example is a string of length L which isfixed at its ends.All such systems can be described by systems of partial differential equations of the type(χmay be a vector)∂χ∂t=Aχ,(3)where A is a linear operator acting only on the spatial variables.Because of thefiniteness of the system the time evolution is only determined if some boundary conditions are prescribed.The search for solutions periodic in time leads to a boundary value problem in the spatial variables.In simple cases it is of the Sturm-Liouville type.The treatment of such boundary value problems for differential equations played an important role in the development of Hilbert space techniques.A Hilbert space is chosen such that the differential operator becomes sym-metric.Due to the boundary conditions dictated by the physical problem,A becomes a self-adjoint operator on the appropriate Hilbert space and has a pure point spectrum.The eigenfunctions and eigenvalues determine the periodic solutions(1).The definition of self-adjointness is rather subtle from a physicist’s point of view since fairly complicated“domain issues”play an essential role.(See[43] where a mathematical exposition for physicists is given.)The wave equation modeling thefinite string has solutions of various degrees of differentiability. To describe all“realistic situations”,clearly C∞functions should be sufficient. Sometimes it may,however,also be convenient to consider more general solu-tions.From the mathematical point of view the collection of all smooth functions is not a natural setting to study the wave equation because sequences of solutionsLiving Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt8 exist which converge to non-smooth solutions.To establish such powerful state-ments like(2)one has to study the equation on certain subsets of the Hilbert space of square integrable functions.For“nice”equations it usually happens that the eigenfunctions are in fact analytic.They can then be used to gen-erate,for example,all smooth solutions by a pointwise converging series(2). The key point is that we need some mathematical sophistication to obtain the “completeness property”of the eigenfunctions.This picture of“normal modes”changes when we consider“open systems”which can lose energy to infinity.The simplest case are waves on an infinite string.The general solution of this problem isχ(t,x)=A(t−x)+B(t+x)(4) with“arbitrary”functions A and B.Which solutions should we study?Since we have all solutions,this is not a serious question.In more general cases, however,in which the general solution is not known,we have to select a certain class of solutions which we consider as relevant for the physical problem.Let us consider for the following discussion,as an example,a wave equation with a potential on the real line,∂2∂t2χ+ −∂2∂x2+V(x)χ=0.(5)Cauchy dataχ(0,x),∂tχ(0,x)which have two derivatives determine a unique twice differentiable solution.No boundary condition is needed at infinity to determine the time evolution of the data!This can be established by fairly simple PDE theory[116].There exist solutions for which the support of thefields are spatially compact, or–the other extreme–solutions with infinite total energy for which thefields grow at spatial infinity in a quite arbitrary way!From the point of view of physics smooth solutions with spatially compact support should be the relevant class–who cares what happens near infinity! Again it turns out that mathematically it is more convenient to study all solu-tions offinite total energy.Then the relevant operator is again self-adjoint,but now its spectrum is purely“continuous”.There are no eigenfunctions which are square integrable.Only“improper eigenfunctions”like plane waves exist.This expresses the fact that wefind a solution of the form(1)for any realωand by forming appropriate superpositions one can construct solutions which are “almost eigenfunctions”.(In the case V(x)≡0these are wave packets formed from plane waves.)These solutions are the analogs of normal modes for infinite systems.Let us now turn to the discussion of“quasi-normal modes”which are concep-tually different to normal modes.To define quasi-normal modes let us consider the wave equation(5)for potentials with V≥0which vanish for|x|>x0.Then in this case all solutions determined by data of compact support are bounded: |χ(t,x)|<C.We can use Laplace transformation techniques to represent such Living Reviews in Relativity(1999-2)9Quasi-Normal Modes of Stars and Black Holes solutions.The Laplace transformˆχ(s,x)(s>0real)of a solutionχ(t,x)isˆχ(s,x)= ∞0e−stχ(t,x)dt,(6) and satisfies the ordinary differential equations2ˆχ−ˆχ +Vˆχ=+sχ(0,x)+∂tχ(0,x),(7) wheres2ˆχ−ˆχ +Vˆχ=0(8) is the homogeneous equation.The boundedness ofχimplies thatˆχis analytic for positive,real s,and has an analytic continuation onto the complex half plane Re(s)>0.Which solutionˆχof this inhomogeneous equation gives the unique solution in spacetime determined by the data?There is no arbitrariness;only one of the Green functions for the inhomogeneous equation is correct!All Green functions can be constructed by the following well known method. Choose any two linearly independent solutions of the homogeneous equation f−(s,x)and f+(s,x),and defineG(s,x,x )=1W(s)f−(s,x )f+(s,x)(x <x),f−(s,x)f+(s,x )(x >x),(9)where W(s)is the Wronskian of f−and f+.If we denote the inhomogeneity of(7)by j,a solution of(7)isˆχ(s,x)= ∞−∞G(s,x,x )j(s,x )dx .(10) We still have to select a unique pair of solutions f−,f+.Here the information that the solution in spacetime is bounded can be used.The definition of the Laplace transform implies thatˆχis bounded as a function of x.Because the potential V vanishes for|x|>x0,the solutions of the homogeneous equation(8) for|x|>x0aref=e±sx.(11) The following pair of solutionsf+=e−sx for x>x0,f−=e+sx for x<−x0,(12) which is linearly independent for Re(s)>0,gives the unique Green function which defines a bounded solution for j of compact support.Note that for Re(s)>0the solution f+is exponentially decaying for large x and f−is expo-nentially decaying for small x.For small x however,f+will be a linear com-bination a(s)e−sx+b(s)e sx which will in general grow exponentially.Similar behavior is found for f−.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt 10Quasi-Normal mode frequencies s n can be defined as those complex numbers for whichf +(s n ,x )=c (s n )f −(s n ,x ),(13)that is the two functions become linearly dependent,the Wronskian vanishes and the Green function is singular!The corresponding solutions f +(s n ,x )are called quasi eigenfunctions.Are there such numbers s n ?From the boundedness of the solution in space-time we know that the unique Green function must exist for Re (s )>0.Hence f +,f −are linearly independent for those values of s .However,as solutions f +,f −of the homogeneous equation (8)they have a unique continuation to the complex s plane.In [35]it is shown that for positive potentials with compact support there is always a countable number of zeros of the Wronskian with Re (s )<0.What is the mathematical and physical significance of the quasi-normal fre-quencies s n and the corresponding quasi-normal functions f +?First of all we should note that because of Re (s )<0the function f +grows exponentially for small and large x !The corresponding spacetime solution e s n t f +(s n ,x )is therefore not a physically relevant solution,unlike the normal modes.If one studies the inverse Laplace transformation and expresses χas a com-plex line integral (a >0),χ(t,x )=12πi +∞−∞e (a +is )t ˆχ(a +is,x )ds,(14)one can deform the path of the complex integration and show that the late time behavior of solutions can be approximated in finite parts of the space by a finite sum of the form χ(t,x )∼N n =1a n e (αn +iβn )t f +(s n ,x ).