圆锥曲线外文翻译
常用研磨机外文文献翻译、中英文翻译、外文翻译
常用研磨机外文文献翻译、中英文翻译、外文翻译Grinding machine is a crucial n processing method that offers high machining accuracy and can process a wide range of materials。
It is suitable for almost all kinds of material processing。
and can achieve very high n and shape accuracy。
even reaching the limit。
The machining accuracy of grinding device is simple and does not require complex ___.2.Types of Grinding MachinesGrinding machines are mainly used for n grinding of workpiece planes。
cylindrical workpiece surfaces (both inside and outside)。
tapered faces inside。
spheres。
thread faces。
and other types of ___ grinding machines。
including disc-type grinding machines。
shaft-type grinding machines。
ic grinding machines。
and special grinding machines.3.Disc-type Grinding MachineThe disc-type grinding machine is a type of grinding machine that uses a grinding disc to grind the ___。
阿波罗尼奥斯《圆锥曲线》ppt要点
学习生涯
阿波罗尼奥斯年青时到亚历山大跟随欧 几里得的后继者学习，那时是托勒密三 世（246BC—221BC）统治时期，到了 托勒密四世（221BC—205BC）时代， 他在天文学研究方面已颇有名气。 后来到过小亚细亚西岸的帕加马王国居 住与工作，晚年回到亚历山大，并卒于 该城。
贡献
阿波罗尼奥斯的主要成就 是建立了完美的圆锥曲线 论，总结了前人在这方面 的工作，再加上自己的研 究成果，撰成了《圆锥曲 线论，将圆锥曲线的性质 网罗殆尽，几乎使后人没 有插足的余地。
简介
阿波罗尼奥斯（Apollonius) 公元前262年出生于小亚细亚 的玻尔加，公元前190年卒于 古埃及的亚历山大。亚历山大 时期第三位重要的数学家，与 欧几里得、阿基米德齐名，其 贡献涉及几何学和天文学。
生平
《圆锥曲线论》是一部
经典巨作，可以说代表 了希腊几何的最高水平， 直至17世纪笛卡尔、帕 斯卡出场之前，始终无 人能够超越。阿波罗尼 奥斯写此书被后世译者 称为“大几何学家”。
双曲线的建筑方面的应用
双曲线绕虚轴旋转形成单叶双曲面，单 叶双曲面上有两族直母线。在建筑上可 以把钢筋作为两族直母线，使他们构成 单叶双曲面。这样设计的建筑物非常轻 巧又坚固。
单叶双曲面之冷却塔
14
55
12 27
广州电视塔小蛮腰
其设计师是荷兰IBA事务所的马克· 海默 尔和芭芭拉· 库伊特。 有一天，我在厨房把一些弹性橡皮绳绑 在两个椭圆形的木盘之间，一个在底部， 一个在顶部。当我开始旋转顶部椭圆的 时候，一个复杂的形状出现了。我开始 激动起来，要从这个简单的想法开始， 把它发展成一个建筑物。
本节结束
谢谢
39
圆锥曲线的定义
序言
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抛物线——平面内与一定点F和一定直线l的距离相等 的点的轨迹叫做抛物线.点F叫做抛物线的焦点。直线l 叫做抛物线的准线。
2、第二定义 点M（x，y）到定点F的距离与它到定直线l的距离的 比是常数e（e＞0）的点的轨迹，0<e<1时是椭圆; e=1 时是抛物线; e>1时是双曲线.e为离心率。
例1、椭圆
Y
P M
F1 O
F2
X
例倾斜4、角若为过60椭°圆的X_a直_22 线+交Y_b_22椭圆= 于1(aA＞、bB＞两0点)的，左且焦点F 1 、
|AF1|=2|BF1|，求椭圆的离心率。
道桥工程中英文对照外文翻译文献
中英文对照外文翻译文献(文档含英文原文和中文翻译)英文：1.1Approach for analyzing the ultimate strength of concrete filled steel tubular arch bridges with stiffening girderAbstract:A convenient approach is proposed for analyzing the ultimate load carrying capacity of concrete filled steel tubular (CFST) arch bridge with stiffening girders. A fiber model beam element is specially used to simulate the stiffening girder and CFST arch rib. The geometric nonlinearity, material nonlinearity。
influenceoftheconstruction process and the contribution of prestressing reinforcement are all taken into consideration. The accuracy of this method is validated by comparing its results with experimental results. Finally, the ultimate strength of an abnormal CFST arch bridge withstiffening girders is investigated and the effect of construction method is discussed. It is concluded that the construction process has little effect on the ultimate strength of the bridge.Key words: Ultimate strength, Concrete filled steel tubular (CFST) arch bridge, Stiffening girder, Fiber model beam element, Construction processdoi:10.1631/jzus.2007.A0682NTRODUCTIONWith the increasing applications of concrete filled steel tubular (CFST) structures in civil engi-neering in China, arch bridges have become one of the competitive styles in moderate span or long span bridges. Taking the Fuxing Bridge in Hangzhou (Zhao et al., 2004), and Wushan Bridge in Chongqing (Zhang et al., 2003), China, as representatives, the structural configuration, the span and construction scale of such bridges have surpassed those of existing CFST arch bridges in the world. Therefore, it is of great importance to enhance the theoretical level in the design of CFST arch bridges for safety and economy.he calculation of ultimate bearing capacity is a significant issue in design of CFST arch bridges. As an arch structure is primarily subjected to compres-sive forces, the ultimate strength of CFST arch bridge is determined by the stability requirement. A numberof theoretical studies were conducted in the past to investigate the stability and load-carrying capacity of CFST arch bridges. Zeng et al.(2003) studied the load capacity of CFST arch bridge using a composite beam element, involving geometric and material nonlin-earity. Zhang et al.(2006) derived a tangent stiffness matrix for spatial CFSTpole element to consider the geometric and material nonlinearities under largedisplacement by co-rotational coordinate method. Xie et al.(2005) proposed a numerical method to determine the ultimate strength of CFST arch bridges and revealed that the effect of the constitutive relation of confined concrete is not significant. Hu et al.(2006) investigated the effect of Poisson’s ratio of core concrete on the ultimate bearing capacity of a long span CFST arch bridge and found that the bearing capacity is enhanced by 10% if the Poisson’s ratio is variable. On the other hand, many experimental studies on the ultimate strength of naked CFST arch rib or CFST arch bridge model hadbeenconducted. Experimental studies on CFST arch rib under in-plane andout-of-plane loads were carried out by Chen and Chen 2000) and Chen et almetrical nonlinearity was significant for the out-of-plane strength and less significant for the in-plane strength. Cui et al.(2004) introduced a global model test of a CFST arch bridge with span of 308 m, and suggested that the influence of initial stress should be considered.The above papers mainly focused on the ultimate strength of CFST naked arch ribs or CFST arch bridges with floating deck. No attempt was made to study the ultimate strength of CFST arch bridges with stiffening girders whose nonlinear behavior and CFST arch should be simulated due to the redistribution of inner forces between arch ribs and stiffening girders. In general, stiffening girders can be classified into steel girder, PC (prestressing concrete) girder and teel-concrete combination girder. It is most difficult to simulate the nonlinear behavior of PC girder, due to the influence of prestressing reinforcement. In contrast to steel or steel-concrete combination beam, the prestressing reinforcements in PC girders not only offer strength and stiffnessdirectly, but their tension greatly affects the stiffness and distribution of the initial forces in the structure. The aims of this paper are (1) to present an elas-tic-plastic analysis of the ultimate strength of CFST arch bridge with arbitrary stiffening girders;(2) to study the ultimate load-carrying capacity of a complicated CFST arch bridge with abnormal arch ribs and PC stiffening girders; and (3) to investigate the effect of construction methods on the ultimate strength of the structure. ANALYTICAL THEORYElasto-plastic large deformation of PC girder element The elasto-plasic large deformation analysis of PC beam elements is based on the following fundamental assumptions:(1) A plane section originally normal to the neutral axis always remains a plane and normal to the neutral axis during deformation;(2) The shear deformation due to shear stress isneglected;(3)The Saint-Venant torsional principle holds in(4) The effect of shear stress on the stress-strain relationship is ignored. The cross-section of a PC box girder with onesymmetric axis is depicted in Fig.1, where, G and s denote the geometry center and the shear center re-spectively. According to the first and the third as-sumptions listed above, the displacement increments of point A(x,y) in the section can be expressed in terms of the displacement increments at the geometry center and the shear center aswhere Ktoris the coefficient factor which is related to the geometry shape of the girder cross-section.Similar to 3D elastic beam theory, the displacement increment of the girder can be expressed in terms of the nodal displacement increments asin which L denotes the element length, and z is the axial coordinate of the local coordinate system of an element. Then, the displacement vector of any section of the element can be written aswhere ∆u is the displacement vector of any section of the beam element, N is the shape function matrix and ∆ue is the displacement vector of the element node. They are respectively expressed asAccording to Eq.(2), the linear strain can be ex-pressed asin which BL is the linear strain matrix of the element Correspondingly, the nonlinear strain may be expressed aswhere BNL is the nonlinear strain matrix of the ele-mentThe stress increment ∆σ can be approximatedusing the linear strain increment aswhere D is the material property matrix. Neglecting the influence of the shear strain, D can be expressedwhere E(ε) is the tangent modulus of the material which is dependent on the strain state, and G is the elastic shearing modulus regarded as a constant. According to the principle of virtual work, we have in which σ and ∆σ are the stress vector and stress increment of the current state, q and P are the dis-tributed load and concentrated load vector, ∆q and ∆P are the increments of distributed load and concen-trated load, δ∆u and δ∆ε are the virtual displacement and virtual strain, and V isthe volume of the element. Substitute Eqs.(9), (11) and (14) into Eq.(16) and ignore the infinitesimal variable ∆σ∆εN, we have where ∆Fe is the increment of element load vectorcorresponding to ∆ue, the element displacement vec-tor. Kepand Kσare the elasto-plastic and geometric stiffness matrixes of the beam element respectively as followsThe distribution of elastic and plastic zones is non-uniform in the element, and varies during de-formation. It is very difficult to present an explicit expression of the property matrix D for the whole section. Hence, the section is divided into many subareas, as shown in Fig.2, and the fiber model is adopted to calculate the element’s stiffness matrix, i.e.Obviously, if the number of subareas is suffi-ciently large, the result of Eq.