第3章〓近似样条
3．1引言

前面的章节讨论了如何使用控制点(CP)和调和函数(BF)来表征插值样条和混合样条。贝塞尔样条曲线是图形包中用于建模样条曲线的非常流行和广泛使用的工具。然而，贝塞尔样条有两个主要缺点，这一直是设计近似样条的动机。第一个缺点是CP的数量取决于曲线的阶数，即不可能在不增加多项式阶数和计算复杂度的情况下增加CP的数量以实现更平滑的控制。第二个缺点是没有提供局部控制，即移动单个CP的位置将改变整个样条的形状，而不是样条的局部部分。在局部图形中，控制通常是可取的，因为它可以对样条进行小幅调整［Hearn and Baker，1996］。

为了克服这些缺点，已经提出了一种称为B样条(Bspline)的新型样条。B样条代表“基本样条”，它们是真正的近似值，即通常它们不经过任何CP，但在特殊条件下它们可能被迫这样做。本质上，B样条由多个在连接点处具有连续性的曲线段组成。这可以实现局部控制，即当移动CP时，仅更改一组局部曲线段而不是整个曲线。B样条的BF是使用算法(称为Cox de Boor算法)来计算的，该算法以德国数学家CarlWilhelm Reinhold de Boor命名。连接点处的参数变量t的值存储在称为“结向量(KV)”的向量中。若结点值是等间距的，则生成的样条称为均匀B样条； 否则，称为非均匀B样条。当KV值重复时，称为开放均匀B样条。

BF和B样条方程不是单个实体，而是每个段的表达式集合。例如，若有4个段，分别指定为A、B、C和D，则每个BF(记为B)具有4个子分量BA、BB、BC和BD，并表示为B={BA，BB，BC，BD}。若有3个CP，即P0、P1和P2，则将有3个BF： B0、B1和B2，每个都有4个子分量，即B0A，…，B0D，B1A，…，B1D和B2A，…，B2D。通常每行一个BF表示为
B0={B0A，B0B，B0C，B0D}
B1={B1A，B1B，B1C，B1D}
B2={B2A，B2B，B2C，B2D}
类似地，曲线方程P也将具有4个子分量，P={PA，PB，PC，PD}。我们之前已经看到，曲线方程可以表示为CP和BF的乘积，因此子分量可以写为： PA=P0．B0A+P1．B1A+P2．B2A，PB=P0．B0B+P1．B1B+P2．B2B等。注意，子分量对t的不同范围有效，因此不能相加，但需要表示为值矩阵。通常，段A在t0≤ t < t1 范围内有效，段B在t1≤ t < t2范围内有效，以此类推。由于曲线表达式可能非常大，在大多数情况下，方程是垂直而不是水平编写的，每个段的值范围都提到了，如下所示： 
P=PA=P0．B0A+P1．B1A+P2．B2A(t0≤t<t1)［段A］PB=P0．B0B+P1．B1B+P2．B2B(t1≤t<t2)［段B］PC=P0．B0C+P1．B1C+P2．B2C(t2≤t<t3)［段C］
以下各节将说明有关B和P值计算的详细信息。

3．2线性均匀B样条

B样条有两个定义参数： d与样条的阶数有关，n与CP的数量有关。样条的阶数实际上是(d-1)，CP的数量是(n+1)。其他相关参数的推导如下所述。

对于线性B样条，我们从d=2和n=2开始： 

曲线的阶数： d-1=1。

CP数量： n+1=3。

BF数量： n+1=3。 

曲线段数： d+n=4。

KV中的元素数： d+n+1=5。





将曲线段指定为A、B、C和D，CP为P0、P1和P2。将KV中的元素指定为{tk}，其中k在值{0，1，2，3，4}上循环。在这种情况下，KV为T=［t0，t1，t2，t3，t4］。让BF以Bk,d的形式指定。由于有3个CP，曲线的BF由B0,2、B1,2和B2,2给出，更高的k值在此不相关。曲线方程由下式给出： 
P(t)=P0B0,2+P1B1,2+P2B2,2(3．1)


图3．1具有3个CP的线性均匀B样条

图3．1是一个示意图，显示了3个CP，即P0、P1和P2如何与3个BF，即B0,2、B1,2和B2,2，4个段A、B、C和D以及五元KV，即［t0，t1，t2，t3，t4］相关。无论CP的实际位置如何，它们对特定段的影响保持不变。CP中，P0对段A和B施加影响，P1对段B和C施加影响，P2对段C和D施加影响。这提供了B样条的局部控制特性： 改变一个CP的位置只改变样条的两个段，而样条曲线的其余部分保持不变。图中虚线表示影响范围。在本节的其余部分，我们提供了对这一事实的验证。

BF的表达式是使用Cox de Boor算法［Hearn and Baker，1996］来计算的，如下所示： 
Bk,1=1,iftk≤t≤tk+1,else 0
Bk,d=t-tktk+d-1-tkBk,d-1+tk+d-ttk+d-tk+1Bk+1,d-1(3．2)
第一行表明，若特定段的t值在特定范围内，则BF的d=1的一阶项等于1，否则为0。第二行说明高阶项如何计算，对d>1，根据一阶项计算。该算法将在后面说明。需要注意的一点是，该算法本质上是递归的，因为与d相关的高阶项是根据与d-1阶相关的低阶项计算的。因此，即使曲线方程只需要有式(3．1)中所示的二阶项Bk,2，还是首先需要计算一阶项Bk,1。

要找到BF和曲线方程，必须分别分析每一段。但首先，需要KV的数值。让我们做一个简化的假设，KV值如下： t0=0、t1=1、t2=2、t3=3、t4=4。因此KV变为T=［0，1，2，3，4］。在适当时，我们将证明这个假设是正确的。

段A： t0≤ t < t1： 
B0,1=1,B1,1=0,B2,1=0,B3,1=0,B4,1=0B0,2=t-t0t1-t0B0,1+t2-tt2-t1B1,1=tB1,2=t-t1t2-t1B1,1+t3-tt3-t2B2,1=0B2,2=t-t2t3-t2B2,1+t4-tt4-t3B3,1=0
段B： t1≤ t < t2： 
B0,1=0,B1,1=1,B2,1=0,B3,1=0,B4,1=0B0,2=t-t0t1-t0B0,1+t2-tt2-t1B1,1=2-tB1,2=t-t1t2-t1B1,1+t3-tt3-t2B2,1=t-1B2,2=t-t2t3-t2B2,1+t4-tt4-t3B3,1=0
段C： t2≤ t < t3： 
B0,1=0,B1,1=0,B2,1=1,B3,1=0,B4,1=0B0,2=t-t0t1-t0B0,1+t2-tt2-t1B1,1=0B1,2=t-t1t2-t1B1,1+t3-tt3-t2B2,1=3-tB2,2=t-t2t3-t2B2,1+t4-tt4-t3B3,1=t-2
段D： t3≤ t < t4： 
B0,1=0,B1,1=0,B2,1=0,B3,1=1,B4,1=0B0,2=t-t0t1-t0B0,1+t2-tt2-t1B1,1=0

B1,2=t-t1t2-t1B1,1+t3-tt3-t2B2,1=0B2,2=t-t2t3-t2B2,1+t4-tt4-t3B3,1=4-t
注意，每段都有一个特定的k值，但我们在计算BF时仍然在其他可能的值上循环k，因为我们想找出其他CP对那个段的影响。例如，对于段A，B0,2=0，但是我们计算B1,2和B2,2来找出第二个CP和第三个CP对段A的影响。在这种情况下，我们看到B1,2和B2,2为零，表示段A仅受第一个CP即P0控制，不受其他CP即P1和P2影响，见式(3．1)。同样对于段B，k=1，但我们看到B0,2和B1,2都是非零值，这意味着段B受两个CP,即P0和P1的影响。

现在根据式(3．1)，确定曲线公式所需的BF为B0,2、B1,2和B3,2。然而，这些BF对于不同的段具有不同的值。因此，需要将所有这些值收集在一起，指定它们有效的段。这里使用单独的下标A、B、C和D来表示相关段。
B0,2={B0,2A,B0,2B,B0,2C,B0,2D}
B1,2={B1,2A,B1,2B,B1,2C,B1,2D}
B2,2={B2,2A,B2,2B,B2,2C,B2,2D}(3．3)
表3．1总结了BF值的分段计算。如前所述，假设KV为T=［0，1，2，3，4］。


表3．1线性均匀B样条的BF计算



段tBk,1Bk,2

At0≤t< t1
B0,1=1B0,2=t

B1,1=0B1,2=0

B2,1=0B2,2=0

B3,1=0

Bt1≤t<t2
B0,1=0B0,2=2-t

B1,1=1B1,2=t-1

B2,1=0B2,2=0

B3,1=0


Ct2≤t<t3
B0,1=0B0,2=0

B1,1=0B1,2=3-t

B2,1=1B2,2=t-2

B3,1=0


Dt3≤t<t4
B0,1=0B0,2=0

B1,1=0B1,2=0

B2,1=0B2,2=4-t

B3,1=1




将上述值代入式(3．3)，得到： 
B0,2={t,2-t,0,0}
B1,2={0,t-1,3-t,0}B2,2={0,0,t-2,4-t}(3．4)
式(3．4)指定了具有3个CP(由3个垂直行表示)和4个CP(由每行4个元素向量表示)的线性均匀B样条的BF。

式(3．5)显示了BF的另一种表示方式，其中仅指示非零值以及它们有效的段名称和t的范围。
B0,2=t(0≤t<1)A2-t(1≤t<2)B
B1,2=t-1(1≤t<2)B3-t(2≤t<3)C
B2,2=t-2(2≤t<3)C4-t(3≤t<4)D(3．5)
式(3．5)表明B0,2并因此和P0影响分段A和B，B1,2和P1影响分段B和C，而B2,2和P2影响分段C和D，这一事实已由图3．1中的虚线指出。

BF的图如图3．2所示。每个BF具有相同的形状，但相对于前一个向右移动1。因此，每个BF都可以通过将t替换为(t-1)从前一个BF中获得，如式(3．5)所示。同样如式(3．4)所示，每个BF有四个细分，其中两个是非零的。B0,2的第一条曲线对于分段A(0≤t<1)和B(1≤t<2)具有非零部分; B1,2的第二条曲线对于分段B(1≤t<2)和C(2≤t<3)，B2,2的第三条曲线对于段C(2≤t<3)和D(3≤t<4)也具有非零部分。由于BF与CP相关联，这再次提供了对样条局部控制属性的指示，即第一个CP对前两个段A和B有影响，第二个CP对B和C有影响，以此类推。这意味着如果第一个CP改变，它将只影响4条线段中的两条，而样条线的其余部分将保持不变。这与每个BF在整个t范围内有效的插值曲线和混合曲线形成对比。