(15)Here we assume that Re (s n +1)<Re (s n )<0,s n =αn +iβn .The approxi-mation ∼means that if we choose x 0,x 1, and t 0then there exists a constant C (t 0,x 0,x 1, )such that χ(t,x )−N n =1a n e (αn +iβn )t f +(s n ,x ) ≤Ce (−|αN +1|+ )t (16)holds for t >t 0,x 0<x <x 1, >0with C (t 0,x 0,x 1, )independent of t .The constants a n depend only on the data [35]!This implies in particular that all solutions defined by data of compact support decay exponentially in time on spatially bounded regions.The generic leading order decay is determined by the quasi-normal mode frequency with the largest real part s 1,i.e.slowest damping.On finite intervals and for late times the solution is approximated by a finite sum of quasi eigenfunctions (15).It is presently unclear whether one can strengthen (16)to a statement like (2),a pointwise expansion of the late time solution in terms of quasi-normal Living Reviews in Relativity (1999-2)11Quasi-Normal Modes of Stars and Black Holes modes.For one particular potential(P¨o schl-Teller)this has been shown by Beyer[42].Let us now consider the case where the potential is positive for all x,but decays near infinity as happens for example for the wave equation on the static Schwarzschild spacetime.Data of compact support determine again solutions which are bounded[117].Hence we can proceed as before.Thefirst new point concerns the definitions of f±.It can be shown that the homogeneous equation(8)has for each real positive s a unique solution f+(s,x)such that lim x→∞(e sx f+(s,x))=1holds and correspondingly for f−.These functions are uniquely determined,define the correct Green function and have analytic continuations onto the complex half plane Re(s)>0.It is however quite complicated to get a good representation of these func-tions.If the point at infinity is not a regular singular point,we do not even get converging series expansions for f±.(This is particularly serious for values of s with negative real part because we expect exponential growth in x).The next new feature is that the analyticity properties of f±in the complex s plane depend on the decay of the potential.To obtain information about analytic continuation,even use of analyticity properties of the potential in x is made!Branch cuts may occur.Nevertheless in a lot of cases an infinite number of quasi-normal mode frequencies exists.The fact that the potential never vanishes may,however,destroy the expo-nential decay in time of the solutions and therefore the essential properties of the quasi-normal modes.This probably happens if the potential decays slower than exponentially.There is,however,the following way out:Suppose you want to study a solution determined by data of compact support from t=0to some largefinite time t=T.Up to this time the solution is–because of domain of dependence properties–completely independent of the potential for sufficiently large x.Hence we may see an exponential decay of the form(15)in a time range t1<t<T.This is the behavior seen in numerical calculations.The situation is similar in the case ofα-decay in quantum mechanics.A comparison of quasi-normal modes of wave equations and resonances in quantum theory can be found in the appendix,see section9.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt123Quasi-Normal Modes of Black HolesOne of the most interesting aspects of gravitational wave detection will be the connection with the existence of black holes[201].Although there are presently several indirect ways of identifying black holes in the universe,gravitational waves emitted by an oscillating black hole will carry a uniquefingerprint which would lead to the direct identification of their existence.As we mentioned earlier,gravitational radiation from black hole oscillations exhibits certain characteristic frequencies which are independent of the pro-cesses giving rise to these oscillations.These“quasi-normal”frequencies are directly connected to the parameters of the black hole(mass,charge and angu-lar momentum)and for stellar mass black holes are expected to be inside the bandwidth of the constructed gravitational wave detectors.The perturbations of a Schwarzschild black hole reduce to a simple wave equation which has been studied extensively.The wave equation for the case of a Reissner-Nordstr¨o m black hole is more or less similar to the Schwarzschild case,but for Kerr one has to solve a system of coupled wave equations(one for the radial part and one for the angular part).For this reason the Kerr case has been studied less thoroughly.Finally,in the case of Kerr-Newman black holes we face the problem that the perturbations cannot be separated in their angular and radial parts and thus apart from special cases[124]the problem has not been studied at all.3.1Schwarzschild Black HolesThe study of perturbations of Schwarzschild black holes assumes a small per-turbation hµνon a static spherically symmetric background metricds2=g0µνdxµdxν=−e v(r)dt2+eλ(r)dr2+r2 dθ2+sin2θdφ2 ,(17) with the perturbed metric having the formgµν=g0µν+hµν,(18) which leads to a variation of the Einstein equations i.e.δGµν=4πδTµν.(19) By assuming a decomposition into tensor spherical harmonics for each hµνof the formχ(t,r,θ,φ)= mχ m(r,t)r Y m(θ,φ),(20)the perturbation problem is reduced to a single wave equation,for the func-tionχ m(r,t)(which is a combination of the various components of hµν).It should be pointed out that equation(20)is an expansion for scalar quantities only.From the10independent components of the hµνonly h tt,h tr,and h rr transform as scalars under rotations.The h tθ,h tφ,h rθ,and h rφtransform asLiving Reviews in Relativity(1999-2)13Quasi-Normal Modes of Stars and Black Holes components of two-vectors under rotations and can be expanded in a series of vector spherical harmonics while the components hθθ,hθφ,and hφφtransform as components of a2×2tensor and can be expanded in a series of tensor spher-ical harmonics(see[202,212,152]for details).There are two classes of vector spherical harmonics(polar and axial)which are build out of combinations of the Levi-Civita volume form and the gradient operator acting on the scalar spherical harmonics.The difference between the two families is their parity. Under the parity operatorπa spherical harmonic with index transforms as (−1) ,the polar class of perturbations transform under parity in the same way, as(−1) ,and the axial perturbations as(−1) +11.Finally,since we are dealing with spherically symmetric spacetimes the solution will be independent of m, thus this subscript can be omitted.The radial component of a perturbation outside the event horizon satisfies the following wave equation,∂2∂t χ + −∂2∂r∗+V (r)χ =0,(21)where r∗is the“tortoise”radial coordinate defined byr∗=r+2M log(r/2M−1),(22) and M is the mass of the black hole.For“axial”perturbationsV (r)= 1−2M r ( +1)r+2σMr(23)is the effective potential or(as it is known in the literature)Regge-Wheeler potential[173],which is a single potential barrier with a peak around r=3M, which is the location of the unstable photon orbit.The form(23)is true even if we consider scalar or electromagnetic testfields as perturbations.The parameter σtakes the values1for scalar perturbations,0for electromagnetic perturbations, and−3for gravitational perturbations and can be expressed asσ=1−s2,where s=0,1,2is the spin of the perturbingfield.For“polar”perturbations the effective potential was derived by Zerilli[212]and has the form V (r)= 1−2M r 2n2(n+1)r3+6n2Mr2+18nM2r+18M3r3(nr+3M)2,(24)1In the literature the polar perturbations are also called even-parity because they are characterized by their behavior under parity operations as discussed earlier,and in the same way the axial perturbations are called odd-parity.We will stick to the polar/axial terminology since there is a confusion with the definition of the parity operation,the reason is that to most people,the words“even”and“odd”imply that a mode transforms underπas(−1)2n or(−1)2n+1respectively(for n some integer).However only the polar modes with even have even parity and only axial modes with even have odd parity.If is odd,then polar modes have odd parity and axial modes have even parity.Another terminology is to call the polar perturbations spheroidal and the axial ones toroidal.This definition is coming from the study of stellar pulsations in Newtonian theory and represents the type offluid motions that each type of perturbation induces.Since we are dealing both with stars and black holes we will stick to the polar/axial terminology.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt14where2n=( −1)( +2).