(19) will approach the exact solution. The value of Kep is calculated using numerical integration, with Di being regarded as i. To compute the geometric stiffness matrix Kσ, the normal stress should be expressed in terms of axial force and bending moment, which actually has very little contribution to the geometric stiffness, so where N is the axial force, and A is the sectional area. Prestressing reinforcement element The reinforced bars parallel to the beam axis may be regarded as fibers, whose contributions to the stiffness could be readily accounted for in Eq.(19). The contributions to the stiffness from those not par-allel to the beam and the prestressing reinforcement (PR), will however be calculated in the following section. The displacement increment of two ends of the prestressing reinforcement in Fig.3 can be expressed by Eq.(21):n which kep and kσare respectively the elasto-plastic and the geometric stiffness matrixes, ∆δis the nodal displacement vector, and ∆f is the nodal force vector of the prestressingreinforcementelement in the local coordinate system. According to Fig.4, ∆δand ∆f can be written in the form Then the stiffness matrix ep( k + k)σof the rein-accordingly. CFST arch rib, steel girder or steel-concrete girder element The fiber model mentioned above can also be used to simulate the CFST arch rib, steel stiffening girder or steel-concrete composite stiffening girder, with similar elasto-plastic stiffness matrix and stiff-ness equation. The detailed description of the deduction can be found in (Xie et al., 2005). However, for the CFST arch rib, the stress-strain relation of structure is very complex due to the com-bined influence of the confined concrete and outer steel tube. In this paper, the following stress-strain relation considering the confinement effect of the steel tube ring (Han, 2000) is adopted: where σytand σycare the yield strengths of the tension and compression sides of the steel tube respectively, βt and βc are the corresponding coefficients. Fig.5b depicted the bilinear stress-strain relationship con-The secondary modulus of the steel tube tendency of local buckling of the steel tube, is assumed to be 1% of the initial elastic modulus. Hanger element The mechanical behavior of cables such as that of hangers and tie bars, is similar to that of truss ele-ments, except that cables cannot bear compressive elasto-plastic computation theory of flexible cable considering the effect of sag was presented by (Xie eal., 1998). In most bridges, however, sag has little fect on the mechanical behavior of hangers. Hence, hangers of arch bridges are treated as elasto-plastic trusses with no compression strength, and the stiff-ness equation is expressed by Eq.(22). PROGRAM SCHEME FOR ULTIMATE BEARING CAPACITY CALCULATIOerection without brackets, and consists of many construction stages. Thus, the func-tion of simulating the construction process mustbe taken into account in the developed program for cal-culating ultimate bearing capacity, including the gradual action of load, the step-by-step formation of the structure, the influence of initial displacement and initial stress. The scheme for the program is indicated in Fig.6. The modified arc-length increment tecnique is adopted to solve the resulting nonlinear equation (Crisfield, 1981). VALIDATION OF THE METHOD FOR A PC GIRDER The accuracy of computation of the ultimate strength for CFST element has been confirmed in (Xie et al., 2005). In this paper, the precision of the present theory is checked for a PC girder by comparison with the experimental result. Fig.7 shows the cross-section and reinforcements of the girder, which spans 13 m, with 9 bundles of prestressing reinforcements and 11 branches of nonprestressing reinforced bars. The design strength of the concrete is 22.4 MPa, and those of nonprestressing reinforced bars A and B depicted in Fig.7a are 195 MPa and 280 MPa respectively of which the diameters are 12 mm and 8 mm. The prestressing reinforcement is high-strength low-rela- xation steel strand with design strength of 1860 MPa and the control force of each bundle is Nk=195 kN. More detailed information about the experiment on this PC girder is available in (Chen, 2005). Comparison of the deflection at the midspan is depicted in Fig.8, showing good consistency between he numerical simulation and experimental result. Fig.5 Stress-strain curves of steel tube (a) Yield condition; (b) Stress-strain relationship APPLICATION IN BRIDGE DESIGNThe ultimate strength of Fenghuajiang Bridge in Ningbo, Zhejiang, China is studied involving the effect of construction process to demonstrate the applicability of the present approach in bridge design. Fig.9 shows the design scheme of Fenghuajiang Bridgewhich is a girder and arch combination bridgewith central span of 138 m. The central span of the stiffening girder is made up of steel and PC composite box. The side span of the stiffening girder is made up of PC box. The abnormal CFST arch in the central span is composed of three arches, with one main archrib in the center and two secondary arch ribs. The diameter of the main arch rib is 1.8 m, and those of the other two are 1.5 m. The design strength of the concrete used in the bridge is 22.4 MPa. The arch ribs are linked with steel pipes and I-steel bearing members, forming a truss arch bridge. The main arch and the deck are connected with vertical hangers. The secondary arches and the deck are connected with inclined hangers. To take into account the effect of the construction method on the ultimate bearing capacity, it is assumed that the bridge is constructed by two kinds of methods. In Case I, there is only a construction process, the supporting frames for construction falling once after the completion of the whole bridge. In Case II, there are two construction processes, as shown in Fig.10. The first process is construction ofthe PC girder on the supporting frames. The second process is to fix the steel girder, assemble the arch rib, and tension the tie-bar and hangers to separate the steel girder from the frame. Prestressing reinforcements in the girder are properly simulated in construction stages, but the reinforced bars are not modelled due to their large number. The elasto-plastic mechanical behaviors of CFST arch ribs, hanger, bearing member, steel pipe, tie-bar, etc. are analyzed.The ultimate strength analysis process is shown in Fig.11. First,the initial stress of the established bridge is calculated under dead load and prestressing force including initial tension of the hangers, the tie and prestressing reinforcements. Then the stress and isplacement under live load are computed. At last,The out-of-plane deformation curves at the quarter points of the main arch rib are shown in Fig.14. The vertical axis denotes the load coefficient µ which does not contain the original dead load and live load exerted in Figs.11a and 11b. When 3.1≤µ≤3.2, the nonlinear behavior of the arch rib becomes obvious in the lateral direction. As shown in the figure, the buckling modes in both cases are antisymmetric out-of-plane, and the buckling load factor of the arch rib is about 4.1 considering the initial dead and live load.A comparison of the lateral and vertical deforMations at the quarter point of the main arch between two cases is shown in Fig.15, showing that the deviation of the load-displacement curves of the two cases is very small, indicating that the influence of the construction method on the stability strength is very slight. Besides, when out-of-plane buckling occurs, the bridge still has certain vertical stiffness.CONCLUSIONIn analyzing the ultimate strength of the CFST arch bridges with stiffening girders, simulating the nonlinear behavior of stiffening girders is as impor-tant as that of the CFST arch rib due to the redistribution of inner force between arch ribs and stiffening girders. In this paper, an analytical approach for estimating the ultimate bearing capacity of CFST arch bridge with stiffening girder is proposed, which takes account of the effects of material and geometric nonlinearity and the contribution of prestressing reinforcement. Based on the fiber beam element theory,the degrees of freedom of the whole structure can be reduced, making it very feasible to predict the ultimate strength of the complex structure. The accuracy of the present method was examined by comparison with the experimental results for a PC girder.To demonstrate the applicability of the present approach in bridge design, the ultimate strength of an abnormal CFST arch bridge with stiffening girder is studied considering the effect of construction process. The result shows that the construction process influences the initial internal force of the bridge significantly. But it has little effect on the ultimate strength of the bridge. Therefore, the relatively accurate stability strength can be obtained by ignoring the influence of the construction process.ReferencesChen, H.Z., 2005. Research of Calculation and Analysis of PCBox Girder Structure with Long Span. Ph.D Thesis,Zhejiang University (in Chinese).Chen, B.C., Chen, Y.J., 2000. Experimental study on me-chanic behaviors of concrete-filled steel tubular rib archunder in-plane loads. Engineering Mechanics,17(2):44-50 (in Chinese).Chen, B.C., Wei, J.G., Lin, J.Y., 2006. Experimental study on concrete filled steel tubular (single tube) arch with onerib under spatial loads. Engineering Mechanics,23(5):99-106 (in Chinese).Crisfield, M.A., 1981. A fast incremental iterative solution procedure that handles “snap through”. Computer and Structures, 13(1-3):55-62. [doi:10.1016/0045-7949(81) 90108-5]Cui, J., Sun, B.N., Lou, W.J., Yang, L.X., 2004. Model test study on concrete-filled steel tube truss arch bridge.Engineering Mechanics, 21(5):83-86 (in Chinese).e, X., Chen, H.Z., Li, H., Song, S.R., 2005. Numerical analysis of ultimate strength of concrete filled steel tu- bular arch bridges. Journal of Zhejiang University SCI-ENCE, 6A(8):859-868. [doi:10.1631/jzus.2005.A0859]Zeng, G.F., Fan, L.C., Zhang, G.Y., 2003. Load capacity analysis of concrete filled steel tube arch bridge with the composite beam element. Journal of the China RailwaySociety, 25(5):97-102 (in Chinese).Zhang, Z.A., Sun, Y., Wang, M.Q., 2003. Key technique in theerection process of the rib steel pipe truss segments forWushan Yangze River bridge. Highway, 12:26-32 (in Chinese).Zhang, Y., Shao, X.D., Cai, S.B., Hu, J.H., 2006. Spatial nonlinear finite element analysis for long-span trussedCFST arch bridge. China Journal of Highway andTransport, 19(4):65-70 (in Chinese).Zhao, L.Q., Xu, R.H., Zheng, X.Z., 2004. Overall design of thefourth Qiantangjiang River Bridge in Hangzhou. BridgeConstruction, 1:27-30 (in Chinese).翻译：分析钢管混凝土拱桥与加劲梁的极限强度的方法摘要：提出的方法是分析和研究负载承载能力的终极钢管混凝土钢管混凝土（加劲梁与钢管混凝土拱桥）。