图3．2线性均匀B样条的BF


样条方程是其四段方程的集合： 
P(t)=PA(0≤t<1)PB(1≤t<2)PC(2≤t<3)PD(3≤t<4)(3．6)
其中，
PA=P0B0,2A+P1B1,2A+P2B2,2APB=P0B0,2B+P1B1,2B+P2B2,2BPC=P0B0,2C+P1B1,2C+P2B2,2CPD=P0B0,2D+P1B1,2D+P2B2,2D(3．7)
将表3．1中的BF值代入式(3．6)，可以得到：
P(t)=P0t(0≤t<1)P0(2-t)+P1(t-1)(1≤t<2)P1(3-t)+P2(t-2)(2≤t<3)P2(4-t)(3≤t<4)(3．8)
式(3．8)表示具有3个CP和4个段的线性均匀B样条方程。这4部分是4个段的方程。

例3．1找到具有CP(2,-3)、(5，5)和(8,-1)的线性均匀B样条的方程。还要编写一个程序来绘制BF。

解： 

根据式(3．8)，代入给定CP的值： 
x(t)=2t(0≤t<1)3t-1(1≤t<2)3t-1(2≤t<3)-8t+32(3≤t<4)
y(t)=-3t(0≤t<1)8t-11(1≤t<2)-6t+17(2≤t<3)t-4(3≤t<4)

MATLAB Code 3．1

clear all; format compact; clc;

t0 = 0; t1 = 1; t2 = 2; t3 = 3; t4 = 4;

T = ［t0, t1, t2, t3, t4］;

syms t P0 P1 P2;



%Segment A

B01 = 1; B11 = 0; B21 = 0; B31 = 0; B41 = 0; B51 = 0; B61 = 0;

S1 = (t - t0)/(t1 - t0); S2 = (t2 - t)/(t2 - t1); B02 = S1*B01 + S2*B11; B02A = B02;

S1 = (t - t1)/(t2 - t1); S2 = (t3 - t)/(t3 - t2); B12 = S1*B11 + S2*B21; B12A = B12;

S1 = (t - t2)/(t3 - t2); S2 = (t4 - t)/(t4 - t3); B22 = S1*B21 + S2*B31; B22A = B22;



%Segment B

B01 = 0; B11 = 1; B21 = 0; B31 = 0; B41 = 0; B51 = 0; B61 = 0;

S1 = (t - t0)/(t1 - t0); S2 = (t2 - t)/(t2 - t1); B02 = S1*B01 + S2*B11; B02B = B02;

S1 = (t - t1)/(t2 - t1); S2 = (t3 - t)/(t3 - t2); B12 = S1*B11 + S2*B21; B12B = B12;

S1 = (t - t2)/(t3 - t2); S2 = (t4 - t)/(t4 - t3); B22 = S1*B21 + S2*B31; B22B = B22;



%Segment C

B01 = 0; B11 = 0; B21 = 1; B31 = 0; B41 = 0; B51 = 0; B61 = 0;

S1 = (t - t0)/(t1 - t0); S2 = (t2 - t)/(t2 - t1); B02 = S1*B01 + S2*B11; B02C = B02;

S1 = (t - t1)/(t2 - t1); S2 = (t3 - t)/(t3 - t2); B12 = S1*B11 + S2*B21; B12C = B12;

S1 = (t - t2)/(t3 - t2); S2 = (t4 - t)/(t4 - t3); B22 = S1*B21 + S2*B31; B22C = B22;



%Segment D

B01 = 0; B11 = 0; B21 = 0; B31 = 1; B41 = 0; B51 = 0; B61 = 0;

S1 = (t - t0)/(t1 - t0); S2 = (t2 - t)/(t2 - t1); B02 = S1*B01 + S2*B11; B02D = B02;

S1 = (t - t1)/(t2 - t1); S2 = (t3 - t)/(t3 - t2); B12 = S1*B11 + S2*B21; B12D = B12;

S1 = (t - t2)/(t3 - t2); S2 = (t4 - t)/(t4 - t3); B22 = S1*B21 + S2*B31; B22D = B22;

fprintf('Blending functions :＼n');

B02 = ［B02A, B02B, B02C, B02D］

B12 = ［B12A, B12B, B12C, B12D］

B22 = ［B22A, B22B, B22C, B22D］

fprintf('＼n');

fprintf('General Equation of Curve :＼n');

P = P0*B02 + P1*B12 + P2*B22

fprintf('＼n');

x0 = 2; x1 = 5; x2 = 8;

y0 = -3; y1 = 5; y2 = -1;

fprintf('Actual Equation :＼n');

x = subs(P, (［P0, P1, P2］), (［x0, x1, x2］))

y = subs(P, (［P0, P1, P2］), (［y0, y1, y2］))



%plotting BFs

tta = linspace(t0, t1);

ttb = linspace(t1, t2);

ttc = linspace(t2, t3);

ttd = linspace(t3, t4);

B02aa = subs(B02A, t, tta);

B02bb = subs(B02B, t, ttb);

B02cc = subs(B02C, t, ttc);

B02dd = subs(B02D, t, ttd);

B12aa = subs(B12A, t, tta);

B12bb = subs(B12B, t, ttb);

B12cc = subs(B12C, t, ttc);

B12dd = subs(B12D, t, ttd);

B22aa = subs(B22A, t, tta);

B22bb = subs(B22B, t, ttb);

B22cc = subs(B22C, t, ttc);

B22dd = subs(B22D, t, ttd);

figure

plot(tta, B02aa, 'k-', ttb,B02bb, 'k--', ttc,B02cc, 'k-.', ttd,B02dd, 'k:'); hold on;

plot(tta, B12aa, 'k-', ttb,B12bb, 'k--', ttc,B12cc, 'k-.', ttd,B12dd, 'k:');

plot(tta, B22aa, 'k-', ttb,B22bb, 'k--', ttc,B22cc, 'k-.', ttd,B22dd, 'k:'); hold off;

xlabel('t'); ylabel('B');

legend('A', 'B', 'C', 'D');

text(0.6, 0.9, 'B02'); text(1.6, 0.9, 'B12'); text(2.6, 0.9, 'B22');

注解

有关BF的图，请参见图3.2。



3．3改变控制点的数量

设计B样条的目标之一是进行局部控制，我们在3.2节中已经看到这一事实是合理的，因为不同的CP只影响特定的曲线段而不是整个曲线。另一个目标是使CP的数量与曲线的阶数无关。为了研究这一点，让我们现在将CP的数量增加1个，同时保持与以前相同的阶次，并找出这样的组合是否会产生有效的曲线方程。

对于这种情况，我们从d=2和n=3开始： 

曲线的阶数： d-1=1。

CP数量： n+1=4。

曲线段数： d+n=5。



图3．3具有4个CP的线性均匀

B样条

将曲线段指定为A、B、C、D和E(见图3．3)，CP为P0、P1、P2和P3。

KV中的元素数量： d+n+1=6。

将KV中的元素指定为T={tk}，其中k在值{0，1，2，3，4，5}上循环。在这种情况下，T=［t0，t1，t2，t3，t4，t5］。

BF的数量与CP的数量相同，即4。将BF指定为Bk,d。在这种情况下，BF是B0,2、B1,2、B2,2和B3,2。较高的k值无关紧要，因此不需要计算。曲线方程由下式给出： 

P(t)=P0B0,2+P1B1,2+P2B2,2+P3B3,2(3．9)
如果遵循3.2节中概述的过程，读者将能够验证这确实产生了有效的B样条。因此，它被留作练习。下面给出最终的BF和曲线方程作为参考。与式(3．5)和式(3．8)进行比较以查看差异。
B0,2=t(0≤t<1)2-t(1≤t<2)
B1,2=t-1(1≤t<2)3-t(2≤t<3)
B2,2=t-2(2≤t<3)4-t(3≤t<4)
B3,2=t-3(3≤t<4)5-t(4≤t<5)
P(t)=P0t(0≤t<1)P0(2-t)+P1(t-1)(1≤t<2)P1(3-t)+P2(t-2)(2≤t<3)P2(4-t)+P3(t-3)(3≤t<4)
P3(5-t)(4≤t<5)
3．4二次均匀B样条

为了生成二次B样条，我们从d=3和n=3开始： 

曲线的阶数： d-1=2。

CP数量： n+1=4。

BF数量： n+1=4。

曲线段数： d+n=6。

KV中的元素数： d+n+1=7。



图3．4具有4个CP的二次均匀B样条

设曲线段为A、B、C、D、E和F，CP为P0、P1、P2和P3(见图3．4)。

对于k={0，1，2，3，4，5，6}，令KV为T={tk}。在这种情况下，T=［t0，t1，t2，t3，t4，t5，t6］。

设BF为B0,3、B1,3、B2,3和B3,3。

曲线方程为
P(t)=P0B0,3+P1B1,3+P2B2,3+P3B3,3(3．10)
如前所述，我们假设KV为T=［0，1，2，3，4，5，6］。根据Cox de Boor算法的第一个条件，一阶项B0,3、B0,1、B1,1、B2,1、B3,1和B4,1将为0或1。根据算法的第二个条件，二阶项计算如下： 
B0,2=(t-0)B0,1+(2-t)B1,1B1,2=(t-1)B1,1+(3-t)B2,1B2,2=(t-2)B2,1+(4-t)B3,1B3,2=(t-3)B3,1+(5-t)B4,1B4,2=(t-4)B4,1+(6-t)B5,1(3．11)
这里也有三阶项，可由二阶项计算得出： 
B0,3=12(t-0)B0,2+12(3-t)B1,2B1,3=12(t-1)B1,2+12(4-t)B2,2B2,3=12(t-2)B2,2+12(5-t)B3,2B3,3=12(t-3)B3,2+12(6-t)B4,2(3．12)
由于有6部分，每个BF包含6个子分量： 
B0,3={B0,3A,B0,3B,B0,3C,B0,3D,B0,3E,B0,3F}B1,3={B1,3A,B1,3B,B1,3C,B1,3D,B1,3E,B1,3F}B2,3={B2,3A,B2,3B,B2,3C,B2,3D,B2,3E,B2,3F}B3,3={B3,3A,B3,3B,B3,3C,B3,3D,B3,3E,B3,3F}(3．13)
表3．2总结了BF值的计算。