(25) Chandrasekhar[54]has shown that one can transform the equation(21)for “axial”modes to the corresponding one for“polar”modes via a transforma-tion involving differential operations.It can also be shown that both forms are connected to the Bardeen-Press[38]perturbation equation derived via the Newman-Penrose formalism.The potential V (r∗)decays exponentially near the horizon,r∗→−∞,and as r−2∗for r∗→+∞.From the form of equation(21)it is evident that the study of black hole perturbations will follow the footsteps of the theory outlined in section2.Kay and Wald[117]have shown that solutions with data of compact sup-port are bounded.Hence we know that the time independent Green function G(s,r∗,r ∗)is analytic for Re(s)>0.The essential difficulty is now to obtain the solutions f±(cf.equation(10))of the equations2ˆχ−ˆχ +Vˆχ=0,(26) (prime denotes differentiation with respect to r∗)which satisfy for real,positives:f+∼e−sr∗for r∗→∞,f−∼e+r∗x for r∗→−∞.(27) To determine the quasi-normal modes we need the analytic continuations of these functions.As the horizon(r∗→∞)is a regular singular point of(26),a representation of f−(r∗,s)as a converging series exists.For M=12it reads:f−(r,s)=(r−1)s∞n=0a n(s)(r−1)n.(28)The series converges for all complex s and|r−1|<1[162].(The analytic extension of f−is investigated in[115].)The result is that f−has an extension to the complex s plane with poles only at negative real integers.The representation of f+is more complicated:Because infinity is a singular point no power series expansion like(28)exists.A representation coming from the iteration of the defining integral equation is given by Jensen and Candelas[115],see also[159]. It turns out that the continuation of f+has a branch cut Re(s)≤0due to the decay r−2for large r[115].The most extensive mathematical investigation of quasi-normal modes of the Schwarzschild solution is contained in the paper by Bachelot and Motet-Bachelot[35].Here the existence of an infinite number of quasi-normal modes is demonstrated.Truncating the potential(23)to make it of compact support leads to the estimate(16).The decay of solutions in time is not exponential because of the weak decay of the potential for large r.At late times,the quasi-normal oscillations are swamped by the radiative tail[166,167].This tail radiation is of interest in its Living Reviews in Relativity(1999-2)。
unity 球谐函数系数
unity 球谐函数系数球谐函数系数（Spherical Harmonic Coefficients）是用于描述球面上的函数的一种数学工具。
球谐函数系数在计算机图形学、计算机视觉和计算机动画等领域中被广泛应用，用于描述和分析球面上的光照、形状、纹理等信息。
球谐函数是一组具有特定正交性质的函数，可以被用于展开球面上的任意函数。
球谐函数的定义涉及到勒让德多项式（Legendre polynomials）和三角函数的乘积，由此产生了球谐函数的一组特殊解。
球谐函数的形式如下：Y(l, m, θ, φ) = A(l, m) P(l, m)(cos(θ)) e^(imφ)其中，l和m分别表示球谐函数的次数和度数（degree），θ表示极角，φ表示方位角。
A(l, m)是球谐函数的系数，P(l, m)是勒让德多项式。
球谐函数系数通常用于展开球面上的光照信息。
光照可以被表示为一个函数，该函数取决于球面上的方向。
通过将光照函数展开为一组球谐函数的线性组合，可以得到一组球谐函数系数。
这些系数描述了光照在球面上各个方向的强度。
在计算机图形学中，球谐函数系数被用于实现全局照明和环境光遮蔽等效果。
除了光照，球谐函数系数还可以用于描述球面上的形状和纹理信息。
通过将球面上的形状函数或纹理函数展开为一组球谐函数的线性组合，可以得到一组球谐函数系数。
这些系数反映了球面上各个方向的形状或纹理特征。
在计算机视觉和计算机动画中，球谐函数系数被用于实现形状分析、图像压缩和纹理合成等应用。
计算球谐函数系数的过程通常涉及到对球谐函数的积分计算。
由于球谐函数是一组正交函数，可以通过勒让德多项式的正交性质进行系数的计算。
在计算中，常用的方法是通过预计算一组球谐函数在球面上的采样点的值，然后利用这些值进行系数的计算。
总结起来，球谐函数系数是用于描述球面上的函数的一种数学工具。
它可以用于展开球面上的光照、形状、纹理等信息。
球谐函数系数常用于计算机图形学、计算机视觉和计算机动画等领域。
利用球谐系数计算函数值及利用EGM球谐系数计算重力异常
利⽤球谐系数计算函数值及利⽤EGM球谐系数计算重⼒异常球谐分析（如重⼒场）是将地球表⾯观测的某个物理量f(theta,lambda)展开成球谐函数的级数：其中，theta为余纬，lambda：经度⼀般地，Pnm为完全归⼀化的缔合勒让德多项式，其与⽆归⼀化的缔合勒让德多项式的Pnm0的关系为：Pnm=(-1)^m*sqrt(k*(2N+1)(N-M)!/(N+M)!)Pnm0其中k=1,当 m=0k=2,当m>0在Matlab中，有现成的缔合勒让德多项式：-----------------------------------------------------------------------------------LEGENDRE Associated Legendre function.P = LEGENDRE(N,X) computes the associated Legendre functionsof degree N and order M = 0, 1, ..., N, evaluated for each elementof X. N must be a scalar integer and X must contain real valuesbetween -1 <= X <= 1.If X is a vector, P is an (N+1)-by-L matrix, where L = length(X).The P(M+1,i) entry corresponds to the associated Legendre functionof degree N and order M evaluated at X(i).There are three possible normalizations, LEGENDRE(N,X,normalize)where normalize is 'unnorm','sch' or 'norm'.The default, unnormalized associated Legendre functions are:P(N,M;X) = (-1)^M * (1-X^2)^(M/2) * (d/dX)^M { P(N,X) },where P(N,X) is the Legendre polynomial of degree N. Note thatthe first row of P is the Legendre polynomial evaluated at X(the M == 0 case).SP = LEGENDRE(N,X,'sch') computes the Schmidt semi-normalizedassociated Legendre functions SP(N,M;X). These functions arerelated to the unnormalized associated Legendre functionsP(N,M;X) by:SP(N,M;X) = P(N,X), M = 0= (-1)^M * sqrt(2*(N-M)!/(N+M)!) * P(N,M;X), M > 0因此，由sch正交化得到⼤地测量中的完全正交化，需要乘以renorm,其中renorm为sqrt(2*N+1)NP = LEGENDRE(N,X,'norm') computes the fully-normalizedassociated Legendre functions NP(N,M;X). These functions areassociated Legendre functions NP(N,M;X). These functions arenormalized such that/1|| [NP(N,M;X)]^2 dX = 1 ,|/-1and are related to the unnormalized associated Legendrefunctions P(N,M;X) by:NP(N,M;X) = (-1)^M * sqrt((N+1/2)*(N-M)!/(N+M)!) * P(N,M;X)因此，由norm正交化得到⼤地测量中的完全正交化，需要乘以renorm,其中renorm=sqrt(2)，当m=0renorm=2,当m>0下⾯的函数使⽤matlab中的缔合勒让德函数legendre来由球谐系数计算球谐函数-----------------------------------------------------------------------------------例⼦1：确定CMB地形(数据来⾃Morelli,1987,nature)下⾯%{ %}注释掉的分别为使⽤matlab legendre函数的sch正交化和norm正交化，然后进⾏modify的结果。
关于Spherical Harmonic
学习笔记--关于Spherical Harmonic xheartblue 2006-2-4关键字:Spherical Harmonic 球面调和函数 球面谐波函数 球形调和函数关联勒让德方程 勒让德多项式 正交多项式 正交函数系Spherical Harmonic 在图形学里,准确的说应该是高级光照技术里, 最近恐怕比较流行. 出于赶时髦的原因,我也在看,无奈数学基础太差了. 只好几乎把所有的时间全用在数学上了.从基本的数学分析看到泛函分析. 名词听的多了,渐渐也对Spherical harmonic 是什么玩意有了一点点了解. 了解而已, 不是理解..写出来, 整理整理思路而已. 望还不了解Spherical Harmonic 何物而来看本文的人自己能带着鄙视的眼光去理解. 否则如果本人理解错误而带坏了小孩概不负责 特此申明.Spherical harmonic 翻译成中文应该叫 球面调和函数, 是调和函数的一种. 所谓调和函数是一类函数(好像是废话). 满足Laplace 方程的的函数u 称为harmonic function. harmonic function 据说有一些很特殊的性质(偶还不了解是什么性质) , 于是有了harmonic analysis 这样的的数学分支.02=∇u 在极坐标系中把拉普拉斯方程表示成极坐标的形式,然后进行分离变量. (这个过程实在是太乱了,打个公式要半天, 具体见工程数学: 数学物理方程和特殊函数). 我们在求解这个方程的时候,会得到一个只和θ有关的方程0sin )1(cot 2222=Θ⎥⎦⎤⎢⎣⎡+++Θ+Θθθθθm n n d d d d 这个方程称为连带勒让德方程或者关联勒让德方程 – Associated Legendre Equation (恩,看来是个法国进口的方程.洋货啊, 看起来都复杂一些) . 给这个方程加一些条件,可以得到一些解. 其中有一些就叫勒让德多项式. 勒让德多项式是正交的.所谓正交的. 表示这个玩意满足 .)01(，其他都为是为j i P P j i j i ==•δ从实变函数和泛函分析的课程可以知道, 在L2空间中, 函数可以展开成关于一组完备的正交函数集, 典型的傅立叶级数就是个例子. 并且展开形式为i i B c f ∑= 其中ci 为系数,在傅立叶级数中,这个就是所谓的傅立叶系数, Bi 为正交函数中的一个.其中.也就是函数在Bi 这个基(向量空间中的基)上的投影, 也就是函数空间中的内积运算. 这样我们可以把任何一个函数都展开成级数.当然这个函数是要满足一定条件的.比如平方可积. ∫×=Bi f c i现在我们回到球面上来. 构造正交函数系ϕθπϕθim m l m l e P m l m l l Y )(cos )!()!(412),(+−+≡. 这玩意就是前面那个Laplace 方程一个解,也就是传说中的spherical harmonic. 那个P 呢,就是勒让德多项式了. 可以验证.他是一个正交的函数系. 而且是完备的. 用它.我们可以把球面上任何一个函数展开成以ϕθπϕθim m l m l e P m l m l l Y )(cos )!()!(412),(+−+≡为基的级数.. 当然,在球面上的遮挡关系也是可以这样的函数 , 球面上各个方向的辐射强度也是这个这样的函数, 同样次表面散射的能力也是个这样的函数, 理论上我们可以把这些函数用spherical harmonic 精确的还原出来. 而我们只需要记录那个Ci就可以了… 不过Ci是无穷多个. 出于人道主义,在一般的real-time rendering中,似乎Ci的个数是16 . 这也就是spherical harmonic 光照,不能模拟高频场景的一个原因了. 同时Ci的计算是比较复杂的. 这样很容易理解为什么DX9里的那几个demo的预处理跟乌龟一样.其实Ci的计算除了慢以外, 还很有技巧, 具体见<Advanced.Lighting.and.Materials.With.Shaders>的相关章节, 准确说是第8章.而关于spherical harmonic则见….. 偶还没有这样的书.不过<数学物理方程和特殊函数>中,有关于勒让德多项式的详细章节了.同样作为工程数学, 它还教你怎么用这个东西来近似球面上的一些函数,,, 比如电荷分布之类的. 类推到图形学上 …. 呵呵, 不说了. 实在没怎么看明白.参考书籍和资料/SphericalHarmonic.html关于spherical harmonic的.<工程数学: 数学物理方程和特殊函数> 高等教育出版社南京工学院编著关于Legendre Polynomial的.<函数论和泛函分析初步> 关于L2空间和L2空间上的傅立叶变换和傅立叶级数<Advanced Lighting and Materials With Shaders > 专门介绍高级光照的(又是废话,看书名就知道) 有人说这本书不好. 不过我觉得似乎这本书里讲SH是讲的最清楚的. 本着厚道的原则,顶一下.。
球谐分析
球谐分析，带谐，田谐，瓣谐球谐函数是拉普拉斯方程的球坐标系形式的解。
球谐函数表示为：球谐分析（如重力场）是将地球表面观测的某个物理量f(theta,lambda)展开成球谐函数的级数：其中，theta为余纬，lambda：经度如重力位可表示为：带谐系数：coefficient of zonal harmonics地球引力位的球谐函数展开式中次为零的位系数。
In themathematicalstudy ofrotational symmetry, the zonal spherical harmonics are specialspherical harmonicsthat are invariant under the rotation through a particular fixed axis. （故m=0，不随经度方向变化）扇谐系数：coefficient of sectorial harmonics地球引力位的球谐函数展开式中阶与次相同的位系数。
田谐：coefficient of tesseral harmonics地球引力位的球谐函数展开式中阶与次不同的位系数。