圆锥曲线论 英文
圆锥曲线论英文Conic sections are a fundamental topic in mathematics that deals with the properties and equations of curves formed by the intersection of a plane with a cone. These curves include the circle, ellipse, parabola, and hyperbola. In this document, we will explore the characteristics and equations of these conic sections.1. Circle:A circle is a conic section formed when a plane intersects a cone at a right angle to its axis. It is defined as the set of all points in a plane that are equidistant from a fixed center point. The equation of a circle with center (h, k) and radius r is given by (x h)^2 + (y k)^2 = r^2.2. Ellipse:An ellipse is formed when a plane intersects a cone at an angle that is less than a right angle. It is defined as the set of all points in a plane for which the sum of the distances from two fixed points (called foci) is constant. The equation of an ellipse with center (h, k), major axis length 2a, and minor axis length 2b is given by ((x h)^2 / a^2) + ((y k)^2 / b^2) = 1.3. Parabola:A parabola is formed when a plane intersects a cone parallel to one of its generating lines. It is defined as the set of all points in a plane that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix). The equation of a parabola with vertex (h, k) and focal length p is given by (x h)^2 = 4p(y k).4. Hyperbola:A hyperbola is formed when a plane intersects a cone at an angle greater than a right angle. It is defined as the set of all points in a plane for which the absolute value of the difference of the distances from two fixed points (called foci) is constant. The equation ofa hyperbola with center (h, k), transverse axis length 2a, and conjugate axis length 2b is given by ((x h)^2 / a^2) ((y k)^2 / b^2) = 1.These equations provide a mathematical representation of the conic sections and allow us to analyze their properties. By manipulating these equations, we can determine important characteristics such as the shape, size, orientation, and position of the conic sections.In addition to their geometric properties, conic sections have various applications in different fields. For example, circles are commonly used in geometry, physics, and engineering to represent objects with rotational symmetry. Ellipses are used in astronomy to describe the orbits of planets and satellites. Parabolas are used in physics to model the trajectory of projectiles and in engineering to design reflectors and antennas. Hyperbolas are used in physics and engineering to describe the behavior of waves and particles.In conclusion, conic sections are a fascinating topic in mathematics with diverse applications in various fields. Understanding the properties and equations of circles, ellipses, parabolas, and hyperbolas allows us to analyze and solve problems involving these curves. By studying conic sections, we gain valuable insights into the fundamental principles of geometry and their practical applications.。
外文文献翻译原文+译文
外文文献翻译原文Analysis of Con tin uous Prestressed Concrete BeamsChris BurgoyneMarch 26, 20051、IntroductionThis conference is devoted to the development of structural analysis rather than the strength of materials, but the effective use of prestressed concrete relies on an appropriate combination of structural analysis techniques with knowledge of the material behaviour. Design of prestressed concrete structures is usually left to specialists; the unwary will either make mistakes or spend inordinate time trying to extract a solution from the various equations.There are a number of fundamental differences between the behaviour of prestressed concrete and that of other materials. Structures are not unstressed when unloaded; the design space of feasible solutions is totally bounded；in hyperstatic structures, various states of self-stress can be induced by altering the cable profile, and all of these factors get influenced by creep and thermal effects. How were these problems recognised and how have they been tackled?Ever since the development of reinforced concrete by Hennebique at the end of the 19th century (Cusack 1984), it was recognised that steel and concrete could be more effectively combined if the steel was pretensioned, putting the concrete into compression. Cracking could be reduced, if not prevented altogether, which would increase stiffness and improve durability. Early attempts all failed because the initial prestress soon vanished, leaving the structure to be- have as though it was reinforced; good descriptions of these attempts are given by Leonhardt (1964) and Abeles (1964).It was Freyssineti’s observations of the sagging of the shallow arches on three bridges that he had just completed in 1927 over the River Allier near Vichy which led directly to prestressed concrete (Freyssinet 1956). Only the bridge at Boutiron survived WWII (Fig 1). Hitherto, it had been assumed that concrete had a Young’s modulus which remained fixed, but he recognised that the de- ferred strains due to creep explained why the prestress had been lost in the early trials. Freyssinet (Fig. 2) also correctly reasoned that high tensile steel had to be used, so that some prestress would remain after the creep had occurred, and alsothat high quality concrete should be used, since this minimised the total amount of creep. The history of Freyssineti’s early prestressed concrete work is written elsewhereFigure1:Boutiron Bridge,Vic h yFigure 2: Eugen FreyssinetAt about the same time work was underway on creep at the BRE laboratory in England ((Glanville 1930) and (1933)). It is debatable which man should be given credit for the discovery of creep but Freyssinet clearly gets the credit for successfully using the knowledge to prestress concrete.There are still problems associated with understanding how prestressed concrete works, partly because there is more than one way of thinking about it. These different philosophies are to some extent contradictory, and certainly confusing to the young engineer. It is also reflected, to a certain extent, in the various codes of practice.Permissible stress design philosophy sees prestressed concrete as a way of avoiding cracking by eliminating tensile stresses; the objective is for sufficient compression to remain after creep losses. Untensionedreinforcement, which attracts prestress due to creep, is anathema. This philosophy derives directly from Freyssinet’s logic and is primarily a working stress concept.Ultimate strength philosophy sees prestressing as a way of utilising high tensile steel as reinforcement. High strength steels have high elastic strain capacity, which could not be utilised when used as reinforcement; if the steel is pretensioned, much of that strain capacity is taken out before bonding the steel to the concrete. Structures designed this way are normally designed to be in compression everywhere under permanent loads, but allowed to crack under high live load. The idea derives directly from the work of Dischinger (1936) and his work on the bridge at Aue in 1939 (Schonberg and Fichter 1939), as well as that of Finsterwalder (1939). It is primarily an ultimate load concept. The idea of partial prestressing derives from these ideas.The Load-Balancing philosophy, introduced by T.Y. Lin, uses prestressing to counter the effect of the permanent loads (Lin 1963). The sag of the cables causes an upward force on the beam, which counteracts the load on the beam. Clearly, only one load can be balanced, but if this is taken as the total dead weight, then under that load the beam will perceive only the net axial prestress and will have no tendency to creep up or down.These three philosophies all have their champions, and heated debates take place between them as to which is the most fundamental.2、Section designFrom the outset it was recognised that prestressed concrete has to be checked at both the working load and the ultimate load. For steel structures, and those made from reinforced concrete, there is a fairly direct relationship between the load capacity under an allowable stress design, and that at the ultimate load under an ultimate strength design. Older codes were based on permissible stresses at the working load; new codes use moment capacities at the ultimate load. Different load factors are used in the two codes, but a structure which passes one code is likely to be acceptable under the other.For prestressed concrete, those ideas do not hold, since the structure is highly stressed, even when unloaded. A small increase of load can cause some stress limits to be breached, while a large increase in load might be needed to cross other limits. The designer has considerable freedom to vary both the working load and ultimate load capacities independently; both need to be checked.A designer normally has to check the tensile and compressive stresses, in both the top and bottom fibre of the section, for every load case. The critical sections are normally, but not always, the mid-span and the sections over piers but other sections may become critical ，when the cable profile has to be determined.The stresses at any position are made up of three components, one of which normally has a different sign from the other two; consistency of sign convention is essential.If P is the prestressing force and e its eccentricity, A and Z are the area of the cross-section and its elastic section modulus, while M is the applied moment, then where ft and fc are the permissible stresses in tension and compression.c