表3．2二次均匀B样条的BF计算



段tBk,1Bk,2Bk,3

A0≤t<1
B0,1=1B0,2=tB0,3=(1/2)t2

B1,1=0B1,2=0B1,3=0


A0≤t<1
B2,1=0B2,2=0B2,3=0

B3,1=0B3,2=0B3,3=0

B4,1=0B4,2=0

B5,1=0

B1≤t<2
B0,1=0B0,2=2-tB0,3=-t2+3t-(3/2)

B1,1=1B1,2=t-1B1,3=(1/2)(t-1)2

B2,1=0B2,2=0B2,3=0

B3,1=0B3,2=0B3,3=0

B4,1=0B4,2=0

B5,1=0


C2≤t<3
B0,1=0B0,2=0B0,3=(1/2)(t-3)2

B1,1=0B1,2=3-tB1,3=-t2+5t-(11/2)

B2,1=1B2,2=t-2B2,3=(1/2)(t-2)2

B3,1=0B3,2=0B3,3=0

B4,1=0B4,2=0

B5,1=0


D3≤t<4
B0,1=0B0,2=0B0,3=0

B1,1=0B1,2=0B1,3=(1/2)(t-4)2

B2,1=0B2,2=4-tB2,3=-t2+7t-(23/2)

B3,1=1B3,2=t-3B3,3=(1/2)(t-3)2

B4,1=0B4,2=0

B5,1=0


E4≤t<5
B0,1=0B0,2=0B0,3=0

B1,1=0B1,2=0B1,3=0

B2,1=0B2,2=0B2,3=(1/2)(t-5)2

B3,1=0B3,2=5-tB3,3=-t2+9t-(39/2)

B4,1=1B4,2=t-4

B5,1=0


F5≤t<6
B0,1=0B0,2=0B0,3=0

B1,1=0B1,2=0B1,3=0

B2,1=0B2,2=0B2,3=0

B3,1=0B3,2=0B3,3=(1/2)(t-6)2

B4,1=0B4,2=6-t

B5,1=1



将上述值代入式(3．13)，得到： 
B0,3=(1/2)t2(0≤t<1)A-t2+3t-(3/2)(1≤t<2)B(1/2)(t-3)2(2≤t<3)C

B1,3=(1/2)(t-1)2(1≤t<2)B-t2+5t-(11/2)(2≤t<3)C(1/2)(t-4)2(3≤t<4)DB2,3=(1/2)(t-2)2(2≤t<3)C-t2+7t-(23/2)(3≤t<4)D(1/2)(t-5)2(4≤t<5)EB3,3=(1/2)(t-3)2(3≤t<4)D-t2+9t-(39/2)(4≤t<5)E(1/2)(t-6)2(5≤t<6)F(3．14)

式(3．14)表示具有4个CP和6个段的二次均匀B样条的BF。BF的图如图3．5所示。每个BF具有相同的形状，但相对于前一个BF向右移动1。因此，每个BF可以通过将t替换为(t-1)从前一个BF中获得。每个BF有6个细分，其中3个是非零的。B0,3的第一条曲线对于段A(0≤t<1)、B(1≤t<2)、C(2≤t<3)具有非零部分，B1,3的第二条曲线对于段B(1≤t<2)、C(2≤t<3)、D(3≤t<4)具有非零部分，B2,3的第三条曲线对于段C(2≤t<3)、D(3≤t<4)、E(4≤t<5)具有非零部分，B3,3的第四条曲线对于段D(3≤t<4)、E(4≤t<5)、F(5≤t<6)具有非零部分。



图3．5二次均匀B样条的BF


样条方程是其六段方程的集合： 
P(t)=PA(0≤t<1)PB(1≤t<2)PC(2≤t<3)PD(3≤t<4)PE(4≤t<5)PF(5≤t<6)(3．15)
其中，
PA=P0B0,3A+P1B1,3A+P2B2,3A+P3B3,3APB=P0B0,3B+P1B1,3B+P2B2,3B+P3B3,3BPC=P0B0,3C+P1B1,3C+P2B2,3C+P3B3,3CPD=P0B0,3D+P1B1,3D+P2B2,3D+P3B3,3DPE=P0B0,3E+P1B1,3E+P2B2,3E+P3B3,3EPF=P0B0,3F+P1B1,3F+P2B2,3F+P3B3,3F(3．16)
将表3．2中的BF值代入式(3．15)并指定它们的有效范围，可以得到： 
P(t)=P0(1/2)t2(0≤t<1)P0(-t2+3t-3/2)+P1(1/2)(t-1)2(1≤t<2)P0(1/2)(t-3)2+P1(-t2+5t-11/2)+P2(1/2)(t-2)2(2≤t<3)P1(1/2)(t-4)2+P2(-t2+7t-23/2)+P3(1/2)(t-3)2(3≤t<4)P2(1/2)(t-5)2+P3(-t2+9t-39/2)(4≤t<5)P3(1/2)(t-6)2(5≤t<6)(3．17)
式(3．17)表示具有4个CP和6个段的二次均匀B样条方程。方程的6部分代表曲线的6个段。

例3．2求具有CP，即P0(1，2)、P1(4，1)、P2(6，5)和P3(8,-1)的均匀二次B样条方程。还要编写一个程序来绘制BF和实际曲线。

解： 

根据式(3．17)，代入给定CP的值(见图3．6)： 
x(t)=t2/2(0≤t<1)t2-t+1/2(1≤t<2)-t2/2+5t-11/2(2≤t<3)2t-1(3≤t<4)-5t2+42t-81(4≤t<5)4(t-6)2(5≤t<6)
y(t)=t2(0≤t<1)-3t2/2+5t-5/2(1≤t<2)5t2/2-11t+27/2(2≤t<3)-5t2+34t-54(3≤t<4)7t2/2-34t+82(4≤t<5)-(1/2)(t-6)2(5≤t<6)


图3．6例3．2的绘图



MATLAB Code 3．2

clear all; format compact; clc;

t0 = 0; t1 = 1; t2 = 2; t3 = 3; t4 = 4; t5 = 5; t6 = 6;

T = ［t0, t1, t2, t3, t4, t5, t6］;

syms t P0 P1 P2 P3;



%Segment A

B01 = 1; B11 = 0; B21 = 0; B31 = 0; B41 = 0; B51 = 0; B61 = 0;

S1 = (t - t0)/(t1 - t0); S2 = (t2 - t)/(t2 - t1); B02 = S1*B01 + S2*B11; B02A = B02;

S1 = (t - t1)/(t2 - t1); S2 = (t3 - t)/(t3 - t2); B12 = S1*B11 + S2*B21; B12A = B12;

S1 = (t - t2)/(t3 - t2); S2 = (t4 - t)/(t4 - t3); B22 = S1*B21 + S2*B31; B22A = B22;

S1 = (t - t3)/(t4 - t3); S2 = (t5 - t)/(t5 - t4); B32 = S1*B31 + S2*B41; B32A = B32;

S1 = (t - t4)/(t5 - t4); S2 = (t6 - t)/(t6 - t5); B42 = S1*B41 + S2*B51; B42A = B42;

S1 = (t - t0)/(t2 - t0); S2 = (t3 - t)/(t3 - t1); B03 = S1*B02 + S2*B12; B03A = B03;

S1 = (t - t1)/(t3 - t1); S2 = (t4 - t)/(t4 - t2); B13 = S1*B12 + S2*B22; B13A = B13;

S1 = (t - t2)/(t4 - t2); S2 = (t5 - t)/(t5 - t3); B23 = S1*B22 + S2*B32; B23A = B23;

S1 = (t - t3)/(t5 - t3); S2 = (t6 - t)/(t6 - t4); B33 = S1*B32 + S2*B42; B33A = B33;



%Segment B

B01 = 0; B11 = 1; B21 = 0; B31 = 0; B41 = 0; B51 = 0; B61 = 0;

S1 = (t - t0)/(t1 - t0); S2 = (t2 - t)/(t2 - t1); B02 = S1*B01 + S2*B11; B02B = B02;

S1 = (t - t1)/(t2 - t1); S2 = (t3 - t)/(t3 - t2); B12 = S1*B11 + S2*B21; B12B = B12;

S1 = (t - t2)/(t3 - t2); S2 = (t4 - t)/(t4 - t3); B22 = S1*B21 + S2*B31; B22B = B22;

S1 = (t - t3)/(t4 - t3); S2 = (t5 - t)/(t5 - t4); B32 = S1*B31 + S2*B41; B32B = B32;

S1 = (t - t4)/(t5 - t4); S2 = (t6 - t)/(t6 - t5); B42 = S1*B41 + S2*B51; B42B = B42;

S1 = (t - t0)/(t2 - t0); S2 = (t3 - t)/(t3 - t1); B03 = S1*B02 + S2*B12; B03B = B03;

S1 = (t - t1)/(t3 - t1); S2 = (t4 - t)/(t4 - t2); B13 = S1*B12 + S2*B22; B13B = B13;

S1 = (t - t2)/(t4 - t2); S2 = (t5 - t)/(t5 - t3); B23 = S1*B22 + S2*B32; B23B = B23;

S1 = (t - t3)/(t5 - t3); S2 = (t6 - t)/(t6 - t4); B33 = S1*B32 + S2*B42; B33B = B33;



%Segment C

B01 = 0; B11 = 0; B21 = 1; B31 = 0; B41 = 0; B51 = 0; B61 = 0;

S1 = (t - t0)/(t1 - t0); S2 = (t2 - t)/(t2 - t1); B02 = S1*B01 + S2*B11; B02C = B02;

S1 = (t - t1)/(t2 - t1); S2 = (t3 - t)/(t3 - t2); B12 = S1*B11 + S2*B21; B12C = B12;

S1 = (t - t2)/(t3 - t2); S2 = (t4 - t)/(t4 - t3); B22 = S1*B21 + S2*B31; B22C = B22;

S1 = (t - t3)/(t4 - t3); S2 = (t5 - t)/(t5 - t4); B32 = S1*B31 + S2*B41; B32C = B32;

S1 = (t - t4)/(t5 - t4); S2 = (t6 - t)/(t6 - t5); B42 = S1*B41 + S2*B51; B42C = B42;

S1 = (t - t0)/(t2 - t0); S2 = (t3 - t)/(t3 - t1); B03 = S1*B02 + S2*B12; B03C = B03;