The Laplace spherical harmonics can be visualized by considering their "nodal lines", that is, the set of points on the sphere where.Nodal lines of are composed of circles: some are latitudes and others are longitudes.One can determine the number of nodal lines of each type by counting the number of zer os of in the latitudinal and longitudinal directions independently.For the latitudinal direction, the associated Legendre polynomials possess ℓ−|m| zeros, whereas for the longitudin al direction, the trigonometric sin and cos functions possess 2|m| zeros.When the spherical harmonic order m is zero(upper-left in the figure), the spherical harm onic functions do not depend upon longitude, and are referred to as zonal. Such spherical harmonics are a special case ofzonal spherical functions.When ℓ= |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral.For the other cases, the functionscheckerthe sphere, and they are referred to as tesseral. More general spherical harmonics of degree ℓare not necessarily those of the Laplace basis, and their nodal sets can be of a fairly general kind.[10]360阶（EGM96）分辨率为0.5分的来历：纬向180°、360=0.5°。
spherical harmonic
1 Introduction
Just as the Fourier basis represents an important tool for evaluation of convolutions in a one- or two dimensional space, the spherical harmonic basis is a similar tool but defined on the surface of a sphere. Spherical harmonics have already been used in the field of computer graphics, especially to model BRDF and incident radiance as well as BRDF inference [1, 2, 3, 8, 9]. But spherical harmonics have just recently become feasible to be used in real time computer graphics, especially in enhancing the dynamic lighting of scenes in real time as will be shown later in this work. The motivation of this pro-seminar is to demystify spherical harmonics in a similar way as Robin Green did [12] but with more mathematical background on the functions themselves and less focus on the actual applications, which in Green’s case was a technique called spherical harmonic lighting. Although the spherical harmonics are not the easiest mathematical functions this is an attempt at explaining and illustrating them as plausible as possible - without leaving out the critical mathematical relationships. Spherical harmonics are sometimes called the swiss army knife of mathematical physics and this metaphor is extensible to computer graphics to a certain degree, as the attentive reader will hopefully understand at the end of this work.
Generation Of Optical Harmonics
GENERATION OF OPTICAL HARMONICS*P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich The Harrison M. Randall Laboratory of Physics, The University of Michigan, Ann Arbor, Michigan(Received July 21, 1961)The development of pulsed ruby optical masers1,2 has made possible the production of monochromatic (6943 A) light beams which, when focussed,exhibit electric fields of the order of 105 volts/cm.The possibility of exploiting this extraordinary intensity for the production of optical harmonics from suitable nonlinear materials is most appealing.In this Letter we present a brief discussion of the requisite analysis and a description of experiments in which we have observed the second harmonic (at ~3472 A) produced upon projection of an intense beam of 6943A light through crystalline quartz.A suitable material for the production of optical harmonics must have a nonlinear dielectric coefficient and be transparent to both the fundamental optical frequency and the desired overtones. Since all dielectrics are nonlinear in high enough fields,this suggests the feasibility of utilizing materials such as quartz and glass. The dependence of polarization of a dielectric upon electric field E may be expressed schematically bywhere E1, E2…are of the order of magnitude of atomic electric fields(~108 esu). If E is sinusoidal in time, the presence in Eq. (1) of terms ofquadratic or higher degree will result in P containing harmonics of the fundamental frequency.Direct-current polarizations should accompany the even harmonics.Let P be that part of P which is quadratic in E;That is, P is a linear function of the components of the symmetric tensor EE. The eighteen coefficients which occur in this function are subject to restrictions due to the point symmetry of the medium. These restrictions are, in fact, identical with those governing the piezoelectric coefficients. In particular,P necessarily vanishes in a material such as glass which is isotropic or contains a center of inversion. For crystalline quartz, however, there are two independent to Efficients αand βin terms of which(z is the threefold, or optic, axis; x a twofold axis). If a light beam traverses quartz in one of the three principal directions, Eqs. (2) predict the results summarized in Table I. The second-harmonic light should be absent in the first case, dependent upon incident polarization in the second case, and independent of this polarization in the third. If an intense beam of monochromatic light is focussed into a region of volume V, there should occur an intensity I of second harmonic given (in Gaussian units) bywhere ωis the angular frequency of the second harmonic, c the velocity of light, and υan effective "volume of coherence"; that is, the size of a region within the sample in which there is phase coherence of the p excitation. (This volume may in practice be much smaller thanV .) An estimate of v is governed by several considerations. Forexample, it is probably of no greater extent in the propagation direction than ~ 12212[()]n n n λ-⨯-，where 1n and 2n are the indices of refraction forthe fundamental and second harmonic frequencies, respectively, and 2λ is the wavelength of the second harmonic. The lateral extent of thisvolume is determined in large part by the coherence characteristics of the optical maser. The situation for a maser of the gas discharge 3 type is clearly more favorable in this respect than that for the ruby device.l,2 For a coherence volume of 10-11 cm 3, which we think may be realistic in our case, Eq. (1) indicates that second harmonic intensities as high as a fraction of a percent of the fundamental could be achieved.In the experiments we have used a commercially available rubyoptical maser 4 which produces approximately 3 joules of 6943A light in a one millisecond pulse. This light is passed through a red filter for the elimination of the xenon flash background and is then brought to a focus inside a crystalline quartz sample. The emergent beam is analyzed by a quartz prism spectrometer equipped with red insensitive Eastman Type 103 spectrographic plates. A reproduction of the first plate in which there was an unambiguous indication of second harmonic (3472 A) is shown in Fig. 1.[ FIG.1. A direct reproduction of the first plate in which there was an indication of second harmonic. The wavelength scale is in units of 100 A. The arrow at 3472 A indicates the small but dense image produced by the second harmonic. The image of the primary beam at 6943 A is very large due to halation.]This plate was exposed to only one "shot" from the optical maser. We believe the following two facts, among others, rule out the possibility of artifact:(1) The light at 3472 A disappears when the quartz is removed or is replaced by glass.(2) The light at 3472 A exhibits the expected dependence on polarization and orientation summarized in Table I.Considerations of the photographic image density and the efficiency of the optical system lead us to believe that the order of 1011 second harmonic photons were generated within the quartz sample per pulse.The production of a second harmonic should be observable in isotropic materials such as glass if a strong bias field were applied to the sample. This bias could be oscillatory, thus producing sidebands on the fundamental frequency and the harmonics.We would like to thank the staff of Trion Instruments, Inc., for their valuable and sustained cooperation in this work.*This work was supported in part by the U. S. Atomic Energy Commission.1T. H. Maiman, Nature 187, 493 (1960).2R. J. Collins et al.，Phys. Rev. Letters 5，303 (1960).3A. Javan, W. R. Bennet, and D. R. Herriott, Phys. Rev. Letters 6，106 (1961). Even though the intensity of the gas device is very low compared with ruby masers, the gain in coherence volume and the potential improvement of focussing suggest that the gas maser may be comparable or even superior as a source for optical harmonics.4Trion Instruments, Inc.，Model No. TO-3000.。
