五点求导分段 三次多项式曲线光滑法

• The basic principle of this method is to set up a third order polynomial equation for a curve between two successive points, and it is demanded that the whole line has the continuous first order derivatives to insure the smoothness of the line. • The first order derivatives on each of the two successive points is determined by this point plus last two points and next two points (5 points altogether).
y=q0+q1z+q2z2+q3z3
In these equations, pi, qi (i=0,1,2,3) are constants, z is the parameter, when the curve is being generated from Pi (xi, yi) to Pi+1 (xi+1, yi+1), z is changing from 0 to 1.
• The stipulated geometric condition for the tangential line passing through the middle point:
Third Order Polynomials with Derivatives from Five Points (2)
are uncertain, let w2=w3=1
Third Order Polynomials with Derivatives from Five Points (5)
Let the third order polynomial equations between the two successive points Pi and Pi+1 be: x=p0+p1z+p2z2+p3z3
w 2b 2 w 3b 3 t3 w 2a 2 w 3a 3 wh e rea i x i 1 x i b i y i 1 y i w 2 S 34 (i 2, 3) w 3 S12
S 34 a 3b 4 a 4b 3 S12 a1b 2 a 2b1
P1 P5
P2 P4 P3 C A B D
P2C P4 D C A DB
Third Order Polynomials with Derivatives from Five Points (3)
• Through mathematical derivation, the derivative t3 on the point P3 can be expressed as:
Third Order Polynomials with Derivatives from Three Points (2)
• The derivative on every given point is determined by the relative positions of the last and next points to this point. Therefore, three successive points are used to calculate the derivative of the curve on the middle point • Let the three points be Pi-1, Pi, Pi+1, their coordinates are (xi-1, yi-1), (xi, yi), (xi+1, yi+1) respectively
Third Order Polynomials with Derivatives from Five Points (6)
These equations should satisfy the following conditions:
When z=0, x=xi, y=yi, dx/dz=rcosθi, dy/dz=rsinθi,
Requirements to a Line Smoothing Method
• Although it is a string of line segments, the visual impression of it should be a smooth line • Not only the pieces of a line between the given adjacent points should be connected, the derivatives of at least the first order must be continuous at the junction points • The figures of the lines on the two sides of a junction point and near to it should be symmetric • Be able to avoid self-intersection as much as possible
When z=1, x=xi+1, y=yi+1, dx/dz=rcosθi+1, dy/dz=rsinθi+1.
Where r=[(xi+1-xi)2+(yi+1-yi)2]1/2
θi Pi+1 θi+1
Leabharlann Baidu
Pi
Third Order Polynomials with Derivatives from Five Points (7)
Third Order Polynomials with Derivatives from Five Points (4)
In order to be b0 convenient, we use θ cosθ3 and sinθ3 for t3 a0 (i.e., tgθ3), and the numerator and the sin b / a 2 b 2 cos a / a 2 b 2 3 0 0 0 3 0 0 0 denominator of the whe rea w a w a b w b w b 0 2 2 3 3 0 2 2 3 3 expression for t3 are w 2 a3b 4 a 4b3 w 3 a1b 2 a 2b1 seen as the two a i x i 1 x i b i y i 1 y i (i 1, 2, 3, 4) edges of a right triangle: When w2=w3=0, cos θ3 and sin θ3
Third Order Polynomials with Derivatives from Five Points (8)
• Supplement of two points on each end for an open line – Suppose that the first three given points (x3,y3), (x4, y4), (x5, y5), and the two points to be supplied (x2,y2) and (x1,y1) are all on the following line: x=g0+g1z+g2z2 y=h0+h1z+h2z2
Third Order Polynomials with Derivatives from Five Points (10)
• Advantages: – To be strict mathematically – Relatively simple in calculation – The resulted curve can pass through every given point exactly while has the continuous first order derivatives everywhere on the line – When given points are relatively dense, the resulted curve will be satifactory
From these conditions, pi and qi (i=0,1,2,3) can be uniquely defined: p0=xi p1=rcosθi p2=3(xi+1-xi)-r(cosθi+1+2cosθi) p3=-2(xi+1-xi)+r(cosθi+1+cosθi) q0=yi q1=rsinθi q2=3(yi+1-yi)-r(sinθi+1+2sinθi) q3=-2(yi+1-yi)+r(sinθi+1+sinθi)
Third Order Polynomials with Derivatives from Five Points (9)
where gk, hk (k=0,1,2) are constants, z is the parameter, and suppose that when z=j, x=xj, y=yj (j=1,2,3,4,5), then: x2=3x3-3x4+x5 x1=3x2-3x3+x4 y2=3y3-3y4+y5 y1=3y2-3y3+y4 The supplement of the points after the last given point is similar to this.
Piecewise Line Smoothing Using Third Order Polynomials with Derivatives from Five Points
五点求导分段 三次多项式曲线光滑法
Third Order Polynomials with Derivatives from Five Points (1)
Piecewise Line Smoothing Using Third Order Polynomials with Derivatives from Three Points
三点求导分段 三次多项式曲线光滑法
Third Order Polynomials with Derivatives from Three Points (1)
The basic principle of this method is similar with the method used in the piecewise line interpolation using the three order polynomial with derivatives from five points. The main difference lies in the method of determination for the derivatives.
Third Order Polynomials with Derivatives from Five Points (11)
• Disadvantages: – The graphic representation would not be ideal at the place of the sudden turning of the curve – For the successive zigzag meanders, the curve sometimes intersects with itself