S1 = (t - t1)/(t3 - t1); S2 = (t4 - t)/(t4 - t2); B13 = S1*B12 + S2*B22; B13C = B13;

S1 = (t - t2)/(t4 - t2); S2 = (t5 - t)/(t5 - t3); B23 = S1*B22 + S2*B32; B23C = B23;

S1 = (t - t3)/(t5 - t3); S2 = (t6 - t)/(t6 - t4); B33 = S1*B32 + S2*B42; B33C = B33;


%Segment D

B01 = 0; B11 = 0; B21 = 0; B31 = 1; B41 = 0; B51 = 0; B61 = 0;

S1 = (t - t0)/(t1 - t0); S2 = (t2 - t)/(t2 - t1); B02 = S1*B01 + S2*B11; B02D = B02;

S1 = (t - t1)/(t2 - t1); S2 = (t3 - t)/(t3 - t2); B12 = S1*B11 + S2*B21; B12D = B12;

S1 = (t - t2)/(t3 - t2); S2 = (t4 - t)/(t4 - t3); B22 = S1*B21 + S2*B31; B22D = B22;

S1 = (t - t3)/(t4 - t3); S2 = (t5 - t)/(t5 - t4); B32 = S1*B31 + S2*B41; B32D = B32;

S1 = (t - t4)/(t5 - t4); S2 = (t6 - t)/(t6 - t5); B42 = S1*B41 + S2*B51; B42D = B42;

S1 = (t - t0)/(t2 - t0); S2 = (t3 - t)/(t3 - t1); B03 = S1*B02 + S2*B12; B03D = B03;

S1 = (t - t1)/(t3 - t1); S2 = (t4 - t)/(t4 - t2); B13 = S1*B12 + S2*B22; B13D = B13;

S1 = (t - t2)/(t4 - t2); S2 = (t5 - t)/(t5 - t3); B23 = S1*B22 + S2*B32; B23D = B23;

S1 = (t - t3)/(t5 - t3); S2 = (t6 - t)/(t6 - t4); B33 = S1*B32 + S2*B42; B33D = B33;



%Segment E

B01 = 0; B11 = 0; B21 = 0; B31 = 0; B41 = 1; B51 = 0; B61 = 0;

S1 = (t - t0)/(t1 - t0); S2 = (t2 - t)/(t2 - t1); B02 = S1*B01 + S2*B11; B02E = B02;

S1 = (t - t1)/(t2 - t1); S2 = (t3 - t)/(t3 - t2); B12 = S1*B11 + S2*B21; B12E = B12;

S1 = (t - t2)/(t3 - t2); S2 = (t4 - t)/(t4 - t3); B22 = S1*B21 + S2*B31; B22E = B22;

S1 = (t - t3)/(t4 - t3); S2 = (t5 - t)/(t5 - t4); B32 = S1*B31 + S2*B41; B32E = B32;

S1 = (t - t4)/(t5 - t4); S2 = (t6 - t)/(t6 - t5); B42 = S1*B41 + S2*B51; B42E = B42;

S1 = (t - t0)/(t2 - t0); S2 = (t3 - t)/(t3 - t1); B03 = S1*B02 + S2*B12; B03E = B03;

S1 = (t - t1)/(t3 - t1); S2 = (t4 - t)/(t4 - t2); B13 = S1*B12 + S2*B22; B13E = B13;

S1 = (t - t2)/(t4 - t2); S2 = (t5 - t)/(t5 - t3); B23 = S1*B22 + S2*B32; B23E = B23;

S1 = (t - t3)/(t5 - t3); S2 = (t6 - t)/(t6 - t4); B33 = S1*B32 + S2*B42; B33E = B33;



%Segment F

B01 = 0; B11 = 0; B21 = 0; B31 = 0; B41 = 0; B51 = 1; B61 = 0;

S1 = (t - t0)/(t1 - t0); S2 = (t2 - t)/(t2 - t1); B02 = S1*B01 + S2*B11; B02F = B02;

S1 = (t - t1)/(t2 - t1); S2 = (t3 - t)/(t3 - t2); B12 = S1*B11 + S2*B21; B12F = B12;

S1 = (t - t2)/(t3 - t2); S2 = (t4 - t)/(t4 - t3); B22 = S1*B21 + S2*B31; B22F = B22;

S1 = (t - t3)/(t4 - t3); S2 = (t5 - t)/(t5 - t4); B32 = S1*B31 + S2*B41; B32F = B32;

S1 = (t - t4)/(t5 - t4); S2 = (t6 - t)/(t6 - t5); B42 = S1*B41 + S2*B51; B42F = B42;

S1 = (t - t0)/(t2 - t0); S2 = (t3 - t)/(t3 - t1); B03 = S1*B02 + S2*B12; B03F = B03;

S1 = (t - t1)/(t3 - t1); S2 = (t4 - t)/(t4 - t2); B13 = S1*B12 + S2*B22; B13F = B13;

S1 = (t - t2)/(t4 - t2); S2 = (t5 - t)/(t5 - t3); B23 = S1*B22 + S2*B32; B23F = B23;

S1 = (t - t3)/(t5 - t3); S2 = (t6 - t)/(t6 - t4); B33 = S1*B32 + S2*B42; B33F = B33;

fprintf('Blending functions :＼n');

B03 = ［B03A, B03B, B03C, B03D, B03E, B03F］; B03 = simplify(B03)

B13 = ［B13A, B13B, B13C, B13D, B13E, B13F］; B13 = simplify(B13)

B23 = ［B23A, B23B, B23C, B23D, B23E, B23F］; B23 = simplify(B23)

B33 = ［B33A, B33B, B33C, B33D, B33E, B33F］; B33 = simplify(B33)

fprintf('＼n');

fprintf('General Equation of Curve :＼n');

P = P0*B03 + P1*B13 + P2*B23 + P3*B33


fprintf('＼n');

x0 = 1; x1 = 4; x2 = 6; x3 = 8;

y0 = 2; y1 = 1; y2 = 5; y3 = -1;

fprintf('Actual Equation :＼n');

y = subs(P, (［P0, P1, P2, P3］), (［y0, y1, y2, y3］)); y = simplify(y)



%plotting BF

tta = linspace(t0, t1);

ttb = linspace(t1, t2);

ttc = linspace(t2, t3);

ttd = linspace(t3, t4);

tte = linspace(t4, t5);

ttf = linspace(t5, t6);

B03aa = subs(B03A, t, tta);

B03bb = subs(B03B, t, ttb);

B03cc = subs(B03C, t, ttc);

B03dd = subs(B03D, t, ttd);

B03ee = subs(B03E, t, tte);

B03ff = subs(B03F, t, ttf);

B13aa = subs(B13A, t, tta);

B13bb = subs(B13B, t, ttb);

B13cc = subs(B13C, t, ttc);

B13dd = subs(B13D, t, ttd);

B13ee = subs(B13E, t, tte);

B13ff = subs(B13F, t, ttf);

B23aa = subs(B23A, t, tta);

B23bb = subs(B23B, t, ttb);

B23cc = subs(B23C, t, ttc);

B23dd = subs(B23D, t, ttd);

B23ee = subs(B23E, t, tte);

B23ff = subs(B23F, t, ttf);

B33aa = subs(B33A, t, tta);

B33bb = subs(B33B, t, ttb);

B33cc = subs(B33C, t, ttc);

B33dd = subs(B33D, t, ttd);

B33ee = subs(B33E, t, tte);

B33ff = subs(B33F, t, ttf);

figure,

plot(tta, B03aa, 'k-', ttb, B03bb, 'k--', ttc, B03cc, 'k-.', ...

ttd, B03dd, 'b-', tte, B03ee, 'b--', ttf, B03ff, 'b-.');

hold on;

plot(tta, B13aa, 'k-', ttb, B13bb, 'k--', ttc, B13cc, 'k-.', ...

ttd, B13dd, 'b-', tte, B13ee, 'b--', ttf, B13ff, 'b-.');

plot(tta, B23aa, 'k-', ttb, B23bb, 'k--', ttc, B23cc, 'k-.', ...

ttd, B23dd, 'b-', tte, B23ee, 'b--', ttf, B23ff, 'b-.');

plot(tta, B33aa, 'k-', ttb, B33bb, 'k--', ttc, B33cc, 'k-.', ...

ttd, B33dd, 'b-', tte, B33ee, 'b--', ttf, B33ff, 'b-.');

xlabel ('t'); ylabel('B'); title('B03 - B13 - B23 - B33');

legend('A', 'B', 'C', 'D', 'E', 'F');

hold off;



%plotting curve

xa = x(1); ya = y(1);

xb = x(2); yb = y(2);

xc = x(3); yc = y(3);

xd = x(4); yd = y(4);

xe = x(5); ye = y(5);

xf = x(6); yf = y(6);

xaa = subs(xa, t, tta); yaa = subs(ya, t, tta);

xbb = subs(xb, t, ttb); ybb = subs(yb, t, ttb);

xcc = subs(xc, t, ttc); ycc = subs(yc, t, ttc);

xdd = subs(xd, t, ttd); ydd = subs(yd, t, ttd);

xee = subs(xe, t, tte); yee = subs(ye, t, tte);

xff = subs(xf, t, ttf); yff = subs(yf, t, ttf);

X = ［x0, x1, x2, x3］; Y = ［y0, y1, y2, y3］;

figure

subplot(131),

plot(tta, xaa, 'k-', ttb, xbb, 'k--', ttc, xcc, 'k-.', ...

ttd, xdd, 'b-', tte, xee, 'b--', ttf, xff, 'b-.');

xlabel ('t'); ylabel('x'); axis square; title('t  x');

subplot(132),

plot(tta, yaa, 'k-', ttb, ybb, 'k--', ttc, ycc, 'k-.', ...

ttd, ydd, 'b-', tte, yee, 'b--', ttf, yff, 'b-.')

xlabel ('t'); ylabel('y'); axis square; title('t  y');

subplot(133),

plot(xaa, yaa, 'k-', xbb, ybb, 'k--', xcc, ycc, 'k-.', ...

xdd, ydd, 'b-', xee, yee, 'b--', xff, yff, 'b-.'); hold on;

scatter(X, Y, 20, 'r', 'filled');

xlabel ('x'); ylabel('y'); axis square; grid; title('x  y');

axis(［0 9 -2 6］);

d = 0.5;

text(x0, y0+d, 'P_0');

text(x1, y1-d, 'P_1');

text(x2, y2-d, 'P_2');

text(x3, y3+d, 'P_3');

hold off;