e t f ZM Z P A P f ≤-+≤Thus, for any combination of P and M , the designer already has four in- equalities to deal with.The prestressing force differs over time, due to creep losses, and a designer isusually faced with at least three combinations of prestressing force and moment;• the applied moment at the time the prestress is first applied, before creep losses occur,• the maximum applied moment after creep losses, and• the minimum applied moment after creep losses.Figure 4: Gustave MagnelOther combinations may be needed in more complex cases. There are at least twelve inequalities that have to be satisfied at any cross-section, but since an I-section can be defined by six variables, and two are needed to define the prestress, the problem is over-specified and it is not immediately obvious which conditions are superfluous. In the hands of inexperienced engineers, the design process can be very long-winded. However, it is possible to separate out the design of the cross-section from the design of the prestress. By considering pairs of stress limits on the same fibre, but for different load cases, the effects of the prestress can be eliminated, leaving expressions of the form:rangestress e Perm issibl Range Mom entZ These inequalities, which can be evaluated exhaustively with little difficulty, allow the minimum size of the cross-section to be determined.Once a suitable cross-section has been found, the prestress can be designed using a construction due to Magnel (Fig.4). The stress limits can all be rearranged into the form:()M fZ PA Z e ++-≤1 By plotting these on a diagram of eccentricity versus the reciprocal of the prestressing force, a series of bound lines will be formed. Provided the inequalities (2) are satisfied, these bound lines will always leave a zone showing all feasible combinations of P and e. The most economical design, using the minimum prestress, usually lies on the right hand side of the diagram, where the design is limited by the permissible tensile stresses.Plotting the eccentricity on the vertical axis allows direct comparison with the crosssection, as shown in Fig. 5. Inequalities (3) make no reference to the physical dimensions of the structure, but these practical cover limits can be shown as wellA good designer knows how changes to the design and the loadings alter the Magnel diagram. Changing both the maximum andminimum bending moments, but keeping the range the same, raises and lowers the feasible region. If the moments become more sagging the feasible region gets lower in the beam.In general, as spans increase, the dead load moments increase in proportion to the live load. A stage will be reached where the economic point (A on Fig.5) moves outside the physical limits of the beam; Guyon (1951a) denoted the limiting condition as the critical span. Shorter spans will be governed by tensile stresses in the two extreme fibres, while longer spans will be governed by the limiting eccentricity and tensile stresses in the bottom fibre. However, it does not take a large increase in moment ，at which point compressive stresses will govern in the bottom fibre under maximum moment.Only when much longer spans are required, and the feasible region moves as far down as possible, does the structure become governed by compressive stresses in both fibres.3、Continuous beamsThe design of statically determinate beams is relatively straightforward; the engineer can work on the basis of the design of individual cross-sections, as outlined above. A number of complications arise when the structure is indeterminate which means that the designer has to consider, not only a critical section,but also the behaviour of the beam as a whole. These are due to the interaction of a number of factors, such as Creep, Temperature effects and Construction Sequence effects. It is the development of these ideas whichforms the core of this paper. The problems of continuity were addressed at a conference in London (Andrew and Witt 1951). The basic principles, and nomenclature, were already in use, but to modern eyes concentration on hand analysis techniques was unusual, and one of the principle concerns seems to have been the difficulty of estimating losses of prestressing force.3.1 Secondary MomentsA prestressing cable in a beam causes the structure to deflect. Unlike the statically determinate beam, where this motion is unrestrained, the movement causes a redistribution of the support reactions which in turn induces additional moments. These are often termed Secondary Moments, but they are not always small, or Parasitic Moments, but they are not always bad.Freyssinet’s bridge across the Marne at Luzancy, started in 1941 but not completed until 1946, is often thought of as a simply supported beam, but it was actually built as a two-hinged arch (Harris 1986), with support reactions adjusted by means of flat jacks and wedges which were later grouted-in (Fig.6). The same principles were applied in the later and larger beams built over the same river.Magnel built the first indeterminate beam bridge at Sclayn, in Belgium (Fig.7) in 1946. The cables are virtually straight, but he adjusted the deck profile so that the cables were close to the soffit near mid-span. Even with straight cables the sagging secondary momentsare large; about 50% of the hogging moment at the central support caused by dead and live load.The secondary moments cannot be found until the profile is known but the cablecannot be designed until the secondary moments are known. Guyon (1951b) introduced the concept of the concordant profile, which is a profile that causes no secondary moments; es and ep thus coincide. Any line of thrust is itself a concordant profile.The designer is then faced with a slightly simpler problem; a cable profile has to be chosen which not only satisfies the eccentricity limits (3) but is also concordant. That in itself is not a trivial operation, but is helped by the fact that the bending moment diagram that results from any load applied to a beam will itself be a concordant profile for a cable of constant force. Such loads are termed notional loads to distinguish them from the real loads on the structure. Superposition can be used to progressively build up a set of notional loads whose bending moment diagram gives the desired concordant profile.3.2 Temperature effectsTemperature variations apply to all structures but the effect on prestressed concrete beams can be more pronounced than in other structures. The temperature profile through the depth of a beam (Emerson 1973) can be split into three components for the purposes of calculation (Hambly 1991). The first causes a longitudinal expansion, which is normally released by the articulation of the structure; the second causes curvature which leads to deflection in all beams and reactant moments in continuous beams, while the third causes a set of self-equilibrating set of stresses across the cross-section.The reactant moments can be calculated and allowed-for, but it is the self- equilibrating stresses that cause the main problems for prestressed concrete beams. These beams normally have high thermal mass which means that daily temperature variations do not penetrate to the core of the structure. The result is a very non-uniform temperature distribution across the depth which in turn leads to significant self-equilibrating stresses. If the core of the structure is warm, while the surface is cool, such as at night, then quite large tensile stresses can be developed on the top and bottom surfaces. However, they only penetrate a very short distance into the concrete and the potential crack width is very small. It can be very expensive to overcome the tensile stress by changing the section or the prestress。
圆锥曲线 柯西不等式
圆锥曲线柯西不等式【中英文版】Task Title: Circle Curves and Cauchy InequalityTask Title: 圆锥曲线与柯西不等式In mathematics, a circle curve, also known as a conic section, is a curve obtained by cutting a cone by a plane.The three most common types of conic sections are circles, ellipses, and parabolas.These curves have various applications in physics, engineering, and mathematics.在数学中，圆锥曲线也称为椭圆曲线，是通过将圆锥与一个平面切割而得到的曲线。
最常见的三种圆锥曲线是圆、椭圆和抛物线。
这些曲线在物理学、工程学和数学中有着各种应用。
Cauchy inequality is a fundamental result in linear algebra that relates the lengths of vectors to the lengths of their projections onto a subspace.It states that for any two vectors u and v of an inner product space, the following inequality holds: ||u + v||^2 ≤ ||u||^2 + ||v||^2.柯西不等式是线性代数中的基本结果，它将向量的长度与它们在子空间上的投影长度相关联。
它表明，对于任意两个内积空间中的向量u和v，以下不等式成立：||u + v||^2 ≤ ||u||^2 + ||v||^2。
高二英语圆锥曲线单选题30题