角谱法计算声场代码
角谱法计算声场代码
角谱法（Spherical Harmonic Analysis，SHA）是一种用于计
算声场的方法，它基于球谐函数展开声场，使得声场的空间分布可
以用一组球谐函数系数来描述。
这种方法在声学领域中被广泛应用，特别是在声场分析和声学信号处理中。
要使用角谱法计算声场，首先需要将声场信号进行球谐函数分解。
这可以通过在空间中的不同方向上进行声压或声速的测量，并
将这些测量结果转换成球谐函数系数。
然后，可以根据具体的声场
问题，选择适当的球谐函数展开阶数，以保证得到足够精确的声场
描述。
在实际计算中，可以使用各种数学软件或编程语言来实现角谱
法计算声场的代码。
例如，MATLAB、Python的SciPy库、C++等都
可以用于实现角谱法计算声场的代码。
在编写代码时，需要考虑如
何进行球谐函数系数的计算、如何选择合适的展开阶数、如何进行
声场重构等问题。
另外，在实际应用中，还需要考虑声场的边界条件、声源分布、介质特性等因素，这些都会影响到角谱法计算声场的代码实现。
因
此，在编写代码时需要综合考虑这些因素，以确保计算结果的准确性和可靠性。
总之，角谱法是一种强大的声场计算方法，通过适当的代码实现，可以对复杂的声场问题进行准确的分析和计算。
希望这个回答能够帮助到你理解角谱法计算声场的代码实现。
A survey of content based 3d shape retrieval methods
A Survey of Content Based3D Shape Retrieval MethodsJohan W.H.Tangelder and Remco C.VeltkampInstitute of Information and Computing Sciences,Utrecht University hanst@cs.uu.nl,Remco.Veltkamp@cs.uu.nlAbstractRecent developments in techniques for modeling,digitiz-ing and visualizing3D shapes has led to an explosion in the number of available3D models on the Internet and in domain-specific databases.This has led to the development of3D shape retrieval systems that,given a query object, retrieve similar3D objects.For visualization,3D shapes are often represented as a surface,in particular polygo-nal meshes,for example in VRML format.Often these mod-els contain holes,intersecting polygons,are not manifold, and do not enclose a volume unambiguously.On the con-trary,3D volume models,such as solid models produced by CAD systems,or voxels models,enclose a volume prop-erly.This paper surveys the literature on methods for con-tent based3D retrieval,taking into account the applicabil-ity to surface models as well as to volume models.The meth-ods are evaluated with respect to several requirements of content based3D shape retrieval,such as:(1)shape repre-sentation requirements,(2)properties of dissimilarity mea-sures,(3)efficiency,(4)discrimination abilities,(5)ability to perform partial matching,(6)robustness,and(7)neces-sity of pose normalization.Finally,the advantages and lim-its of the several approaches in content based3D shape re-trieval are discussed.1.IntroductionThe advancement of modeling,digitizing and visualizing techniques for3D shapes has led to an increasing amount of3D models,both on the Internet and in domain-specific databases.This has led to the development of thefirst exper-imental search engines for3D shapes,such as the3D model search engine at Princeton university[2,57],the3D model retrieval system at the National Taiwan University[1,17], the Ogden IV system at the National Institute of Multimedia Education,Japan[62,77],the3D retrieval engine at Utrecht University[4,78],and the3D model similarity search en-gine at the University of Konstanz[3,84].Laser scanning has been applied to obtain archives recording cultural heritage like the Digital Michelan-gelo Project[25,48],and the Stanford Digital Formae Urbis Romae Project[75].Furthermore,archives contain-ing domain-specific shape models are now accessible by the Internet.Examples are the National Design Repos-itory,an online repository of CAD models[59,68], and the Protein Data Bank,an online archive of struc-tural data of biological macromolecules[10,80].Unlike text documents,3D models are not easily re-trieved.Attempting tofind a3D model using textual an-notation and a conventional text-based search engine would not work in many cases.The annotations added by human beings depend on language,culture,age,sex,and other fac-tors.They may be too limited or ambiguous.In contrast, content based3D shape retrieval methods,that use shape properties of the3D models to search for similar models, work better than text based methods[58].Matching is the process of determining how similar two shapes are.This is often done by computing a distance.A complementary process is indexing.In this paper,indexing is understood as the process of building a datastructure to speed up the search.Note that the term indexing is also of-ten used for the identification of features in models,or mul-timedia documents in general.Retrieval is the process of searching and delivering the query results.Matching and in-dexing are often part of the retrieval process.Recently,a lot of researchers have investigated the spe-cific problem of content based3D shape retrieval.Also,an extensive amount of literature can be found in the related fields of computer vision,object recognition and geomet-ric modelling.Survey papers to this literature have been provided by Besl and Jain[11],Loncaric[50]and Camp-bell and Flynn[16].For an overview of2D shape match-ing methods we refer the reader to the paper by Veltkamp [82].Unfortunately,most2D methods do not generalize di-rectly to3D model matching.Work in progress by Iyer et al.[40]provides an extensive overview of3D shape search-ing techniques.Atmosukarto and Naval[6]describe a num-ber of3D model retrieval systems and methods,but do not provide a categorization and evaluation.In contrast,this paper evaluates3D shape retrieval meth-ods with respect to several requirements on content based 3D shape retrieval,such as:(1)shape representation re-quirements,(2)properties of dissimilarity measures,(3)ef-ficiency,(4)discrimination abilities,(5)ability to perform partial matching,(6)robustness,and(7)necessity of posenormalization.In section2we discuss several aspects of3D shape retrieval.The literature on3D shape matching meth-ods is discussed in section3and evaluated in section4. 2.3D shape retrieval aspectsIn this section we discuss several issues related to3D shape retrieval.2.1.3D shape retrieval frameworkAt a conceptual level,a typical3D shape retrieval frame-work as illustrated byfig.1consists of a database with an index structure created offline and an online query engine. Each3D model has to be identified with a shape descrip-tor,providing a compact overall description of the shape. To efficiently search a large collection online,an indexing data structure and searching algorithm should be available. The online query engine computes the query descriptor,and models similar to the query model are retrieved by match-ing descriptors to the query descriptor from the index struc-ture of the database.The similarity between two descriptors is quantified by a dissimilarity measure.Three approaches can be distinguished to provide a query object:(1)browsing to select a new query object from the obtained results,(2) a direct query by providing a query descriptor,(3)query by example by providing an existing3D model or by creating a3D shape query from scratch using a3D tool or sketch-ing2D projections of the3D model.Finally,the retrieved models can be visualized.2.2.Shape representationsAn important issue is the type of shape representation(s) that a shape retrieval system accepts.Most of the3D models found on the World Wide Web are meshes defined in afile format supporting visual appearance.Currently,the most common format used for this purpose is the Virtual Real-ity Modeling Language(VRML)format.Since these mod-els have been designed for visualization,they often contain only geometry and appearance attributes.In particular,they are represented by“polygon soups”,consisting of unorga-nized sets of polygons.Also,in general these models are not“watertight”meshes,i.e.they do not enclose a volume. By contrast,for volume models retrieval methods depend-ing on a properly defined volume can be applied.2.3.Measuring similarityIn order to measure how similar two objects are,it is nec-essary to compute distances between pairs of descriptors us-ing a dissimilarity measure.Although the term similarity is often used,dissimilarity corresponds to the notion of dis-tance:small distances means small dissimilarity,and large similarity.A dissimilarity measure can be formalized by a func-tion defined on pairs of descriptors indicating the degree of their resemblance.Formally speaking,a dissimilarity measure d on a set S is a non-negative valued function d:S×S→R+∪{0}.Function d may have some of the following properties:i.Identity:For all