注解

…： 将当前命令或函数调用继续到下一行。

simplify： 通过解决所有交集和嵌套来简化方程。



3．5结向量值的证明

现在，是时候为选择我们迄今为止一直使用的［0，1，2，3，…］的KV值提供理由了。让我们首先选择其他一组值并观察我们得到的结果。因此，让我们假设对于均匀二次B样条，将KV更改为T=［0，5，10，15，20，25，30］。我们可以任意选择任何一组值，唯一的约束是值之间的间距应该是一致的，这也是对均匀B样条的要求。因此，在这种情况下，我们选择了一个间距为早期值5倍的KV。如果我们按照与3．4节相同的步骤进行并计算BF，会得到如下所示的结果： 
B0,3=(1/50)t2(0≤t<1)A-t2/25+3t/5-(3/2)(1≤t<2)B(1/50)(t-15)2(2≤t<3)CB1,3=(1/50)(t-5)2(1≤t<2)B-t2/25+t-(11/2)(2≤t<3)C(1/50)(t-20)2(3≤t<4)DB2,3=(1/50)(t-10)2(2≤t<3)C-t2/25+7t/5-(23/2)(3≤t<4)D(1/50)(t-25)2(4≤t<5)EB3,3=(1/50)(t-15)2(3≤t<4)D-t2/25+9t/5-(39/2)(4≤t<5)E(1/50)(t-30)2(5≤t<6)F(3．18)
与式(3．14)比较可知，t已被t/5替换，即所有t值都已缩放5倍。所以现在要获得相同的B值需要t值比之前大5倍。这反映在图3．7中的BF图中，与图3．5中的BF图进行比较，它显示的t轴要扩大5倍。



图3．7BF随KV的变化而变化


将新的KV插入例3．2后，我们可以观察到对参数曲线的影响。可以看到下面显示的x(t)和y(t)的值受到类似于BF的影响，即这些值已按5倍进行了缩放。
x(t)=t2/50(0≤t<5)t2/25-t/5+1/2(5≤t<10)-t2/50+t-11/2(10≤t<15)2t/5-1(15≤t<20)-t2/5+42t/5-81(20≤t<25)(4/25)(t-30)2(25≤t<30)
y(t)=t2/25(0≤t<5)-3t2/50+t-5/2(5≤t<10)t2/10-11t/5+27/2(10≤t<15)-t2/5+34t/5-54(15≤t<20)7t2/50-34t/5+82(20≤t<25)-(1/50)(t-30)2(25≤t<30)
在绘制实际曲线时，可以看到xt和yt图受到类似影响，即t值已更改5倍。请参见图3．8，不同的线型表示分段间隔。然而，由于x和y值保持不变，xy图与以前完全相同，因为它不受t值的任何变化的影响，只要它以相同的量均匀地影响x和y值。例如，对于图3．6，有t=2，x=2．5，y=1．5，而对于图3．8，有t=10，x=2．5，y=1．5，这意味着x与y保持不变，即不受影响。这对所有点都是正确的，因此曲线的xy图与以前相同。显然，对于任何改变t的比例因子都是如此。



图3．8曲线方程随KV变化的变化


鼓励读者使用不同的KV值生成图表并验证结果，可以通过使用MATLAB Code 3．2并简单地更改程序第二行中的KV来轻松地做到这一点。

这使我们得出结论，空间域中的实际曲线与KV值无关，因为它不依赖于t值的比例因子。因此，习惯上选择最小的t值，即0，1，2，…以降低方程的复杂性，但实际上，我们可以为KV选择任何值并得到相同的结果。

3．6二次开放均匀B样条

在开放均匀样条曲线中，KV是均匀的，除了重复d次的末端，其中(d-1)是曲线的次数。重复值称为多重性。多重性意味着Cox de Boor项的分母在许多情况下都为零。因此，这里需要做一个重要的假设： 除以零被视为零。

考虑一个d=3和n=3的二次开放均匀B样条。让KV选择为T={1，1，1，2，3，3，3}。如前所述，进行分段分析，获得的结果列于表3．3中。


表3．3二次开放均匀B样条的BF计算



段tBk,1Bk,2Bk,3

A0≤t<1
B0,1=1B0,2=0B0,3=0

B1,1=0B1,2=0B1,3=0

B2,1=0B2,2=0B2,3=0

B3,1=0B3,2=0B3,3=0

B4,1=0B4,2=0

B5,1=0


B1≤t<2
B0,1=0B0,2=0B0,3=0

B1,1=1B1,2=0B1,3=0

B2,1=0B2,2=0B2,3=0

B3,1=0B3,2=0B3,3=0

B4,1=0B4,2=0

B5,1=0


C2≤t<3
B0,1=0B0,2=0B0,3=(t-2)2

B1,1=0B1,2=2-tB1,3=-(3/2)t2+5t-(7/2)

B2,1=1B2,2=t-1B2,3=(1/2)(t-1)2

B3,1=0B3,2=0B3,3=0

B4,1=0B4,2=0

B5,1=0


D3≤t<4
B0,1=0B0,2=0B0,3=0

B1,1=0B1,2=0B1,3=(1/2)(t-3)2

B2,1=0B2,2=3-tB2,3=-(3/2)t2+7t-(15/2)

B3,1=1B3,2=t-2B3,3=(t-2)2

B4,1=0B4,2=0

B5,1=0



E4≤t<5
B0,1=0B0,2=0B0,3=0

B1,1=0B1,2=0B1,3=0

B2,1=0B2,2=0B2,3=0

B3,1=0B3,2=0B3,3=0

B4,1=1B4,2=0

B5,1=0


F5≤t<6
B0,1=0B0,2=0B0,3=0

B1,1=0B1,2=0B1,3=0


F5 ≤ t < 6
B2,1=0B2,2=0B2,3=0

B3,1=0B3,2=0B3,3=0

B4,1=0B4,2=0

B5,1=1



将上述值代入式(3．13)，得到： 
B0,3=(t-2)2(2≤t<3)CB1,3=-(3/2)t2+5t-(7/2)(2≤t<3)C(1/2)(t-3)2(3≤t<4)DB2,3=(1/2)(t-1)2(2≤t<3)C-(3/2)t2+7t-(15/2)(3≤t<4)DB3,3=(t-2)2(3≤t<4)D(3．19)
式(3．19)表示具有4个CP的二次开放均匀B样条的BF。BF的图如图3．9所示。仅存在与段C(2≤t<3)和D(3≤t<4)相关的部分。



图3．9二次开放均匀B样条的BF


曲线方程是通过将BF值代入式(3．15)得到的： 
P(t)=0(0≤t<1)0(1≤t<2)P0(t-2)2+P1-32t2+5t-72+P212(t-1)2(2≤t<3)P112(t-3)2+P2-32t2+7t-152+P3(t-2)2(3≤t<4)0(4≤t<5)0(5≤t<6)
(3．20)

图3．10中的曲线图显示了多重性对样条曲线的影响： 它迫使曲线实际通过第一个和最后一个CP。这创建了一条只有两条线段C和D的样条曲线，而其他线段A、B、E和F不存在。因此，我们可以得出结论，若KV具有重复值，则近似样条可以表现得像插值样条或混合样条那样。



图3．10开放均匀二次B样条曲线


例3．3求具有CP，即P0(1，2)、P1(4，1)、P2(6，5)和P3(8,-1)的开放均匀二次B样条方程。还要编写一个程序来绘制BF和实际曲线。

解： 

根据式(3．20)，代入给定CP的值得到： 
x(t)=0(0≤t<1)0(1≤t<2)-2t2+10t-7(2≤t<3)t2-2t+5(3≤t<4)0(4≤t<5)0(5≤t<6)
y(t)=0(0≤t<1)0(1≤t<2)3t2-8t+7(2≤t<3)-8t2+36t-37(3≤t<4)0(4≤t<5)0(5≤t<6)
MATLAB Code 3．3

该代码与MATLAB Code 3．2几乎完全相同，除了在计算每个BF之前进行额外检查以避免被零除的情况。如果存在被零除条件，则表达式将替换为0； 否则，应按正常流程计算。说明如下。

D =(t1 - t0); if D = = 0, S1 = 0; else S1 = (t - t0)/D; end;

D =(t2 - t1); if D = = 0, S2 = 0; else S2 = (t - t0)/D; end;

B02 = S1*B01 + S2*B11; B02A = B02;

3．7二次非均匀B样条

在非均匀样条中，KV不均匀，即结点元素之间的间距不同。这使得BF不对称，并且它们倾向于在间距较小的地方聚集在一起，这具有将曲线拉向相应CP的效果。图3．11显示了二次非均匀B样条的BF。不对称的程度取决于KV中的间距。



图3．11二次非均匀B样条的BF


例3．4求具有CP，即P0(1，2)、P1(4，1)、P2(6，5)和P3(8,-1)的二次非均匀B样条方程，假设KV为T=［0，1，3，5，15，18，20］。

解： 

使用分段分析： 
x(t)=t2/3(0≤t<1)5t2/24+t/4-1/8(1≤t<2)-7t2/24+13t/4-37/8(2≤t<3)-t2/780+9t/26+137/52(3≤t<4)-38t2/65+232t/13-1672/13(4≤t<5)4(t-20)2/5(5≤t<6)
y(t)=2t2/3(0≤t<1)-11t2/24+9t/4-9/8(1≤t<2)7t2/24-9t/4+45/8(2≤t<3)-t2/390+9t/13+95/26(3≤t<4)43t2/195-98t/13+830/13(4≤t<5)-(t-20)2/10(5≤t<6)