高二英语圆锥曲线单选题30题1. Which of the following is not a conic section? A. Circle. B. Parabola.C. Rectangle.D. Ellipse. Answer: C. Rectangle is not a conic section. Conic sections are curves obtained by intersecting a cone with a plane. Rectangle is a quadrilateral and not a conic section.2. The standard equation of a circle with center (h,k) and radius r is?A. (x - h)^2 + (y - k)^2 = r^2.B. (x + h)^2 + (y + k)^2 = r^2.C. (x - h)^2 - (y - k)^2 = r^2.D. (x + h)^2 - (y + k)^2 = r^2. Answer: A. The standard equation of a circle with center (h,k) and radius r is (x - h)^2 + (y - k)^2 = r^2.3. What is the eccentricity of a circle? A. 0. B. 1. C. 2. D. Undefined. Answer: A. The eccentricity of a circle is 0. Eccentricity is a measure of how much a conic section deviates from being circular. Since a circle is perfectly circular, its eccentricity is 0.4. Which conic section has one focus? A. Circle. B. Parabola. C. Ellipse. D. Hyperbola. Answer: B. Parabola has one focus. A circle has no foci. An ellipse has two foci. A hyperbola has two foci.5. The equation of a parabola with vertex at the origin and axis of symmetry along the x-axis is of the form? A. y^2 = 4ax. B. x^2 = 4ay. C. y^2 = -4ax. D. x^2 = -4ay. Answer: A. The equation of a parabola with vertex at the origin and axis of symmetry along the x-axis is of the form y^2 = 4ax. If the axis of symmetry is along the y-axis, the equation is x^2= 4ay. Negative signs are used for parabolas opening to the left or down.6. Which of the following is the equation of an ellipse?A. y = x²B. x² + y² = 1C. y = 1/xD. y = 2x.Answer: B. An ellipse is defined by the equation x²/a² + y²/b² = 1. In this case, x² + y² = 1 is in the form of an ellipse equation with a = b = 1.7. The equation x²/16 + y²/9 = 1 represents what conic section?A. CircleB. ParabolaC. EllipseD. Hyperbola.Answer: C. The equation x²/a² + y²/b² = 1 represents an ellipse. Here, a² = 16 and b² = 9, so it is an ellipse.8. What is the standard equation of a hyperbola with transverse axis along the x-axis?A. x²/a² - y²/b² = 1B. x²/a² + y²/b² = 1C. y²/a² - x²/b² = 1D. y²/a² + x²/b² = 1.Answer: A. The standard equation of a hyperbola with transverse axisalong the x-axis is x²/a² - y²/b² = 1.9. The equation y² = 8x represents what conic section?A. CircleB. ParabolaC. EllipseD. Hyperbola.Answer: B. The equation y² = 2px represents a parabola opening right or left. Here, 2p = 8, so it is a parabola.10. Which equation represents a circle?A. x² + y² - 6x + 8y = 0B. x² - y² = 1C. y = x²D. y = 1/x.Answer: A. The equation of a circle is (x - h)² + (y - k)² = r². By completing the square, x² + y² - 6x + 8y = 0 can be rewritten as (x - 3)² + (y + 4)² = 25, which is the equation of a circle.11. Which of the following is a property of an ellipse?A. The sum of the distances from any point on the ellipse to two fixed points is constant.B. The distance from any point on the curve to a fixed point is equal to the distance from that point to a fixed line.C. The ratio of the distances from a point on the curve to a fixed pointand a fixed line is a constant greater than 1.D. The set of all points in a plane that are equidistant from a fixed line and a fixed point.Answer: A. An ellipse is defined by the property that the sum of the distances from any point on the ellipse to two fixed points (the foci) is constant.12. In a hyperbola, the distance between the two foci is 2c. If the distance between the two vertices is 2a, then which of the following is true?A. c = aB. c > aC. c < aD. There is no definite relationship between c and a.Answer: B. In a hyperbola, c is always greater than a. The distance between the foci is greater than the distance between the vertices.13. The eccentricity of a circle is:A. 0B. 1C. Between 0 and 1D. Greater than 1Answer: A. A circle is a special case of an ellipse where the two foci coincide. The eccentricity of a circle is 0.14. For an ellipse with major axis length 2a and minor axis length 2b,the area is given by:A. πabB. 2πabC. πa²b²D. π(a + b)Answer: A. The area of an ellipse is πab, where a is half the length of the major axis and b is half the length of the minor axis.15. The standard equation of a parabola with vertex at the origin and focus on the positive y-axis is:A. y² = 4axB. x² = 4ayC. y² = -4axD. x² = -4ayAnswer: B. When the vertex is at the origin and the focus is on the positive y-axis, the standard equation of a parabola is x² = 4ay.16. The line y = 2x + 1 intersects the ellipse x²/9 + y²/4 = 1. How many points of intersection are there?Answer: Substituting y = 2x + 1 into the equation of the ellipse, we get x²/9 + (2x + 1)²/4 = 1. After simplification, we get a quadratic equation. Solving it, we find two solutions. So there are two points of intersection.17. Which of the following lines is tangent to the parabola y² = 8x?A. y = x + 2B. y = 2x + 1C. y = x - 2D. y = 2x - 1Answer: For a line to be tangent to a parabola, the discriminant of the quadratic equation obtained by substituting the equation of the line into the equation of the parabola should be zero. Substituting each option into y² = 8x and checking the discriminant, we find that y = 2x - 1 is tangent.18. The line y = kx + 3 intersects the hyperbola x² - y² = 1. If there is exactly one point of intersection, what is the value of k?Answer: Substituting y = kx + 3 into the equation of the hyperbola, we get x² - (kx + 3)² = 1. Simplifying and setting the discriminant equal to zero since there is exactly one point of intersection, we can solve for k.19. Determine if the line 2x - y + 5 = 0 intersects the ellipse 2x² + 3y² = 12.Answer: Substituting y = 2x + 5 into the equation of the ellipse and simplifying. Then checking if the resulting equation has solutions determines if they intersect.20. A line passes through the point (1,2) and intersects the parabola y² = 4x. What is the equation of the line if the point of intersection is the vertex of the parabola?Answer: The vertex of the parabola y² = 4x is (0,0). The equation of the line passing through (1,2) and (0,0) can be found using the two-pointform of a line.21. In a rectangular coordinate system, the equation of an ellipse is x²/16 + y²/9 = 1. A point P on the ellipse is 2 units away from one focus. The distance from point P to the other focus is ( ).Solution: For an ellipse, the sum of the distances from any point on the ellipse to the two foci is constant and equal to the length of the major axis. Here, a² = 16, so a = 4. The sum of the distances from a point on the ellipse to the two foci is 2a = 8. Since the point is 2 units away from one focus, the distance to the other focus is 8 - 2 = 6. The answer is 6.22. The hyperbola x²/9 - y²/16 = 1 has focal length ( ).Solution: For a hyperbola x²/a² - y²/b² = 1, the focal length c is given by c² = a² + b². Here, a² = 9 and b² = 16, so c² = 9 + 16 = 25. Thus, c = 5. The focal length is 2c = 10. The answer is 10.23. The parabola y² = 8x has focus coordinates ( ).Solution: For the parabola y² = 2px, the focus coordinates are (p/2, 0). Here, 2p = 8, so p = 4. Thus, the focus coordinates are (2, 0). The answer is (2, 0).24. If a point moves so that the sum of its distances from two fixed points is constant, then the locus of the point is ( ).Solution: If a point moves so that the sum of its distances from two fixed points is constant, then the locus of the point is an ellipse. The answer is an ellipse.25. The asymptotes of the hyperbola x²/4 - y²/9 = 1 are ( ).Solution: For a hyperbola x²/a² - y²/b² = 1, the asymptotes are y = ±(b/a)x. Here, a² = 4 and b² = 9, so a = 2 and b = 3. The asymptotes are y = ±(3/2)x. The answer is y = ±(3/2)x.26. Which of the following is true about the asymptotes of a hyperbola?Answer: The asymptotes of a hyperbola are two lines that the hyperbola approaches as x and y approach infinity or negative infinity. The equation of the asymptotes for a hyperbola of the form (x^2/a^2) - (y^2/b^2) = 1 is y = ±(b/a)x.27. If the equation of an ellipse is (x^2/25) + (y^2/16) = 1, what is the length of the major axis?Answer: The major axis of an ellipse is along the x-axis when the larger denominator is under x^2. In this case, a^2 = 25, so a = 5. The length of the major axis is 2a = 10.28. What is the eccentricity of a parabola?Answer: The eccentricity of a parabola is always 1.29. For a hyperbola, as the eccentricity increases, what happens to the shape?Answer: As the eccentricity of a hyperbola increases, the hyperbola becomes more elongated or "thinner".30. If the focus of a parabola is (3,0) and the directrix is x = -3, what is the equation of the parabola?Answer: Since the focus is (3,0) and the directrix is x = -3, the parabola opens to the right. The equation of a parabola opening to the right with focus (p,0) and directrix x = -p is y^2 = 4px. Here, p = 3, so the equation is y^2 = 12x.。
高二英语圆锥曲线练习题30题
高二英语圆锥曲线练习题30题1.The equation of an ellipse is x²/25 + y²/16 = 1. The length of the major axis is:A.5B.10C.4D.8答案：B。
本题考查椭圆的标准方程及性质。
对于椭圆方程x²/a² + y²/b² = 1，a 表示长半轴长，2a 为长轴长。
在本题中a² = 25，所以a = 5，长轴长为2a = 10。
选项A 是长半轴长；选项C、D 不符合椭圆长轴长的计算结果。
2.The foci of the ellipse x²/9 + y²/4 = 1 are at:A.(±√5,0)B.(±2,0)C.(±3,0)D.(±4,0)答案：A。
根据椭圆方程x²/a² + y²/b² = 1，c² = a² - b²，这里a² = 9，b² = 4，可得c² = 9 - 4 = 5，c = ±√5。
焦点在x 轴上，坐标为(±c,0)，即(±√5,0)。
选项B、C、D 计算错误。
3.An ellipse has the equation x²/16 + y²/9 = 1. The eccentricity of the ellipse is:A.√7/4B.4/5C.3/4D.5/4答案：C。
根据椭圆离心率公式 e = c/a，由方程可得a² = 16，b² = 9，c² = a² - b² = 16 - 9 = 7，c = √7，a = 4，所以e = √7/4。
选项A、B、D 计算错误。
亚里士多德 圆锥曲线论
亚里士多德圆锥曲线论在古希腊哲学家和科学家亚里士多德的数学著作中，有一篇被称为《圆锥曲线论》的著作，其中详细阐述了圆锥曲线的基本性质和公式。
圆锥曲线是一种非常重要的数学概念，其几何形状可以用来描述天文学、机械学、电子学等多个领域的现象和应用。
在本文中，我们将对亚里士多德的圆锥曲线论进行介绍和讨论。
圆锥曲线的定义圆锥曲线是指一个圆锥体被一个平面截割后所得到的曲线。
根据截割平面与圆锥体的位置关系，圆锥曲线可以分为三类：椭圆、双曲线和抛物线。
其中，椭圆是指截割平面与圆锥体的轴线平行，双曲线是指截割平面与圆锥体的轴线有交点而不垂直，而抛物线则是指截割平面与圆锥体的轴线垂直。
圆锥曲线的基本性质由于圆锥曲线的定义和形状的不同，其性质和公式也会有所不同。
在此，我们列举出各个类型圆锥曲线的基本性质。
椭圆椭圆是一个有限的曲线，其与两个焦点的距离之和等于常数2a，即|PF1|+|PF2|=2a。
其中，焦点F1、F2又称为焦点。
椭圆的长半轴a和短半轴b定义如下：椭圆图椭圆图椭圆的面积和周长公式为：•面积：$S=\\pi a b$•周长：$C=2\\pi\\sqrt{\\frac{a^2+b^2}{2}}$双曲线双曲线是一个非常特殊的曲线，其与两个焦点的距离之差等于常数2a，即|PF1|−|PF2|=2a。
其中，焦点F1、F2又称为焦点。
双曲线的长半轴a和短半轴b定义如下：双曲线图双曲线图双曲线的面积和周长公式为：•面积：$S=\\pi ab$•周长：$C=2\\pi\\sqrt{\\frac{a^2+b^2}{2}}$抛物线抛物线是一个无限的曲线，其与焦点的距离等于常数p，即|PF|=p。
其中，焦点F又称为焦点。
抛物线的几个重要点的坐标定义如下：•抛物顶点的坐标为(0,0)•焦点的坐标为(0,p)•直线x=-p称为抛物线的对称轴抛物线图抛物线图抛物线的面积和周长公式为：•面积：$S=\\frac{4}{3}\\pi a^2$•周长：$C=\\infty$圆锥曲线的应用圆锥曲线的应用非常广泛，其中一些重要的应用包括：天文学天体运动的数学模型需要使用椭圆、双曲线和抛物线的形状和公式。
高中数学 第二章 圆锥曲线与方程 2.1 椭圆的历史素材 北师大版选修11
椭圆的历史与应用历史Apollonius所著的八册《圆锥曲线》(Conics)集其大成，可以说是古希腊几何学一个登峰造极的精擘之作。
今日大家熟知的 ellipse（椭圆）、parabola（抛物线）、hyperbola （双曲线）这些名词，都是 Apollonius 所发明的。
当时对于这种既简朴又完美的曲线的研究，乃是纯粹从几何学的观点，研讨和圆密切相关的这种曲线；它们的几何乃是圆的几何的自然推广，在当年这是一种纯理念的探索，并不寄望也无从预期它们会真的在大自然的基本结构中扮演著重要的角色。
此事一直到十六、十七世纪之交，开普勒（Kepler）行星运行三定律的发现才知道行星绕太阳运行的轨道，乃是一种以太阳为其一焦点的椭圆。
开普勒三定律乃是近代科学开天辟地的重大突破，它不但开创了天文学的新纪元，而且也是牛顿万有引力定律的根源所在。
由此可见，圆锥截线不单单是几何学家所爱好的精简事物，它们也是大自然的基本规律中所自然选用的精要之一。
手工画法绘法一画长轴AB，短轴CD，AB和CD互垂平分于O点。
⑵：连接AC。
⑶：以O为圆心，OA为半径作圆弧交OC延长线于E点。
⑷：以C为圆心，CE为半径作圆弧与AC交于F点。
⑸：作AF的垂直平分线交CD延长线于G点，交AB于H点。
⑹：截取H，G对于O点的对称点H’，G’⑺：H，H’为长轴圆心，分别以HB、H‘A为半径；G，G’为短轴原心，分别以GC、G ‘D为半径。
用一根线或者细铜丝，铅笔，2个图钉或大头针画椭圆的方法：先画好长短轴的十字线，在长轴上以圆点为中心先找2个大于短轴半径的点，一个点先用图钉或者大头针栓好线固定住，另一个点的线先不要固定，用笔带住线去找长短轴的4个顶点，此步骤需要多次定位，直到都正好能于顶点吻合后固定住这2个点，用笔带住线，直接画出椭圆：）使用细铜丝最好，因为线的弹性较大画出来不一定准确！绘法二椭圆的焦距│FF'│（Z）定义，为已知椭圆所构成的长轴X(ab）与短轴Y(cd）则以长轴一端A为圆心短轴Y为半径画弧，从长轴另一段点B引出与弧相切的线段则为该椭圆焦距，求证公式为2√{(Z/2）^2+(Y/2）^2}+Z=X+Z（平面内与两定点F、F'的距离的和等于常数2a （2a>|FF'|）的动点P的轨迹叫做椭圆），可演变为z=√x^2-y^2(x>y>0）。
【高中数学】圆锥曲线常见条件翻译转化
6
−32 +4 −6
解得 y=0,或 y=﹣ 2 ∴B(
,
).