x∈S,d(x,x)=0.ii.Positivity:For all x=y in S,d(x,y)>0.iii.Symmetry:For all x,y∈S,d(x,y)=d(y,x).iv.Triangle inequality:For all x,y,z∈S,d(x,z)≤d(x,y)+d(y,z).v.Transformation invariance:For a chosen transforma-tion group G,for all x,y∈S,g∈G,d(g(x),g(y))= d(x,y).The identity property says that a shape is completely similar to itself,while the positivity property claims that dif-ferent shapes are never completely similar.This property is very strong for a high-level shape descriptor,and is often not satisfied.However,this is not a severe drawback,if the loss of uniqueness depends on negligible details.Symmetry is not always wanted.Indeed,human percep-tion does not alwaysfind that shape x is equally similar to shape y,as y is to x.In particular,a variant x of prototype y,is often found more similar to y then vice versa[81].Dissimilarity measures for partial matching,giving a small distance d(x,y)if a part of x matches a part of y, do not obey the triangle inequality.Transformation invariance has to be satisfied,if the com-parison and the extraction process of shape descriptors have to be independent of the place,orientation and scale of the object in its Cartesian coordinate system.If we want that a dissimilarity measure is not affected by any transforma-tion on x,then we may use as alternative formulation for (v):Transformation invariance:For a chosen transforma-tion group G,for all x,y∈S,g∈G,d(g(x),y)=d(x,y).When all the properties(i)-(iv)hold,the dissimilarity measure is called a metric.Other combinations are possi-ble:a pseudo-metric is a dissimilarity measure that obeys (i),(iii)and(iv)while a semi-metric obeys only(i),(ii)and(iii).If a dissimilarity measure is a pseudo-metric,the tri-angle inequality can be applied to make retrieval more effi-cient[7,83].2.4.EfficiencyFor large shape collections,it is inefficient to sequen-tially match all objects in the database with the query object. Because retrieval should be fast,efficient indexing search structures are needed to support efficient retrieval.Since for query by example the shape descriptor is computed online, it is reasonable to require that the shape descriptor compu-tation is fast enough for interactive querying.2.5.Discriminative powerA shape descriptor should capture properties that dis-criminate objects well.However,the judgement of the sim-ilarity of the shapes of two3D objects is somewhat sub-jective,depending on the user preference or the application at hand.E.g.for solid modeling applications often topol-ogy properties such as the numbers of holes in a model are more important than minor differences in shapes.On the contrary,if a user searches for models looking visually sim-ilar the existence of a small hole in the model,may be of no importance to the user.2.6.Partial matchingIn contrast to global shape matching,partial matching finds a shape of which a part is similar to a part of another shape.Partial matching can be applied if3D shape mod-els are not complete,e.g.for objects obtained by laser scan-ning from one or two directions only.Another application is the search for“3D scenes”containing an instance of the query object.Also,this feature can potentially give the user flexibility towards the matching problem,if parts of inter-est of an object can be selected or weighted by the user. 2.7.RobustnessIt is often desirable that a shape descriptor is insensitive to noise and small extra features,and robust against arbi-trary topological degeneracies,e.g.if it is obtained by laser scanning.Also,if a model is given in multiple levels-of-detail,representations of different levels should not differ significantly from the original model.2.8.Pose normalizationIn the absence of prior knowledge,3D models have ar-bitrary scale,orientation and position in the3D space.Be-cause not all dissimilarity measures are invariant under ro-tation and translation,it may be necessary to place the3D models into a canonical coordinate system.This should be the same for a translated,rotated or scaled copy of the model.A natural choice is tofirst translate the center to the ori-gin.For volume models it is natural to translate the cen-ter of mass to the origin.But for meshes this is in gen-eral not possible,because they have not to enclose a vol-ume.For meshes it is an alternative to translate the cen-ter of mass of all the faces to the origin.For example the Principal Component Analysis(PCA)method computes for each model the principal axes of inertia e1,e2and e3 and their eigenvaluesλ1,λ2andλ3,and make the nec-essary conditions to get right-handed coordinate systems. These principal axes define an orthogonal coordinate sys-tem(e1,e2,e3),withλ1≥λ2≥λ3.Next,the polyhe-dral model is rotated around the origin such that the co-ordinate system(e x,e y,e z)coincides with the coordinatesystem(e1,e2,e3).The PCA algorithm for pose estimation is fairly simple and efficient.However,if the eigenvalues are equal,prin-cipal axes may switch,without affecting the eigenvalues. Similar eigenvalues may imply an almost symmetrical mass distribution around an axis(e.g.nearly cylindrical shapes) or around the center of mass(e.g.nearly spherical shapes). Fig.2illustrates the problem.3.Shape matching methodsIn this section we discuss3D shape matching methods. We divide shape matching methods in three broad cate-gories:(1)feature based methods,(2)graph based meth-ods and(3)other methods.Fig.3illustrates a more detailed categorization of shape matching methods.Note,that the classes of these methods are not completely disjoined.For instance,a graph-based shape descriptor,in some way,de-scribes also the global feature distribution.By this point of view the taxonomy should be a graph.3.1.Feature based methodsIn the context of3D shape matching,features denote ge-ometric and topological properties of3D shapes.So3D shapes can be discriminated by measuring and comparing their features.Feature based methods can be divided into four categories according to the type of shape features used: (1)global features,(2)global feature distributions,(3)spa-tial maps,and(4)local features.Feature based methods from thefirst three categories represent features of a shape using a single descriptor consisting of a d-dimensional vec-tor of values,where the dimension d isfixed for all shapes.The value of d can easily be a few hundred.The descriptor of a shape is a point in a high dimensional space,and two shapes are considered to be similar if they are close in this space.Retrieving the k best matches for a3D query model is equivalent to solving the k nearest neighbors -ing the Euclidean distance,matching feature descriptors can be done efficiently in practice by searching in multiple1D spaces to solve the approximate k nearest neighbor prob-lem as shown by Indyk and Motwani[36].In contrast with the feature based methods from thefirst three categories,lo-cal feature based methods describe for a number of surface points the3D shape around the point.For this purpose,for each surface point a descriptor is used instead of a single de-scriptor.3.1.1.Global feature based similarityGlobal features characterize the global shape of a3D model. Examples of these features are the statistical moments of the boundary or the volume of the model,volume-to-surface ra-tio,or the Fourier transform of the volume or the boundary of the shape.Zhang and Chen[88]describe methods to com-pute global features such as volume,area,statistical mo-ments,and Fourier transform coefficients efficiently.Paquet et al.[67]apply bounding boxes,cords-based, moments-based and wavelets-based descriptors for3D shape matching.Corney et al.[21]introduce convex-hull based indices like hull crumpliness(the ratio of the object surface area and the surface area of its convex hull),hull packing(the percentage of the convex hull volume not occupied by the object),and hull compactness(the ratio of the cubed sur-face area of the hull and the squared volume of the convex hull).Kazhdan et al.