MATLAB Code 3．4

与MATLAB Code 3．2相同，但KV有所变化。

3．8三次均匀B样条

三次均匀B样条的处理方式与二次均匀B样条的处理方式大致相同，但由Cox de Boor算法生成的四阶BF项产生了额外的复杂性。

为了生成三次均匀B样条，我们需要从d=4和n=4开始： 

曲线的阶数： d-1=3。

CP数量： n+1=5。

曲线段数： d+n=8。

KV中的元素数： d+n+1=9。

设曲线段为A、B、C、D、E、F、G和H，CP为P0、P1、P2、P3和P4(见图3．12)。



图3．12具有5个CP的三次均匀B样条


对于k={0，1，2，3，4，5，6，7，8}，令KV为T={tk}。在这种情况下，T=［t0，t1，t2，t3，t4，t5，t6，t7，t8］。

设BF为B0,4、B1,4、B2,4、B3,4和B4,4。

曲线方程由下式给出： 
P(t)=P0B0,4+P1B1,4+P2B2,4+P3B3,4+P4B4,4(3．21)
如前所述，我们假设KV为T=［0，1，2，3，4，5，6，7，8］。根据第一个Cox de Boor算法的条件，一阶项B0,1，B1,1，B2,1，B3,1，B4,1，B5,1，B6,1，B7,1和B8,1将为0或1。二阶项计算如下： 
B0,2=(t-0)B0,1+(2-t)B1,1B1,2=(t-1)B1,1+(3-t)B2,1B2,2=(t-2)B2,1+(4-t)B3,1B3,2=(t-3)B3,1+(5-t)B4,1B4,2=(t-4)B4,1+(6-t)B5,1B5,2=(t-5)B5,1+(7-t)B6,1B6,2=(t-6)B6,1+(8-t)B7,1B7,2=(t-7)B7,1+(9-t)B8,1(3．22)
三阶项由二阶项计算得出： 
B0,3=(1/2)(t-0)B0,2+(1/2)(3-t)B1,2B1,3=(1/2)(t-1)B1,2+(1/2)(4-t)B2,2B2,3=(1/2)(t-2)B2,2+(1/2)(5-t)B3,2B3,3=(1/2)(t-3)B3,2+(1/2)(6-t)B4,2B4,3=(1/2)(t-4)B4,2+(1/2)(7-t)B5,2B5,3=(1/2)(t-5)B5,2+(1/2)(8-t)B6,2B6,3=(1/2)(t-6)B6,2+(1/2)(9-t)B7,2(3．23)
四阶项由三阶项计算得出： 
B0,4=(1/3)(t-0)B0,3+(1/3)(4-t)B1,3B1,4=(1/3)(t-1)B1,3+(1/3)(5-t)B2,3B2,4=(1/3)(t-2)B2,3+(1/3)(6-t)B3,3B3,4=(1/3)(t-3)B3,3+(1/3)(7-t)B4,3B4,4=(1/3)(t-4)B4,3+(1/3)(8-t)B5,3(3．24)
由于有8个段，每个BF包含8个子分量： 
B0,4={B0,4A,B0,4B,B0,4C,B0,4D,B0,4E,B0,4F,B0,4G,B0,4H}B1,4={B1,4A,B1,4B,B1,4C,B1,4D,B1,4E,B1,4F,B1,4G,B1,4H}B2,4={B2,4A,B2,4B,B2,4C,B2,4D,B2,4E,B2,4F,B2,4G,B2,4G}
B3,4={B3,4A,B3,4B,B3,4C,B3,4D,B3,4E,B3,4F,B3,4G,B3,4H}B4,4={B4,4A,B4,4B,B4,4C,B4,4D,B4,4E,B4,4F,B4,4G,B4,4H}(3．25)
表3．4总结了对BF值的计算。


表3．4三次均匀B样条的BF计算



段tBk,1Bk,2Bk,3Bk,4

A0≤t<1
B0,1=1B0,2=tB0,3=(1/2)t2B0,4=(1/6)t3

B1,1=0B1,2=0B1,3=0

B2,1=0B2,2=0B2,3=0


A0≤t<1
B3,1=0B3,2=0B3,3=0

B4,1=0B4,2=0

B5,1=0

B6,1=0

B7,1=0


B1≤t<2
B0,1=0B0,2=2-tB0,3=-t2+3t-(3/2)B0,4=-(1/3)t3+2t2-2t+(2/3)

B1,1=1B1,2=t-1B1,3=(1/2)(t-1)2B1,4=(1/6)(t-1)3

B2,1=0B2,2=0B2,3=0

B3,1=0B3,2=0B3,3=0

B4,1=0B4,2=0

B5,1=0

B6,1=0

B7,1=0


C2≤t<3
B0,1=0B0,2=0B0,3=(1/2)(t-3)2B0,4=-(1/2)t3-4t2+10t-(22/3)

B1,1=0B1,2=3-tB1,3=-t2+5t-(11/2)B1,4=-(1/2)t3+(7/2)t2-(15/2)t+(31/6)

B2,1=1B2,2=t-2B2,3=(1/2)(t-2)2B2,4=(1/6)(t-2)3

B3,1=0B3,2=0B3,3=0

B4,1=0B4,2=0

B5,1=0

B6,1=0

B7,1=0


D3≤t<4
B0,1=0B0,2=0B0,3=0B0,4=-(1/6)(t-4)3

B1,1=0B1,2=0B1,3=(1/2)(t-4)2B1,4=（1/2）t3-(11/2)t2+(39/2)t-(131/6)

B2,1=0B2,2=4-tB2,3=-t2+7t-(23/2)B2,4=-(1/2)t3+(5/2)t2-16t+(50/3)

B3,1=1B3,2=t-3B3,3=(1/2)(t-3)2B3,4=(1/6)(t-3)3

B4,1=0B4,2=0

B5,1=0

B6,1=0

B7,1=0


E4≤t<5
B0,1=0B0,2=0B0,3=0

B1,1=0B1,2=0B1,3=0B1,4=-(1/6)(t-5)3

B2,1=0B2,2=0B2,3=(1/2)(t-5)2B2,4=(1/2)t3-(7/2)t2+32t-(142/3)

B3,1=0B3,2=5-tB3,3=-t2+9t-(39/2)B3,4=-(1/2)t3+(13/2)t2-(55/2)t+(229/6)

B4,1=1B4,2=t-4B4,3=(1/2)(t-4)2B4,4=(1/6)(t-4)3

B5,1=0

B6,1=0

B7,1=0


F5≤t<6
B0,1=0B0,2=0B0,3=0

B1,1=0B1,2=0B1,3=0

B2,1=0B2,2=0B2,3=0B2,4=-(1/6)(t-6)3

B3,1=0B3,2=0B3,3=(1/2)(t-6)2B3,4=(1/2)t3-(17/2)t2+(95/2)t-(521/6)

B4,1=0B4,2=6-tB4,3=-t2+11t-(59/2)B4,4=-(1/2)t3+8t2-42t+(218/3)

B5,1=1B5,2=t-5B5,3=(1/2)(t-5)2

B6,1=0

B7,1=0


G6≤t<7
B0,1=0B0,2=0B0,3=0

B1,1=0B1,2=0B1,3=0

B2,1=0B2,2=0B2,3=0

B3,1=0B3,2=0B3,3=0B3,4=-(1/6)(t-7)3

B4,1=0B4,2=0B4,3=(1/2)(t-7)2B4,4=(1/2)t3-10t2+66t-(430/3)

B5,1=0B5,2=7-tB5,3=-t2+13t-(83/2)

B6,1=1B6,2=t-6B6,3=(1/2)(t-6)2

B7,1=0


H7≤t<8
B0,1=0B0,2=0B0,3=0

B1,1=0B1,2=0B1,3=0

B2,1=0B2,2=0B2,3=0

B3,1=0B3,2=0B3,3=0

B4,1=0B4,2=0B4，3=0B4,4=-(1/6)(t-8)3

B5,1=0B5,2=0B5,3=(1/2)(t-8)2

B6,1=0B6,2=8-tB6,3=-(1/2)t2+7t-24

B7,1=1B7,2=t-7



将上述值代入式(3．25)，可以得到： 

B0,4=(1/6)t3(0≤t<1)A-(1/3)t3+2t2-2t+(2/3)(1≤t<2)B-(1/2)t3-4t2+10t-(22/3)(2≤t<3)C-(1/6)(t-4)3(3≤t<4)D
B1,4=(1/6)(t-1)3(1≤t<2)B-(1/2)t3+(7/2)t2-(15/2)t+(31/6)(2≤t<3)C(1/2)t3-(11/2)t2+(39/2)t-(131/6)(3≤t<4)D-(1/6)(t-5)3(4≤t<5)E
B2,4=(1/6)(t-2)3(2≤t<3)C-(1/2)t3+(5/2)t2-16t+(50/3)(3≤t<4)D(1/2)t3-(7/2)t2+32t-(142/3)(4≤t<5)E-(1/6)(t-6)3(5≤t<6)F
B3,4=(1/6)(t-3)3(3≤t<4)D-(1/2)t3+(13/2)t2-(55/2)t+(229/6)(4≤t<5)E(1/2)t3-(17/2)t2+(95/2)t-(521/6)(5≤t<6)F-(1/6)(t-7)3(6≤t<7)G
B4,4=(1/6)(t-4)3(4≤t<5)E-(1/2)t3+8t2-42t+(218/3)(5≤t<6)F(1/2)t3-10t2+66t-(430/3)(6≤t<7)G-(1/6)(t-8)3(7≤t<8)H(3．26)
式(3．26)表示具有5个CP和8个段的三次均匀B样条的BF。BF的图如图3．13所示。每个BF具有相同的形状，但相对于前一个向右移动1。因此，每个BF可以通过将t替换为(t-1)从前一个BF获得。如式(3．26)所示，每个BF有8个细分，其中4个是非零的。B0,4的第一条曲线对于分段A(0≤t<1)、B(1≤t<2)、C(2≤t<3)和D(3≤t<4)具有非零部分，B1,4的第二条曲线对于分段B(1≤t<2)、C(2≤t<3)、D(3≤t<4)和E(4≤t<5)具有非零部分，B2,4的第三条曲线对于分段C(2≤t<3)、D(3≤t<4)、E(4≤t<5)和F(5≤t<6)具有非零部分，B3,4的第四条曲线对于分段D(3≤t<4)、E(4≤t<5)、F(5≤t<6)和G(6≤t<7)具有非零部分，B4,4的第五条曲线对于分段E(4≤t<5)、F(5≤t<6)、G(6≤t<7)和H(7≤t<8)具有非零部分。由于BF与CP相关联，因此这提供了三次样条的局部控制属性的指示。第一个CP对前4个段A、B、C和D有影响，第二个CP对B、C、D和E有影响，以此类推。这意味着如果第一个CP发生更改，它将仅影响前4个段，而样条的其余部分将保持不变。