3 +4
32 +4 32 +4
∴直线 BQ 的方程为(
令 y=0,解得 x=
2
−32 +4
2
﹣
)(x+1)﹣(
+
1)(y﹣
= 1(0＜a2＜4),
= 1.解得 a2=18(舍去)或 a2=2,
= 1.
(2):依题意,可设直线 l 的方程为 y=kx+2,代入双曲线 C 的方程并整理,
得(1﹣k2)x2﹣4kx﹣6=0.
∵直线 I 与双曲线 C 相交于不同的两点 E、F,
≠ ±1
1 − 2 ≠ 0
⇔�
∴�
2
2
−√3＜＜√3
3
【解析】(1)由题意得
(2)由
设点
,
,解得
,
,得
,
因为直线
,所以椭圆 的方程为
.
,则
,
.
,所以
恒过椭圆内一点
恒成立.
,
由根与系数的关系得:
.
,又因
所以
为点
所以
.
到直线
的面积为
的距离
,
,即
,
.
解得
y
P
【例 5】.圆
x
x2 + y 2 =
4 的切线与轴正半轴,轴正半轴围成
O
一个三角形,当该三角形面
x2 y 2
+
=
1.
6
3
【中档】
【 例 1 】 . 已 知 双 曲 线 :
圆锥曲线的通径
圆锥曲线的通径
圆锥曲线（Conical curve）是一类封闭的曲线，它具有一定的规律性，能够分为三类：以圆锥弧（conical arc）为主的圆锥正曲线，以及圆锥负曲线和交叉曲线
（crossing curve）。
圆锥曲线的始终与法线的夹角都是定值，以封闭的弧形为主要组成
部分，其形状多样，如圆形、椭圆形、抛物线、双抛物线等。
具有特殊钳制应用价值。
圆锥曲线(conical curve)是由曲线而非直线组成的封闭曲线，具有一定的法向角度。
它有三类：正曲线，负曲线和交叉曲线。
正曲线以圆锥弧为基础，它可以是完全圆形，也
可以是椭圆形、抛物线形或双抛物线形等；负曲线可以是二次曲线，也可以是超曲线等；
而交叉曲线则包含了正负曲线的部分特征。
圆锥曲线的应用广泛，它可用于测量圆周规则的零件，测量零件外部形状和封闭曲线
外形。
它还可以作为制作木模版、机械夹具及木模具时使用。
例如：圆锥曲线可用于箱子
的设计，其内部采用合理的三维网格形式，可从多边形的任何一边任意弯曲而成，从而减
少零件的锻件繁琐和成形材料的浪费。
此外，圆锥正曲线用于拉伸绳，收缩钢，修剪轴承的曲面等，它的精度远高于圆周曲线；圆锥负曲线则主要用于运动曲线上，如飞轮、摩擦轮、悬臂重载运动曲线等；交叉曲
线则常用于零件或构件的耐久性强度测试中。
因此，圆锥曲线是一类十分重要的曲线，应用十分广泛，被用于工程、绘画、制图以
及机械加工中。
圆锥曲线的定义、概念与定理
圆锥曲线的定义、概念与定理圆锥曲线包括椭圆，抛物线，双曲线。
那么你对圆锥曲线的定义了解多少呢?以下是由店铺整理关于圆锥曲线的定义的内容，希望大家喜欢!圆锥曲线的定义几何观点用一个平面去截一个二次锥面，得到的交线就称为圆锥曲线(conic sections)。
通常提到的圆锥曲线包括椭圆，双曲线和抛物线，但严格来讲，它还包括一些退化情形。
具体而言：1) 当平面与二次锥面的母线平行，且不过圆锥顶点，结果为抛物线。
2) 当平面与二次锥面的母线平行，且过圆锥顶点，结果退化为一条直线。
3) 当平面只与二次锥面一侧相交，且不过圆锥顶点，结果为椭圆。
4) 当平面只与二次锥面一侧相交，且不过圆锥顶点，并与圆锥的对称轴垂直，结果为圆。
5) 当平面只与二次锥面一侧相交，且过圆锥顶点，结果为一点。
6) 当平面与二次锥面两侧都相交，且不过圆锥顶点，结果为双曲线(每一支为此二次锥面中的一个圆锥面与平面的交线)。
7) 当平面与二次锥面两侧都相交，且过圆锥顶点，结果为两条相交直线。
代数观点在笛卡尔平面上，二元二次方程的图像是圆锥曲线。
根据判别式的不同，也包含了椭圆、双曲线、抛物线以及各种退化情形。
焦点--准线观点(严格来讲，这种观点下只能定义圆锥曲线的几种主要情形，因而不能算是圆锥曲线的定义。
但因其使用广泛，并能引导出许多圆锥曲线中重要的几何概念和性质)。
给定一点P，一直线L以及一非负实常数e，则到P的距离与L距离之比为e的点的轨迹是圆锥曲线。
根据e的范围不同，曲线也各不相同。
具体如下：1) e=0，轨迹为圆(椭圆的特例);2) e=1(即到P与到L距离相同)，轨迹为抛物线 ;3) 0<e<1，轨迹为椭圆;4) e>1，轨迹为双曲线的一支。
圆锥曲线的概念(以下以纯几何方式叙述主要的圆锥曲线通用的概念和性质，由于大部分性质是在焦点-准线观点下定义的，对于更一般的退化情形，有些概念可能不适用。
)考虑焦点--准线观点下的圆锥曲线定义。
圆锥曲线
1）椭圆（Ellipse)文字语言定义：平面内一个动点到一个定点与一条定直线的距离之比是一个小于1的正常数e。
平面内一个动点到两个定点（焦点）的距离和等于定长2a的点的集合。
定点是椭圆的焦点，定直线是椭圆的准线，常数e是椭圆的离心率。
标准方程：1.中心在原点，焦点在x轴上的椭圆标准方程：（x^2/a^2)+(y^2/b^2)=1其中a>b>0，c>0，c^2=a^2-b^2.2.中心在原点，焦点在y轴上的椭圆标准方程：（x^2/b^2)+(y^2/a^2)=1其中a>b>0，c>0，c^2=a^2-b^2。
参数方程：x=acosθ y=bsinθ（θ为参数,0≤θ≤2π)2）双曲线（Hyperbola)文字语言定义：平面内一个动点到一个定点与一条定直线的距离之比是一个大于1的常数e。
定点是双曲线的焦点，定直线是双曲线的准线，常数e是双曲线的离心率。
标准方程：1.中心在原点，焦点在x轴上的双曲线标准方程：（x^2/a^2)-(y^2/b^2)=1其中a>0,b>0,c^2=a^2+b^2.2.中心在原点，焦点在y轴上的双曲线标准方程：（y^2/a^2)-(x^2/b^2)=1.其中a>0,b>0,c^2=a^2+b^2.参数方程：x=asecθ y=btanθ（θ为参数)直角坐标（中心为原点）：x^2/a^2 - y^2/b^2 = 1 （开口方向为x轴）y^2/a^2 - x^2/b^2 = 1 （开口方向为y轴）3）抛物线（Parabola)文字语言定义：平面内一个动点到一个定点与一条定直线的距离之比是等于1。
定点是抛物线的焦点，定直线是抛物线的准线。
参数方程x=2pt^2 y=2pt (t为参数）t=1/tanθ（tanθ为曲线上点与坐标原点确定直线的斜率）特别地，t可等于0直角坐标y=ax^2+bx+c （开口方向为y轴，a≠0） x=ay^2+by+c （开口方向为x轴，a≠0 )圆锥曲线（二次非圆曲线）的统一极坐标方程为ρ=ep/(1-ecosθ）其中e表示离心率，p为焦点到准线的距离。
桥梁工程中英文对照外文翻译文献
桥梁工程中英文对照外文翻译文献BRIDGE ENGINEERING AND AESTHETICSEvolvement of bridge Engineering,brief reviewAmong the early documented reviews of construction materials and structu re types are the books of Marcus Vitruvios Pollio in the first century B.C.The basic principles of statics were developed by the Greeks , and were exemplifi ed in works and applications by Leonardo da Vinci,Cardeno,and Galileo.In the fifteenth and sixteenth century, engineers seemed to be unaware of this record , and relied solely on experience and tradition for building bridges and aqueduc ts .The state of the art changed rapidly toward the end of the seventeenth cent ury when Leibnitz, Newton, and Bernoulli introduced mathematical formulatio ns. Published works by Lahire (1695)and Belidor (1792) about the theoretical a nalysis of structures provided the basis in the field of mechanics of materials .Kuzmanovic(1977) focuses on stone and wood as the first bridge-building materials. Iron was introduced during the transitional period from wood to steel .According to recent records , concrete was used in France as early as 1840 for a bridge 39 feet (12 m) long to span the Garoyne Canal at Grisoles, but r einforced concrete was not introduced in bridge construction until the beginnin g of this century . Prestressed concrete was first used in 1927.Stone bridges of the arch type (integrated superstructure and substructure) were constructed in Rome and other European cities in the middle ages . Thes e arches were half-circular , with flat arches beginning to dominate bridge wor k during the Renaissance period. This concept was markedly improved at the e nd of the eighteenth century and found structurally adequate to accommodate f uture railroad loads . In terms of analysis and use of materials , stone bridgeshave not changed much ,but the theoretical treatment was improved by introd ucing the pressure-line concept in the early 1670s(Lahire, 1695) . The arch the ory was documented in model tests where typical failure modes were considere d (Frezier,1739).Culmann(1851) introduced the elastic center method for fixed-e nd arches, and showed that three redundant parameters can be found by the us e of three equations of coMPatibility.Wooden trusses were used in bridges during the sixteenth century when P alladio built triangular frames for bridge spans 10 feet long . This effort also f ocused on the three basic principles og bridge design : convenience(serviceabili ty) ,appearance , and endurance(strength) . several timber truss bridges were co nstructed in western Europe beginning in the 1750s with spans up to 200 feet (61m) supported on stone substructures .Significant progress was possible in t he United States and Russia during the nineteenth century ,prompted by the ne ed to cross major rivers and by an abundance of suitable timber . Favorable e conomic considerations included initial low cost and fast construction .The transition from wooden bridges to steel types probably did not begin until about 1840 ,although the first documented use of iron in bridges was the chain bridge built in 1734 across the Oder River in Prussia . The first truss completely made of iron was in 1840 in the United States , followed by Eng land in 1845 , Germany in 1853 , and Russia in 1857 . In 1840 , the first ir on arch truss bridge was built across the Erie Canal at Utica .The Impetus of AnalysisThe theory of structures ,developed mainly in the ninetheenth century,foc used on truss analysis, with the first book on bridges written in 1811. The Wa rren triangular truss was introduced in 1846 , supplemented by a method for c alculating the correcet forces .I-beams fabricated from plates became popular