[42]describe a reflective symmetry de-scriptor as a2D function associating a measure of reflec-tive symmetry to every plane(specified by2parameters) through the model’s centroid.Every function value provides a measure of global shape,where peaks correspond to the planes near reflective symmetry,and valleys correspond to the planes of near anti-symmetry.Their experimental results show that the combination of the reflective symmetry de-scriptor with existing methods provides better results.Since only global features are used to characterize the overall shape of the objects,these methods are not very dis-criminative about object details,but their implementation is straightforward.Therefore,these methods can be used as an activefilter,after which more detailed comparisons can be made,or they can be used in combination with other meth-ods to improve results.Global feature methods are able to support user feed-back as illustrated by the following research.Zhang and Chen[89]applied features such as volume-surface ratio, moment invariants and Fourier transform coefficients for 3D shape retrieval.They improve the retrieval performance by an active learning phase in which a human annotator as-signs attributes such as airplane,car,body,and so on to a number of sample models.Elad et al.[28]use a moments-based classifier and a weighted Euclidean distance measure. Their method supports iterative and interactive database searching where the user can improve the weights of the distance measure by marking relevant search results.3.1.2.Global feature distribution based similarityThe concept of global feature based similarity has been re-fined recently by comparing distributions of global features instead of the global features directly.Osada et al.[66]introduce and compare shape distribu-tions,which measure properties based on distance,angle, area and volume measurements between random surface points.They evaluate the similarity between the objects us-ing a pseudo-metric that measures distances between distri-butions.In their experiments the D2shape distribution mea-suring distances between random surface points is most ef-fective.Ohbuchi et al.[64]investigate shape histograms that are discretely parameterized along the principal axes of inertia of the model.The shape descriptor consists of three shape histograms:(1)the moment of inertia about the axis,(2) the average distance from the surface to the axis,and(3) the variance of the distance from the surface to the axis. Their experiments show that the axis-parameterized shape features work only well for shapes having some form of ro-tational symmetry.Ip et al.[37]investigate the application of shape distri-butions in the context of CAD and solid modeling.They re-fined Osada’s D2shape distribution function by classifying2random points as1)IN distances if the line segment con-necting the points lies complete inside the model,2)OUT distances if the line segment connecting the points lies com-plete outside the model,3)MIXED distances if the line seg-ment connecting the points lies passes both inside and out-side the model.Their dissimilarity measure is a weighted distance measure comparing D2,IN,OUT and MIXED dis-tributions.Since their method requires that a line segment can be classified as lying inside or outside the model it is required that the model defines a volume properly.There-fore it can be applied to volume models,but not to polyg-onal soups.Recently,Ip et al.[38]extend this approach with a technique to automatically categorize a large model database,given a categorization on a number of training ex-amples from the database.Ohbuchi et al.[63],investigate another extension of the D2shape distribution function,called the Absolute Angle-Distance histogram,parameterized by a parameter denot-ing the distance between two random points and by a pa-rameter denoting the angle between the surfaces on which two random points are located.The latter parameter is ac-tually computed as an inner product of the surface normal vectors.In their evaluation experiment this shape distribu-tion function outperformed the D2distribution function at about1.5times higher computational costs.Ohbuchi et al.[65]improved this method further by a multi-resolution ap-proach computing a number of alpha-shapes at different scales,and computing for each alpha-shape their Absolute Angle-Distance descriptor.Their experimental results show that this approach outperforms the Angle-Distance descrip-tor at the cost of high processing time needed to compute the alpha-shapes.Shape distributions distinguish models in broad cate-gories very well:aircraft,boats,people,animals,etc.How-ever,they perform often poorly when having to discrimi-nate between shapes that have similar gross shape proper-ties but vastly different detailed shape properties.3.1.3.Spatial map based similaritySpatial maps are representations that capture the spatial lo-cation of an object.The map entries correspond to physi-cal locations or sections of the object,and are arranged in a manner that preserves the relative positions of the features in an object.Spatial maps are in general not invariant to ro-tations,except for specially designed maps.Therefore,typ-ically a pose normalization is donefirst.Ankerst et al.[5]use shape histograms as a means of an-alyzing the similarity of3D molecular surfaces.The his-tograms are not built from volume elements but from uni-formly distributed surface points taken from the molecular surfaces.The shape histograms are defined on concentric shells and sectors around a model’s centroid and compare shapes using a quadratic form distance measure to compare the histograms taking into account the distances between the shape histogram bins.Vrani´c et al.[85]describe a surface by associating to each ray from the origin,the value equal to the distance to the last point of intersection of the model with the ray and compute spherical harmonics for this spherical extent func-tion.Spherical harmonics form a Fourier basis on a sphere much like the familiar sine and cosine do on a line or a cir-cle.Their method requires pose normalization to provide rotational invariance.Also,Yu et al.[86]propose a descrip-tor similar to a spherical extent function and a descriptor counting the number of intersections of a ray from the ori-gin with the model.In both cases the dissimilarity between two shapes is computed by the Euclidean distance of the Fourier transforms of the descriptors of the shapes.Their method requires pose normalization to provide rotational in-variance.Kazhdan et al.[43]present a general approach based on spherical harmonics to transform rotation dependent shape descriptors into rotation independent ones.Their method is applicable to a shape descriptor which is defined as either a collection of spherical functions or as a function on a voxel grid.In the latter case a collection of spherical functions is obtained from the function on the voxel grid by restricting the grid to concentric spheres.From the collection of spher-ical functions they compute a rotation invariant descriptor by(1)decomposing the function into its spherical harmon-ics,(2)summing the harmonics within each frequency,and computing the L2-norm for each frequency component.The resulting shape descriptor is a2D histogram indexed by ra-dius and frequency,which is invariant to rotations about the center of the mass.This approach offers an alternative for pose normalization,because their method obtains rotation invariant shape descriptors.Their experimental results show indeed that in general the performance of the obtained ro-tation independent shape descriptors is better than the cor-responding normalized descriptors.Their experiments in-clude the ray-based spherical harmonic descriptor proposed by Vrani´c et al.[85].Finally,note that their approach gen-eralizes the method to compute voxel-based spherical har-monics shape descriptor,described by Funkhouser et al.