图3．13具有5个CP的三次均匀B样条的BF


样条方程是其八段方程的集合： 

P(t)=PA(0≤t<1)PB(1≤t<2)PC(2≤t<3)PD(3≤t<4)PE(4≤t<5)PF(5≤t<6)PG(6≤t<7)PH(7≤t<8)(3．27)
其中，
PA=P0B0,4A+P1B1,4A+P2B2,4A+P3B3,4A+P4B4,4APB=P0B0,4B+P1B1,4B+P2B2,4B+P3B3,4B+P4B4,4BPC=P0B0,4C+P1B1,4C+P2B2,4C+P3B3,4C+P4B4,4CPD=P0B0,4D+P1B1,4D+P2B2,4D+P3B3,4D+P4B4,4DPE=P0B0,4E+P1B1,4E+P2B2,4E+P3B3,4E+P4B4,4EPF=P0B0,4F+P1B1,4F+P2B2,4F+P3B3,4F+P4B4,4FPG=P0B0,4G+P1B1,4G+P2B2,4G+P3B3,4G+P4B4,4GPH=P0B0,4H+P1B1,4H+P2B2,4H+P3B3,4H+P4B4,4H(3．28)
将表3．4中的BF值代入式(3．28)，可以得到：

P(t)
=P016t3P0-t33+2t2-2t+23+P116(t-1)3P0-t32-4t2+10t-223+P1-t32+7t22-15t2+316+P216(t-2)3P016(4-t)3+P1t32-11t22+39t2-1316+P2-t32+5t22-16t+503+P316(t-3)3P116(5-t)3+P2t32-7t22+32t-1423+P3-t32+13t22-55t2+2296+P416(t-4)3P216(6-t)3+P3t32-17t22+95t2-5216+P4-t32+8t2-42t+2183P316(7-t)3+P4t32-10t2+66t-4303P416(8-t)3(3．29)
式(3．29)表示具有5个CP的三次均匀B样条曲线方程。方程的8个子分量代表8个段的部分。

例3．5找到具有CP，即(-1，0)、(0，1)、(1，0)、(0,-1)和(-0．5,-0．5)的均匀三次B样条的方程。还要编写一个程序来绘制其BF和实际曲线。

解： 

将得到的给定CP值代入式(3．29)，得到(见图3．14)。


图3．14例3．5的绘图


x(t)=t3/6(0≤t<1)t3/3-2t2+2t-2/3(1≤t<2)2t3/3+3t2-32/3t+6(2≤t<3)-t3/3+t2/2-8t+6(3≤t<4)712t3-5t2+812t-125(4≤t<5)
112t3-2t2+9t-47(5≤t<6)-t3/4+5t2-33t+215/3(6≤t<7)(t-8)3/12(7≤t<8)
y(t)=0(0≤t<1)(t-1)3/6(1≤t<2)-t3/2+7t2/2-15t/2+31/6(2≤t<3)t3/3-4t2+15t-52/3(3≤t<4)t3/4-3t2+11t-12(4≤t<5)-t3/4+9t2/2-53t/2+101/2(5≤t<6)-t3/12+3t2/2-17t/2+29/2(6≤t<7)(t-8)3/12(7≤t<8)

MATLAB Code 3．5

clear all; format compact; clc;

x0 = -1; x1 = 0; x2 = 1; x3 = 0; x4 = -0.5;

y0 = 0; y1 = 1; y2 = 0; y3 = -1; y4 = -0.5;

t0 = 0; t1 = 1; t2 = 2; t3 = 3; t4 = 4; t5 = 5; t6 = 6; t7 = 7; t8 = 8;

T = ［t0, t1, t2, t3, t4, t5, t6, t7, t8］;

syms t P0 P1 P2 P3 P4;

syms B01 B11 B21 B31 B41 B51 B61 B71;

B02 = ((t - t0)/(t1 - t0))*B01 + ((t2 - t)/(t2 - t1))*B11;

B12 = ((t - t1)/(t2 - t1))*B11 + ((t3 - t)/(t3 - t2))*B21;

B22 = ((t - t2)/(t3 - t2))*B21 + ((t4 - t)/(t4 - t3))*B31;

B32 = ((t - t3)/(t4 - t3))*B31 + ((t5 - t)/(t5 - t4))*B41;

B42 = ((t - t4)/(t5 - t4))*B41 + ((t6 - t)/(t6 - t5))*B51;

B52 = ((t - t5)/(t6 - t5))*B51 + ((t7 - t)/(t7 - t6))*B61;

B62 = ((t - t6)/(t7 - t6))*B61 + ((t8 - t)/(t8 - t7))*B71;

B72 = ((t - t7)/(t8 - t7))*B71 + 0;

B03 = ((t - t0)/(t2 - t0))*B02 + ((t3 - t)/(t3 - t1))*B12;

B13 = ((t - t1)/(t3 - t1))*B12 + ((t4 - t)/(t4 - t2))*B22;

B23 = ((t - t2)/(t4 - t2))*B22 + ((t5 - t)/(t5 - t3))*B32;

B33 = ((t - t3)/(t5 - t3))*B32 + ((t6 - t)/(t6 - t4))*B42;

B43 = ((t - t4)/(t6 - t4))*B42 + ((t7 - t)/(t7 - t5))*B52;

B53 = ((t - t5)/(t7 - t5))*B52 + ((t8 - t)/(t8 - t6))*B62;

B63 = ((t - t6)/(t8 - t6))*B62 + 0;

B04 = ((t - t0)/(t3 - t0))*B03 + ((t4 - t)/(t4 - t1))*B13;

B14 = ((t - t1)/(t4 - t1))*B13 + ((t5 - t)/(t5 - t2))*B23;

B24 = ((t - t2)/(t5 - t2))*B23 + ((t6 - t)/(t6 - t3))*B33;

B34 = ((t - t3)/(t6 - t3))*B33 + ((t7 - t)/(t7 - t4))*B43;

B44 = ((t - t4)/(t7 - t4))*B43 + ((t8 - t)/(t8 - t5))*B53;

B54 = ((t - t5)/(t8 - t5))*B53 + 0;



%Segment A

B02A = subs(B02, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B12A = subs(B12, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B22A = subs(B22, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B32A = subs(B32, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B42A = subs(B42, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B52A = subs(B52, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B62A = subs(B62, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B72A = subs(B72, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B03A = subs(B03, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B13A = subs(B13, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B23A = subs(B23, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B33A = subs(B33, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B43A = subs(B43, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B53A = subs(B53, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B63A = subs(B63, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B04A = subs(B04, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B14A = subs(B14, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B24A = subs(B24, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B34A = subs(B34, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B44A = subs(B44, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});

B54A = subs(B54, {B01,B11,B21,B31,B41,B51,B61,B71}, {1,0,0,0,0,0,0,0});



%Segment B

B02B = subs(B02, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B12B = subs(B12, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B22B = subs(B22, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B32B = subs(B32, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B42B = subs(B42, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B52B = subs(B52, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B62B = subs(B62, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B72B = subs(B72, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B03B = subs(B03, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B13B = subs(B13, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B23B = subs(B23, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B33B = subs(B33, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B43B = subs(B43, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B53B = subs(B53, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B63B = subs(B63, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B04B = subs(B04, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B14B = subs(B14, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B24B = subs(B24, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B34B = subs(B34, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B44B = subs(B44, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});

B54B = subs(B54, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,1,0,0,0,0,0,0});



%Segment C

B02C = subs(B02, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B12C = subs(B12, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B22C = subs(B22, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B32C = subs(B32, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B42C = subs(B42, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B52C = subs(B52, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B62C = subs(B62, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B72C = subs(B72, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B03C = subs(B03, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B13C = subs(B13, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B23C = subs(B23, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B33C = subs(B33, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B43C = subs(B43, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B53C = subs(B53, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B63C = subs(B63, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B04C = subs(B04, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B14C = subs(B14, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B24C = subs(B24, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B34C = subs(B34, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B44C = subs(B44, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});

B54C = subs(B54, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,1,0,0,0,0,0});



%Segment D

B02D = subs(B02, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B12D = subs(B12, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B22D = subs(B22, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B32D = subs(B32, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B42D = subs(B42, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B52D = subs(B52, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B62D = subs(B62, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B72D = subs(B72, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B03D = subs(B03, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B13D = subs(B13, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B23D = subs(B23, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B33D = subs(B33, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B43D = subs(B43, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B53D = subs(B53, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B63D = subs(B63, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B04D = subs(B04, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B14D = subs(B14, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B24D = subs(B24, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B34D = subs(B34, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B44D = subs(B44, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});

B54D = subs(B54, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,1,0,0,0,0});



%Segment E

B02E = subs(B02, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B12E = subs(B12, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B22E = subs(B22, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B32E = subs(B32, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B42E = subs(B42, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B52E = subs(B52, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B62E = subs(B62, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B72E = subs(B72, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B03E = subs(B03, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B13E = subs(B13, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B23E = subs(B23, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B33E = subs(B33, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B43E = subs(B43, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B53E = subs(B53, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B63E = subs(B63, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B04E = subs(B04, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B14E = subs(B14, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B24E = subs(B24, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B34E = subs(B34, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B44E = subs(B44, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});

B54E = subs(B54, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,1,0,0,0});



%Segment F

B02F = subs(B02, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B12F = subs(B12, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B22F = subs(B22, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B32F = subs(B32, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B42F = subs(B42, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B52F = subs(B52, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B62F = subs(B62, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B72F = subs(B72, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B03F = subs(B03, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B13F = subs(B13, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B23F = subs(B23, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B33F = subs(B33, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B43F = subs(B43, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B53F = subs(B53, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B63F = subs(B63, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B04F = subs(B04, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B14F = subs(B14, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B24F = subs(B24, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B34F = subs(B34, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B44F = subs(B44, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});

B54F = subs(B54, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,1,0,0});



%Segment G

B02G = subs(B02, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B12G = subs(B12, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B22G = subs(B22, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B32G = subs(B32, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B42G = subs(B42, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B52G = subs(B52, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B62G = subs(B62, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B72G = subs(B72, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B03G = subs(B03, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B13G = subs(B13, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B23G = subs(B23, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B33G = subs(B33, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B43G = subs(B43, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B53G = subs(B53, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B63G = subs(B63, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B04G = subs(B04, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B14G = subs(B14, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B24G = subs(B24, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B34G = subs(B34, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B44G = subs(B44, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});

B54G = subs(B54, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,1,0});



%Segment H

B02H = subs(B02, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B12H = subs(B12, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B22H = subs(B22, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B32H = subs(B32, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B42H = subs(B42, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B52H = subs(B52, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B62H = subs(B62, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B72H = subs(B72, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B03H = subs(B03, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B13H = subs(B13, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B23H = subs(B23, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B33H = subs(B33, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B43H = subs(B43, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B53H = subs(B53, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B63H = subs(B63, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B04H = subs(B04, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B14H = subs(B14, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B24H = subs(B24, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B34H = subs(B34, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B44H = subs(B44, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