in England and were used in short-span bridges.In 1866, Culmann explained the principles of cantilever truss bridges, an d one year later the first cantilever bridge was built across the Main River in Hassfurt, Germany, with a center span of 425 feet (130m) . The first cantilever bridge in the United States was built in 1875 across the Kentucky River.A most impressive railway cantilever bridge in the nineteenth century was the Fir st of Forth bridge , built between 1883 and 1893 , with span magnitudes of 1 711 feet (521.5m).At about the same time , structural steel was introduced as a prime mater ial in bridge work , although its quality was often poor . Several early exampl es are the Eads bridge in St.Louis ; the Brooklyn bridge in New York ; and t he Glasgow bridge in Missouri , all completed between 1874 and 1883.Among the analytical and design progress to be mentioned are the contrib utions of Maxwell , particularly for certain statically indeterminate trusses ; the books by Cremona (1872) on graphical statics; the force method redefined by Mohr; and the works by Clapeyron who introduced the three-moment equation s.The Impetus of New MaterialsSince the beginning of the twentieth century , concrete has taken its place as one of the most useful and important structural materials . Because of the coMParative ease with which it can be molded into any desired shape , its st ructural uses are almost unlimited . Wherever Portland cement and suitable agg regates are available , it can replace other materials for certain types of structu res, such as bridge substructure and foundation elements .In addition , the introduction of reinforced concrete in multispan frames at the beginning of this century imposed new analytical requirements . Structures of a high order of redundancy could not be analyzed with the classical metho ds of the nineteenth century .The importance of joint rotation was already dem onstrated by Manderla (1880) and Bendixen (1914) , who developed relationshi ps between joint moments and angular rotations from which the unknown mom ents can be obtained ,the so called slope-deflection method .More simplification s in frame analysis were made possible by the work of Calisev (1923) , who used successive approximations to reduce the system of equations to one simpl e expression for each iteration step . This approach was further refined and integrated by Cross (1930) in what is known as the method of moment distributi on .One of the most import important recent developments in the area of anal ytical procedures is the extension of design to cover the elastic-plastic range , also known as load factor or ultimate design. Plastic analysis was introduced with some practical observations by Tresca (1846) ; and was formulated by Sa int-Venant (1870) , The concept of plasticity attracted researchers and engineers after World War Ⅰ, mainly in Germany , with the center of activity shifting to England and the United States after World War Ⅱ.The probabilistic approa ch is a new design concept that is expected to replace the classical determinist ic methodology.A main step forward was the 1969 addition of the Federal Highway Adim inistration (FHWA)”Criteria for Reinforced Concrete Bridge Members “ that co vers strength and serviceability at ultimate design . This was prepared for use in conjunction with the 1969 American Association of State Highway Offficials (AASHO) Standard Specification, and was presented in a format that is readil y adaptable to the development of ultimate design specifications .According to this document , the proportioning of reinforced concrete members ( including c olumns ) may be limited by various stages of behavior : elastic , cracked , an d ultimate . Design axial loads , or design shears . Structural capacity is the r eaction phase , and all calculated modified strength values derived from theoret ical strengths are the capacity values , such as moment capacity ,axial load ca pacity ,or shear capacity .At serviceability states , investigations may also be n ecessary for deflections , maximum crack width , and fatigue .Bridge TypesA notable bridge type is the suspension bridge , with the first example bu ilt in the United States in 1796. Problems of dynamic stability were investigate d after the Tacoma bridge collapse , and this work led to significant theoretica l contributions Steinman ( 1929 ) summarizes about 250 suspension bridges bu ilt throughout the world between 1741 and 1928 .With the introduction of the interstate system and the need to provide stru ctures at grade separations , certain bridge types have taken a strong place in bridge practice. These include concrete superstructures (slab ,T-beams,concrete b ox girders ), steel beam and plate girders , steel box girders , composite const ruction , orthotropic plates , segmental construction , curved girders ,and cable-stayed bridges . Prefabricated members are given serious consideration , while interest in box sections remains strong .Bridge Appearance and AestheticsGrimm ( 1975 ) documents the first recorded legislative effort to control t he appearance of the built environment . This occurred in 1647 when the Cou ncil of New Amsterdam appointed three officials . In 1954 , the Supreme Cou rt of the United States held that it is within the power of the legislature to de termine that communities should be attractive as well as healthy , spacious as well as clean , and balanced as well as patrolled . The Environmental Policy Act of 1969 directs all agencies of the federal government to identify and dev elop methods and procedures to ensure that presently unquantified environmenta l amentities and values are given appropriate consideration in decision making along with economic and technical aspects .Although in many civil engineering works aesthetics has been practiced al most intuitively , particularly in the past , bridge engineers have not ignored o r neglected the aesthetic disciplines .Recent research on the subject appears to lead to a rationalized aesthetic design methodology (Grimm and Preiser , 1976 ) .Work has been done on the aesthetics of color ,light ,texture , shape , and proportions , as well as other perceptual modalities , and this direction is bot h theoretically and empirically oriented .Aesthetic control mechanisms are commonly integrated into the land-use re gulations and design standards . In addition to concern for aesthetics at the sta te level , federal concern focuses also on the effects of