[30],which is defined as a binary function on the voxel grid, where the value at each voxel is given by the negatively ex-ponentiated Euclidean Distance Transform of the surface of a3D model.Novotni and Klein[61]present a method to compute 3D Zernike descriptors from voxelized models as natural extensions of spherical harmonics based descriptors.3D Zernike descriptors capture object coherence in the radial direction as well as in the direction along a sphere.Both 3D Zernike descriptors and spherical harmonics based de-scriptors achieve rotation invariance.However,by sampling the space only in radial direction the latter descriptors donot capture object coherence in the radial direction,as illus-trated byfig.4.The limited experiments comparing spherical harmonics and3D Zernike moments performed by Novotni and Klein show similar results for a class of planes,but better results for the3D Zernike descriptor for a class of chairs.Vrani´c[84]expects that voxelization is not a good idea, because manyfine details are lost in the voxel grid.There-fore,he compares his ray-based spherical harmonic method [85]and a variation of it using functions defined on concen-tric shells with the voxel-based spherical harmonics shape descriptor proposed by Funkhouser et al.[30].Also,Vrani´c et al.[85]accomplish pose normalization using the so-called continuous PCA algorithm.In the paper it is claimed that the continuous PCA is better as the conventional PCA and better as the weighted PCA,which takes into account the differing sizes of the triangles of a mesh.In contrast with Kazhdan’s experiments[43]the experiments by Vrani´c show that for ray-based spherical harmonics using the con-tinuous PCA without voxelization is better than using rota-tion invariant shape descriptors obtained using voxelization. Perhaps,these results are opposite to Kazhdan results,be-cause of the use of different methods to compute the PCA or the use of different databases or both.Kriegel et al.[46,47]investigate similarity for voxelized models.They obtain a spatial map by partitioning a voxel grid into disjoint cells which correspond to the histograms bins.They investigate three different spatial features asso-ciated with the grid cells:(1)volume features recording the fraction of voxels from the volume in each cell,(2) solid-angle features measuring the convexity of the volume boundary in each cell,(3)eigenvalue features estimating the eigenvalues obtained by the PCA applied to the voxels of the model in each cell[47],and a fourth method,using in-stead of grid cells,a moreflexible partition of the voxels by cover sequence features,which approximate the model by unions and differences of cuboids,each containing a number of voxels[46].Their experimental results show that the eigenvalue method and the cover sequence method out-perform the volume and solid-angle feature method.Their method requires pose normalization to provide rotational in-variance.Instead of representing a cover sequence with a single feature vector,Kriegel et al.[46]represent a cover sequence by a set of feature vectors.This approach allows an efficient comparison of two cover sequences,by compar-ing the two sets of feature vectors using a minimal match-ing distance.The spatial map based approaches show good retrieval results.But a drawback of these methods is that partial matching is not supported,because they do not encode the relation between the features and parts of an object.Fur-ther,these methods provide no feedback to the user about why shapes match.3.1.4.Local feature based similarityLocal feature based methods provide various approaches to take into account the surface shape in the neighbourhood of points on the boundary of the shape.Shum et al.[74]use a spherical coordinate system to map the surface curvature of3D objects to the unit sphere. By searching over a spherical rotation space a distance be-tween two curvature distributions is computed and used as a measure for the similarity of two objects.Unfortunately, the method is limited to objects which contain no holes, i.e.have genus zero.Zaharia and Prˆe teux[87]describe the 3D Shape Spectrum Descriptor,which is defined as the histogram of shape index values,calculated over an en-tire mesh.The shape index,first introduced by Koenderink [44],is defined as a function of the two principal curvatures on continuous surfaces.They present a method to compute these shape indices for meshes,byfitting a quadric surface through the centroids of the faces of a mesh.Unfortunately, their method requires a non-trivial preprocessing phase for meshes that are not topologically correct or not orientable.Chua and Jarvis[18]compute point signatures that accu-mulate surface information along a3D curve in the neigh-bourhood of a point.Johnson and Herbert[41]apply spin images that are2D histograms of the surface locations around a point.They apply spin images to recognize models in a cluttered3D scene.Due to the complexity of their rep-resentation[18,41]these methods are very difficult to ap-ply to3D shape matching.Also,it is not clear how to define a dissimilarity function that satisfies the triangle inequality.K¨o rtgen et al.[45]apply3D shape contexts for3D shape retrieval and matching.3D shape contexts are semi-local descriptions of object shape centered at points on the sur-face of the object,and are a natural extension of2D shape contexts introduced by Belongie et al.[9]for recognition in2D images.The shape context of a point p,is defined as a coarse histogram of the relative coordinates of the re-maining surface points.The bins of the histogram are de-。
附录B：Mathematica的基本应用
附录B ：Mathematica 的基本应用1. 什么是MathematicaMathematica 是美国Wolfram Research 公司开发的通用科学计算软件，主要用途是科学研究与工程技术中的计算,这里介绍的是第6版（2009年更新为第7版）。
由于它的功能十分强大，使用非常简便，现在已成为大学师生进行教学和科研的有力工具。
它的主要特点有：1)既可以进行程序运行，又可以进行交互式运行。
一句简单的Mathematic 命令常常可以完成普通的c 语言几十甚至几百个语句的工作。
例如解方程：x 4 + x 3 + 3x -5 = 0只要运行下面的命令：Solve[x^4+x^3+3 x-5 0,x] 。
2) 既可以进行任意高精度的数值计算，又可以进行各种复杂的符号演算，如函数的微分、积分、幂级数展开、矩阵求逆等等。
它使许多以前只能靠纸和笔解决的推理工作可以用计算机处理。
例如求不定积分：⎰ x 4 e -2x dx 只要运行下面的命令：Integrate[x^4*Exp[2 x],x]。
3) 既可以进行抽象计算，又可以用图形、动画和声音等形式来具体表现，使人能够直观地把握住研究对象的特性。
例如绘制函数图形：y = e -x /2 cos x , x ∈ [0, π]，只要运行下面的命令： Plot[Exp[x/2]*Cos[x],{x,0,Pi}]。
4) Mathematica 把各种功能有机地结合在一个集成环境里，可以根据需要做不同的操作，给使用者带来极大的方便。
2. Mathematica 的基本功能2.1 基本运算及其对象Mathematica 的基本数值运算有加法、减法、乘法、除法和乘(开)方，分别用运算符“+”、“-”、“*”、“/”和“^”来表示（在不引起误解的情况下，乘号可以省略或用空格代替），例如2.4*3^2 -(5/(6+3))^(1/3)表示3236534.2）（+÷-⨯。
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学习笔记--关于Spherical Harmonic xheartblue 2006-2-4
关键字:
Spherical Harmonic 球面调和函数 球面谐波函数 球形调和函数
关联勒让德方程 勒让德多项式 正交多项式 正交函数系
Spherical Harmonic 在图形学里,准确的说应该是高级光照技术里, 最近恐怕比较流行. 出于赶时髦的原因,我也在看,无奈数学基础太差了. 只好几乎把所有的时间全用在数学上了.从基本的数学分析看到泛函分析. 名词听的多了,渐渐也对Spherical harmonic 是什么玩意有了一点点了解. 了解而已, 不是理解..写出来, 整理整理思路而已. 望还不了解Spherical Harmonic 何物而来看本文的人自己能带着鄙视的眼光去理解. 否则如果本人理解错误而带坏了小孩概不负责 特此申明.
Spherical harmonic 翻译成中文应该叫 球面调和函数, 是调和函数的一种. 所谓调和函数是一类函数(好像是废话). 满足Laplace 方程的的函数u 称为harmonic function. harmonic function 据说有一些很特殊的性质(偶还不了解是什么性质) , 于是有了harmonic analysis 这样的的数学分支.
02=∇u 在极坐标系中把拉普拉斯方程表示成极坐标的形式,然后进行分离变量. (这个过程实在是太乱了,打个公式要半天, 具体见工程数学: 数学物理方程和特殊函数). 我们在求解这个方程
的时候,会得到一个只和θ有关的方程0sin )1(cot 2222=Θ⎥⎦
⎤⎢⎣⎡+++Θ+Θθθθθm n n d d d d 这个方程称为连带勒让德方程或者关联勒让德方程 – Associated Legendre Equation (恩,看来是个法国进口的方程.洋货啊, 看起来都复杂一些) . 给这个方程加一些条件,可以得到一些解. 其中有一些就叫勒让德多项式. 勒让德多项式是正交的.所谓正交的. 表示这个玩意满足 .
)01(，其他都为是为j i P P j i j i ==•δ从实变函数和泛函分析的课程可以知道, 在L2空间中, 函数可以展开成关于一组完备的正交函数集, 典型的傅立叶级数就是个例子. 并且展开形式为i i B c f ∑= 其中ci 为系数,在傅立叶级数中,这个就是所谓的傅立叶系数, Bi 为正交函数中的一个.其中.也就是函数在Bi 这个基(向量空间中的基)上的投影, 也就是函数空间中的内积运算. 这样我们可以把任何一个函数都展开成级数.当然这个函数是要满足一定条件的.比如平方可积. ∫×=Bi f c i
现在我们回到球面上来. 构造正交函数系ϕθπϕθim m l m l e P m l m l l Y )(cos )!
()!(412),(+−+≡. 这玩意就是前面那个Laplace 方程一个解,也就是传说中的spherical harmonic. 那个P 呢,就是勒让德多项式了. 可以验证.他是一个正交的函数系. 而且是完备的. 用它.我们可以把球面上任何一个函数展开成以ϕθπϕθim m l m l e P m l m l l Y )(cos )!
()!(412),(+−+≡为基的级数.. 当然,
在球面上的遮挡关系也是可以这样的函数 , 球面上各个方向的辐射强度也是这个这样的函数, 同样次表面散射的能力也是个这样的函数, 理论上我们可以把这些函数用spherical harmonic 精确的还原出来. 而我们只需要记录那个Ci就可以了… 不过Ci是无穷多个. 出于人道主义,在一般的real-time rendering中,似乎Ci的个数是16 . 这也就是spherical harmonic 光照,不能模拟高频场景的一个原因了. 同时Ci的计算是比较复杂的. 这样很容易理解为什么DX9里的那几个demo的预处理跟乌龟一样.
其实Ci的计算除了慢以外, 还很有技巧, 具体见<Advanced.Lighting.and.Materials.With.Shaders>的相关章节, 准确说是第8章.
而关于spherical harmonic则见….. 偶还没有这样的书.不过<数学物理方程和特殊函数>中,有关于勒让德多项式的详细章节了.同样作为工程数学, 它还教你怎么用这个东西来近似球面上的一些函数,,, 比如电荷分布之类的. 类推到图形学上 …. 呵呵, 不说了. 实在没怎么看明白.
参考书籍和资料
/SphericalHarmonic.html关于spherical harmonic的.
<工程数学: 数学物理方程和特殊函数> 高等教育出版社南京工学院编著关于Legendre Polynomial的.
<函数论和泛函分析初步> 关于L2空间和L2空间上的傅立叶变换和傅立叶级数
<Advanced Lighting and Materials With Shaders > 专门介绍高级光照的(又是废话,看书名就知道) 有人说这本书不好. 不过我觉得似乎这本书里讲SH是讲的最清楚的. 本着厚道的原则,顶一下.。