B54H = subs(B54, {B01,B11,B21,B31,B41,B51,B61,B71}, {0,0,0,0,0,0,0,1});

fprintf('Blending functions :＼n');

B04 = ［B04A, B04B, B04C, B04D, B04E, B04F, B04G, B04H］; B04 = simplify(B04)

B14 = ［B14A, B14B, B14C, B14D, B14E, B14F, B14G, B14H］; B14 = simplify(B14)

B24 = ［B24A, B24B, B24C, B24D, B24E, B24F, B24G, B24H］; B24 = simplify(B24)

B34 = ［B34A, B34B, B34C, B34D, B34E, B34F, B34G, B34H］; B34 = simplify(B34)

B44 = ［B44A, B44B, B44C, B44D, B44E, B44F, B44G, B44H］; B44 = simplify(B44)

fprintf('＼n');

fprintf('General Equation of Curve :＼n');

P = P0*B04 + P1*B14 + P2*B24 + P3*B34 + P4*B44

fprintf('＼n');

fprintf('Actual Equation :＼n');

x = subs(P, (［P0, P1, P2, P3, P4］), (［x0, x1, x2, x3, x4］)); x = simplify(x)

y = subs(P, (［P0, P1, P2, P3, P4］), (［y0, y1, y2, y3, y4］)); y = simplify(y)



%plotting BF

tta = linspace(t0, t1);

ttb = linspace(t1, t2);

ttc = linspace(t2, t3);

ttd = linspace(t3, t4);

tte = linspace(t4, t5);

ttf = linspace(t5, t6);

ttg = linspace(t6, t7);

tth = linspace(t7, t8);

B04aa = subs(B04A, t, tta);

B04bb = subs(B04B, t, ttb);

B04cc = subs(B04C, t, ttc);

B04dd = subs(B04D, t, ttd);

B04ee = subs(B04E, t, tte);

B04ff = subs(B04F, t, ttf);

B04gg = subs(B04G, t, ttg);

B04hh = subs(B04H, t, tth);

B14aa = subs(B14A, t, tta);

B14bb = subs(B14B, t, ttb);

B14cc = subs(B14C, t, ttc);

B14dd = subs(B14D, t, ttd);

B14ee = subs(B14E, t, tte);

B14ff = subs(B14F, t, ttf);

B14gg = subs(B14G, t, ttg);

B14hh = subs(B14H, t, tth);

B24aa = subs(B24A, t, tta);

B24bb = subs(B24B, t, ttb);

B24cc = subs(B24C, t, ttc);

B24dd = subs(B24D, t, ttd);

B24ee = subs(B24E, t, tte);

B24ff = subs(B24F, t, ttf);

B24gg = subs(B24G, t, ttg);

B24hh = subs(B24H, t, tth);

B34aa = subs(B34A, t, tta);

B34bb = subs(B34B, t, ttb);

B34cc = subs(B34C, t, ttc);

B34dd = subs(B34D, t, ttd);

B34ee = subs(B34E, t, tte);

B34ff = subs(B34F, t, ttf);

B34gg = subs(B34G, t, ttg);

B34hh = subs(B34H, t, tth);

B44aa = subs(B44A, t, tta);

B44bb = subs(B44B, t, ttb);

B44cc = subs(B44C, t, ttc);

B44dd = subs(B44D, t, ttd);

B44ee = subs(B44E, t, tte);

B44ff = subs(B44F, t, ttf);

B44gg = subs(B44G, t, ttg);

B44hh = subs(B44H, t, tth);

figure,

plot(tta, B04aa, 'k-', ttb, B04bb, 'k--', ttc, B04cc, 'k-.', ttd, ...

B04dd, 'b-', tte, B04ee, 'b--', ttf, B04ff, 'b-.', ttg, B04gg, 'r-', tth, B04hh, 'r--');

hold on;

plot(tta, B14aa, 'k-', ttb, B14bb, 'k--', ttc, B14cc, 'k-.', ttd, ...

B14dd, 'b-', tte, B14ee, 'b--', ttf, B14ff, 'b-.', ttg, B14gg, 'r-', tth, B14hh, 'r--');

plot(tta, B24aa, 'k-', ttb, B24bb, 'k--', ttc, B24cc, 'k-.', ttd, ...

B24dd, 'b-', tte, B24ee, 'b--', ttf, B24ff, 'b-.', ttg, B24gg, 'r-', tth, B24hh, 'r--');

plot(tta, B34aa, 'k-', ttb, B34bb, 'k--', ttc, B34cc, 'k-.', ttd, ...

B34dd, 'b-', tte, B34ee, 'b--', ttf, B34ff, 'b-.', ttg, B34gg, 'r-', tth, B34hh, 'r--');

plot(tta, B44aa, 'k-', ttb, B44bb, 'k--', ttc, B44cc, 'k-.', ttd, ...

B44dd, 'b-', tte, B44ee, 'b--', ttf, B44ff, 'b-.', ttg, B44gg, 'r-', tth, B44hh, 'r--');

xlabel ('t'); ylabel('B'); title('B04 - B14 - B24 - B34 - B44');

legend('A', 'B', 'C', 'D', 'E', 'F', 'G', 'H');

hold off;



%plotting curve

xa = x(1); ya = y(1);

xb = x(2); yb = y(2);

xc = x(3); yc = y(3);

xd = x(4); yd = y(4);

xe = x(5); ye = y(5);

xf = x(6); yf = y(6);

xg = x(7); yg = y(7);

xh = x(8); yh = y(8);

xaa = subs(xa, t, tta); yaa = subs(ya, t, tta);

xbb = subs(xb, t, ttb); ybb = subs(yb, t, ttb);

xcc = subs(xc, t, ttc); ycc = subs(yc, t, ttc);

xdd = subs(xd, t, ttd); ydd = subs(yd, t, ttd);

xee = subs(xe, t, tte); yee = subs(ye, t, tte);

xff = subs(xf, t, ttf); yff = subs(yf, t, ttf);

xgg = subs(xg, t, ttg); ygg = subs(yg, t, ttg);

xhh = subs(xh, t, tth); yhh = subs(yh, t, tth);

X = ［x0, x1, x2, x3, x4］; Y = ［y0, y1, y2, y3, y4］;

figure

subplot(131), plot(tta, xaa, 'k-', ttb, xbb, 'k--', ttc, xcc, 'k-.', ...

ttd, xdd, 'b-', tte, xee, 'b--', ttf, xff, 'b-.', ttg, xgg, 'r-', tth, xhh, 'r--');

xlabel('t'); ylabel('x'); title('t  x'); axis square;

subplot(132), plot(tta, yaa, 'k-', ttb, ybb, 'k--', ttc, ycc, 'k-.', ...

ttd, ydd, 'b-', tte, yee, 'b--', ttf, yff, 'b-.', ttg, ygg, 'r-', tth, yhh, 'r--');

xlabel('t'); ylabel('y'); title('t  y'); axis square;

subplot(133), plot(xaa, yaa, 'k-', xbb, ybb, 'k--', xcc, ycc, 'k-.', ...

xdd, ydd, 'b-', xee, yee, 'b--', xff, yff, 'b-.', xgg, ygg, 'r-', xhh, yhh, 'r--');

hold on;

scatter(X, Y, 20, 'r', 'filled');

xlabel('x'); ylabel('y'); title('x  y'); axis square; grid;

axis(［-1.5 1.5 -1.5 1.5］);

hold off;

开放均匀样条和非均匀样条的概念与前面讨论的二次样条的概念相似，请读者自己扩展三次样条的概念。

3．9本章小结

以下几点总结了本章讨论的主题： 

 近似样条一般不通过它们的任何CP。

 B样条是为克服贝塞尔样条的缺点而提出的近似样条。

 B样条由多个在连接点处具有连续性的曲线段组成。

 连接点处的参数变量t的值存储在KV中。

 均匀B样条在KV中具有均匀的间隙。

 B样条的BF使用Cox de Boor算法计算。

 B样条有两个定义参数，d与它的阶数有关，n与CP的数量有关。

 CP的数量可以独立于B样条的阶数进行更改。

 BF和B样条方程由于分段而由多个部分组成。

 均匀B样条的BF具有对称的形状，但相互偏移。

 更改CP仅影响特定段而不是整个曲线。

 空间B样条曲线与KV值无关。

 开放均匀B样条具有重复值的KV，称为多重性。

 多重性迫使逼近B样条曲线表现得像混合样条曲线。

 不均匀的B样条在KV中具有不均匀的间距。

 非均匀B样条的BF在形状上是不对称的。

3．10复习题

1． B样条和贝塞尔样条的主要区别是什么？

2． 区分均匀B样条、开放均匀B样条和非均匀B样条。

3． 什么是B样条曲线的KV？

4． 如何使用Cox de Boor算法计算B样条的BF？

5． B样条在什么条件下可以表现得像混合样条？

6． B样条的局部控制属性是什么意思？

7． CP的数量是否可以独立于B样条的次数而改变？

8． 改变KV对空间B样条曲线有何影响？

9． 为什么一个B样条曲线方程有多个子分量？

10． 什么是KV的多重性，它如何影响BF和样条曲线？

3．11练习题

1． 找出与CP，即(1，0)、(-1，1)和(1,-1)相关的线性均匀B样条方程。

2． 导出d=2和n=3的均匀线性B样条的BF。

3． 导出与5个CP相关的二次均匀B样条的BF。

4． 二次的B样条与4个CP相关联。若KV的形式为T=［0，0．2，0．5，0．7，…］，请使用Cox de Boor算法为前两个曲线段A和B找到第一个BF，即B03的表达式。

5． 一阶均匀B样条与3个 CP，即P0、P1和P2相关联。若KV的形式为T=［0，0．4，0．5，0．8，…］，请使用Cox de Boor算法导出第二条曲线段的方程。

6． 非均匀B样条的阶数为1，并与3个CP，即P0、P1和P2相关联。若KV为T=［0，4，5，8，9］，请导出第一和第二曲线段的方程。

7． 找到具有CP，即(2，5)、(4,-1)、(5，8)和(7,-5)的二次均匀B样条的方程。

8． 找到具有CP，即(2，5)、(4,-1)、(5，8)和(7,-5)且KV为T=［0，4、5、8、9、13、15］的二次非均匀B样条的方程。

9． 找到具有CP，即(2，0)、(4，1)、(5，7)、(6,-5)和(8,-1)的三次均匀B样条的前两段的方程。

10． 找到d=4 和n=4且KV为T=［0，1，2，5，6，8，10，12，13］的三次非均匀B样条的前两个BF。