man-constructed enviro nment on human life , with guidelines and criteria directed toward improving quality and appearance in the design process . Good potential for the upgrading of aesthetic quality in bridge superstructures and substructures can be seen in the evaluation structure types aimed at improving overall appearance .Lords and lording groupsThe loads to be considered in the design of substructures and bridge foun dations include loads and forces transmitted from the superstructure, and those acting directly on the substructure and foundation .AASHTO loads . Section 3 of AASHTO specifications summarizes the loa ds and forces to be considered in the design of bridges (superstructure and sub structure ) . Briefly , these are dead load ,live load , iMPact or dynamic effec t of live load , wind load , and other forces such as longitudinal forces , cent rifugal force ,thermal forces , earth pressure , buoyancy , shrinkage and long t erm creep , rib shortening , erection stresses , ice and current pressure , collisi on force , and earthquake stresses .Besides these conventional loads that are ge nerally quantified , AASHTO also recognizes indirect load effects such as fricti on at expansion bearings and stresses associated with differential settlement of bridge components .The LRFD specifications divide loads into two distinct cate gories : permanent and transient .Permanent loadsDead Load : this includes the weight DC of all bridge components , appu rtenances and utilities, wearing surface DW nd future overlays , and earth fill EV. Both AASHTO and LRFD specifications give tables summarizing the unit weights of materials commonly used in bridge work .Transient LoadsVehicular Live Load (LL) Vehicle loading for short-span bridges :considera ble effort has been made in the United States and Canada to develop a live lo ad model that can represent the highway loading more realistically than the H or the HS AASHTO models . The current AASHTO model is still the applica ble loading.桥梁工程和桥梁美学桥梁工程的发展概况早在公元前1世纪，Marcus Vitrucios Pollio 的著作中就有关于建筑材料和结构类型的记载和评述。
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高阶参数多项式逼近圆锥曲线Ga šper Jakli ˇ c ·Jernej Kozak ·Marjeta Krajnc · Vito Vitrih ·Emil Žagar收稿日期：2011年12月22日/修订：2013年2月7日/接受：2013年2月12日/ 网上公布：2013年5月9日©施普林格科学+商业媒体纽约2013摘要：在本文中，特定的形状保持的圆锥曲线参数多项式逼近进行了研究。
该方法是基于对隐含定义平面曲线的参数逼近。
推导出的多项式逼近给出了一个封闭的形式，并提供尽可能高的逼近阶。
关键词：圆锥曲线·参数曲线·隐式曲线·逼近1 介绍圆锥截面是CAGD （计算机辅助几何设计）和许多计算机图形系统默认的标准对象。
椭圆形和双曲线可以在参数形式使用，如三角函数和双曲函数来表示。
与此相反，以抛物线为例，它们没有一个参数多项式参数化，但它们可以被写成二次有理贝塞尔曲线。
在许多应用中，是需要的圆锥截面的参数多项式近似，所以推导准确的多项式逼近是重要的。
圆弧是在圆锥截面中最重要的几何对象实践。
很多论文考虑圆弧段的良好逼近径向误差作为参数距离。
二次贝塞尔逼近被认为是在[16]，以及它们的概括，以立方情况下，可以在[6]和[11]中找到。
四次的情况下进行了系统研究[1,14]和[12]，并五次贝塞尔逼近推导在[7]和[8]。
对圆锥曲线由参数埃尔米特型逼近总成绩奇数多项式曲线中给出[9]和[10]。
然而，结果保持真只渐近，也就是说，对于特定的圆锥截面的小片段。
通过三次参数贝塞尔曲线的椭圆段埃尔米特逼近研究在[5]，并在[4]。
圆形的泰勒式的内插一个有趣的封闭形式的解决方案弧由奇数度参数多项式曲线追溯到[15]。
在该文件中，作者构建了一个明确的公式参数多项式逼近。
结果被后来扩展到[13]中均匀度曲线。
作为一个动机，以改善在[15]和[13]所得到的结果，考虑下面的例子。
采取的特定参数五次多项式逼近单位圆，在给定的[13，15]中像⎪⎪⎭⎫ ⎝⎛+-+-534222221t t t t t （1） 它见图1中用新五次逼近在一起⎪⎪⎭⎫ ⎝⎛+++++++-5342)53()51()51()53(1t t t t t (2) 很明显，后者则具有更好的逼近性质。
这个论文其中一个目标是建立一个通用框架参数的构造的圆锥截面多项式逼近，例如（2）。
主要的问题是在单位圆和单位双曲线的近似，通过隐含给定的122=±y x (3)更精确地，两个非常多项式[]t R y x n n ∈, 必须找到像())(122t y t x n n ε+=± （4）对于一些函数ε是足够小的。
也就是说，一个好的参数多项式中查找隐含的表示（3）的逼近。
两个重要的问题出现了：如何选择一个合适的ε，什么是近似性隐含的ε？该公式的重要性（4）已经在[15]指出了。
即这个形式已经在奇数中成立n n n t t y t x 22221)()(+=+ （5） 然而，在[13]中，已经证明，公式（5）对所有的n ∈N.至少有一个真正的解决方案 它是根据一个特定的图1单位圆（虚线）由下式给出五次参数多项式逼近（1）（黑色），并通过（2）（灰色）给予相同程度的新参数逼近。
单位圆的参数和多项式n x 和n y 的系数被优雅表示在第一和第二的切比雪夫多项式。
然而这样是远离最优解（见图1）。
在本文中，所有形式如（5）和n n n t t y t x 22221)()(±=-被构造在一个封闭的形式，并且相对于最好的该逼近误差进行了详细研究。
所以这种近似是优良的，因为误差指数衰减了n 次。
本文的结构如下。
在第二节中对圆锥截面进行了研究，并且对近似的问题是精确定义。
在下面的部分中，一个结构是所有适当的解决方案概述。
在第4节最好的解决办法是仔细研究。
第5条处理的最佳解决方案的误差分析。
第6节是本文由结论在举出的一些数值例子。
2 圆锥曲线在本节中，隐含定义圆锥曲线参数多项式逼近将予以考虑。
由于抛物线具有多项式参数化，这里只研究椭圆和双曲线。
通过选择适当的坐标系中，它们被给定为12020=⎪⎭⎫ ⎝⎛-±⎪⎭⎫ ⎝⎛-b y y a x x另外，通过一个转换的坐标系和比例，上述等式可以改写到（3）。
考虑的主要问题是要找到一个参数曲线。
[]⎪⎪⎭⎫⎝⎛→)()(,,:2t y t x t R b a r n n （6）其中n x 和n y 是非恒定多项式，使得（4）成立，残差多项式ε最多2n 个。
由于在至少一个点上的圆锥曲线应当插入，让我们选择ε（0）= 0。
为了对近似误差要尽可能地小，ε不应涉及低度条件和经过nt2。
这也意味着，该圆弧的几何接触及其逼近将顺序的2n （见[1]）。
因此，我们正在寻找一种几何泰勒近似，以最高的接触顺序。
此外，不失一般性，我们可以假设1)0(,1)0(,1)0(,1)0(''====n n n n y y x x未知多项式因此可写为,:)(,1:)(22∑∑==++nn n n t b t t y t a t x该变换（4）为().112222222nn nnn t b a t b t t a ++=⎪⎭⎫ ⎝⎛+±⎪⎭⎫ ⎝⎛+∑∑== （7） 该公式实际上是2n-2非线性方程组进行的未知数()()nnb a 22== 和。
它可以进一步用合适的重新参数简化。
让.1:222nnnba A ±=线性参数标定A t t / 和新的变数,,...,2,1,:,:n A b A a === βα （8）取1:,0:11==b a ，将问题转化为寻找两个非常多项式,0,1:)(,1:)(112>+=+=∑∑==ββαnn n n t t y t t x(9)这样.)(1)()(22222nn n n n t b a sign t y t x ±+=± （10） 这个圆锥曲线有两种可能性，因为22nn b a =不可能发生，意味着只有椭圆形的情况。
需要注意的是类似的问题是关系到佩尔方程，并通过契贝谢夫多项式间（[2]）解决方案。
3解决方案求解方程（10）等效于解决1)()(22=±t y t x n n在阶乘环形[]ntt R 2/。
但是，因为有更多的限制（9），该问题不能由经典的代数工具处理。
幸运的是，有一个另一种方式，在该公式的特殊形式（10），使得能够直截了当地得到所有的解决方案，满足特殊要求。
让我们分别考虑每种情况。
在椭圆形的情况下，该公式（10）可以被改写为,sin cos :,)())()())(()((120212ϕϕϕπi e et t iy t x t iy t x i n k nk in n n n +=-=-+∏-=+ （11）其中右手侧的是1+nt2以上C 的因子。
即方程12-=nt可以通过欧拉公式πi e =-1解决并应用应用棣莫弗的标识计算根，.12,...,1,0,112122222-===-++n k eenk ink innππππ从多项式分解对C 的唯一的一个常数因子，并从该因子（11）中出现共轭对的事实，它遵循,1,,)()()(1212=∈-=+∏-=+γγγπσc et t iy t x n k nk i n n k当1±=k σ时，为了内插点（0，1），γ必须取,)1(:1212∏-=+--=n k nk i nkeπσγ这意味着),;(:)1()1()()(1212σπσt p tet iy t x e n k nk i nn n k=--=+∏-=+- （12）当(){}.1,110nn k k -∈=-=σσ这会导致n2个解，但那些具有01=β必须被排除在外。
由于其余的成对出现（n n y x ±,），他们一半要求β1>0。
表1：可以受理的解决方案在所有的三个数对于n =2,3，...，9现在让我们考虑双曲线情况。
类似于在（11）中，表达式（10）可改写为)1)cos(2()1(1))()())(()((11222+--=-=-+∏-=t nk t t tt y t x t y t x n k nn n n n π（13）在22nn b a < )1)212cos(2(1))()())(()((122++-=+=-+∏-=t nk t tt y t x t y t x n k nn n n n π （14）至于22nn b a >，通过使用根的分解中得到的右手侧统一并通过加入共轭复数因素平方得到真正的答案，现在是写的（13）和（14）的右手侧的两个多项式的产物h h q p 和和定义n n y x 和)).()((21)()),()((21)(t q t p t y t q t p t x h h n h h n -±=+=（15） 因为n n y x 和必须n ≤，两个产物h h q p 和必须在n 在内，否则n n y x 或的程度将是过高的。
因此，{})1)cos(2()1(:);(:)(1,...,2,122)1(1+-+Γ=∏-⊆Γ∈--t nk t t t p t p n k n h h nπ（16） 当22nn b a <，在分解（14）中，只有均匀度的因素是可用的。
一个解决方案从而只存在于偶数n ，和{})1)212cos(2(:);(:)(1,...,2,12++-=Γ=∏-⊆Γ∈t nk t t p t p n k n h h π （17） 如在椭圆的情况下，与β1 =0的解，必须排除，并从其余的对（n n y x ±,），那些β1> 0被保留。
允许解的个数用正呈几何级数增长，如可从表1可见用最小逼近特定解决方案的选择 错误和其明确公式将在下一节给出。
4 最佳解决方案对于这两个椭圆和双曲线的情况下，最好的解决方案是一个与最大可能的β1>0。
这可以清楚地从（4）及（8）中体现，由于误差可见，在给定的参数将是最小所以A 尽可能大。
定理1 为椭圆形的情况下，最好的解决方法是:,)1()),;(Im()()),;(Re()(10***-====n k e n e n t p t y t p t x σσσ并为双曲线情况下，最好的办法是:)),;();((21)()),;();((21)(****n h n h n n h n h n t p t p t y t p t p t x Γ--Γ=Γ-+Γ=其中Ph 是用于奇数n 和由（17），用于通过（16）中定义的偶数n ，.1,...,121,21*⎭⎬⎫⎩⎨⎧-+⎥⎦⎥⎢⎣⎢+⎥⎦⎥⎢⎣⎢+=Γn n n n 在所有的情况下啊，证明首先考虑的椭圆情况。