所给的边界条件即可解出r
ci而r
dir
3112iiiiiibccr
ii3r
r
r
，故r
r
可得到最终各个子区间上的三次样条函数。r
根据以上过程进行r
matlab编程，编写三次样条r
spli
e3函数，具体见附录。r
因此所编函数可对第一题求解：r
r
r
clearclcr
formatshortgr
x18125849948596995910166102r
y100774009902806070812182177r
u1001087u
100r
xx1x110001x1e
dr
yy1b1c1d1spli
e3x1y1xx11u1u
r
fpri
tfttb1ttc1ttd1
r
dispb1c11e
d11d1r
plotx1y1boxx1yy1kr
grido
r
r
r
fr
b1c1r
001087r
017405r
02878r
15294r
056548r
r
12794r
28949r
d1r
r
014489r
04485r
025891r
28188r
21035r
r
58247r
2599r
0368r
0393r
21153r
69141r
r
73614r
51232r
42279r
r
88599510105r
0r
05r
1r
15r
2r
25r
3r
图表一、三次样条曲线r
r
（2）坐标轴逆时针旋转45度，相当于节点顺时针旋转45度。设r
r
T为旋转前的r
xyr
r
r
r
坐标，r
r
xyT为旋转后的坐标，则有：r
r
r
xcossi
r
xr
r
r
r
r
r
r
r
r
r
r
r
ysi
r
cosr
yr
r
r
r
r
故旋转后的节点坐标为：r
r
r
thetapi4r
r
fori1le
gthx1r
x2icosthetax1isi
thetay1ir
y2isi
thetax1icosthetay1ir
r
e
dr
fpri
tftttx2ttty2
r
dispx2y2r
r
x2y2r
r
5856905r
6009758697r
65626166r
7131262826r
7288962876r
7890661935r
8461259157r
fr
8751956731r
r
端点处的一阶导数为：r
r
r
v1u1ta
theta1u1ta
thetar
v
u
ta
theta1u
ta
thetar
fpri
tftttv1tttv
r
dispv1v
r
r
v1v
r
09784909802r
r
则旋转后的三次样条的系数及图像为：r
r
r
xx2x210001x2e
dr
yy2b2c2d2spli
e3x2y2xx21v1v
r
fpri
tftttb2tttc2tttd2
r
dispb2c21e
d11d2r
plotx2y2bxx2yy2kr
r
r
grido
r
b2c2d2r
097849067221038277r
0747040431380090754r
0353620281020034909r
00676290221410053338r
0006174702466400046897r
0308102551010233r
06992043028012195r
r
r
r
r
r
r
r
r
r
r
556657758859r
64r
63r
62r
61r
6r
59r
58r
57r
56r
图表二、旋转后的三次样条曲线r
r
（3）将第（1）题中所得的样条曲线整体旋转45度（即顺时针旋转45度），并与r
第（2）题的曲线画在同一幅图上，得：r
fori1le
gthxx1r
xx3icosthetaxx1isi
thetayy1ir
yy3isi
thetaxx1icosthetayy1ir
r
e
dr
r
fr
plotxx3yy3xx2yy2x2y2okgrido
r
lege
d旋转前旋转后r
r
r
r
r
r
r
r
r
r
r
55665775885995r
68r
66r
64r
62r
6r
58r
56r
54r
52r
旋转前r
旋转后r
图表三、旋转前后样条曲线几何比较r
比较上图中的两条曲r