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1、http:/ July 4, 2012四川大学数学学院 徐小湛用二重积分计算旋转体体积用二重积分计算旋转体体积的几何解释的几何解释龙凤古镇1http:/ July 4, 2012四川大学数学学院 徐小湛 设设D是上半平面内的一个有界闭区域。是上半平面内的一个有界闭区域。 将将D绕绕x轴轴旋转一周得一旋转体，求该旋旋转一周得一旋转体，求该旋转体的体积转体的体积V。 我们用元素法来建立旋转体体积的二重我们用元素法来建立旋转体体积的二重积分公式。积分公式。D2http:/ July 4, 2012四川大学数学学院 徐小湛D在区域在区域D的的(x,y)处取一个面积元素处取一个面积元素它到它到x轴的距离
2、是轴的距离是 y （如图）。（如图）。该面积元素绕该面积元素绕x轴轴旋转而成的旋转体的体积约为：旋转而成的旋转体的体积约为：（体积元素）（体积元素）于是整个区域绕于是整个区域绕x轴轴旋转而旋转而成的旋转体的体积为：成的旋转体的体积为：3http:/ July 4, 2012四川大学数学学院 徐小湛D命题命题1：上半平面内一个有界闭区域：上半平面内一个有界闭区域D绕绕x轴轴旋转而成的旋转体的体积为：旋转而成的旋转体的体积为：4http:/ July 4, 2012四川大学数学学院 徐小湛下面来解下面来解释以上公式的几何意以上公式的几何意义5http:/ July 4, 2012四川大学数学学院
3、徐小湛区域区域D中一面积元素中一面积元素 绕绕x轴轴旋转而成旋转而成的旋转体为一的旋转体为一环形体环形体(如图如图)。6http:/ July 4, 2012四川大学数学学院 徐小湛区域区域D中一面积元素中一面积元素 绕绕x轴轴旋转而成旋转而成的旋转体为一环形体的旋转体为一环形体(如图如图)。其体积约为：其体积约为：（体积元素）（体积元素）7http:/ July 4, 2012四川大学数学学院 徐小湛将将dV在在D上二重上二重积分的几何意分的几何意义是将划分是将划分D的的n个面个面积元素分元素分别绕x轴旋旋转而成的旋而成的旋转体体相加，得到整个相加，得到整个D绕x轴旋旋转的旋的旋转体。体。于
4、是整个区域绕于是整个区域绕x轴轴旋转而成的旋转体旋转而成的旋转体的体积为：的体积为：8http:/ July 4, 2012四川大学数学学院 徐小湛以下以下图形形给出了出了这种方法的几何解种方法的几何解释9http:/ July 4, 2012四川大学数学学院 徐小湛10http:/ July 4, 2012四川大学数学学院 徐小湛display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4
5、_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h_20,h_28,h_2_8,h_44,h_4_4,h_66,h_6_6,h_80,h_84,h_8_4,scaling=
6、constrained,color=green);11http:/ July 4, 2012四川大学数学学院 徐小湛 display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,
7、y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h_20,h_28,h_2_8,h_44,h_4_4,h_66,h_6_6,h_80,h_84,h_8_4,scaling=constrained,color=green);12http:/ July 4, 2012四川大学数学学院 徐小湛 display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y0
8、8,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82
9、,y_84,y_8_2,y_8_4,h00,h_20,h_28,h_2_8,h_44,h_4_4,h_66,h_6_6,h_80,h_84,h_8_4,scaling=constrained,color=green);13http:/ July 4, 2012四川大学数学学院 徐小湛14http:/ July 4, 2012四川大学数学学院 徐小湛设想用想用电缆做成一个做成一个圆环体体那么那么这个个圆环体可由体可由电缆中很多中很多圆环形形状的光状的光纤组成成因此，我因此，我们可以把可以把这种种计算旋算旋转体体体体积的方法形象地称的方法形象地称为光纤法光纤法或或电缆法电缆法15http:/ Ju
10、ly 4, 2012四川大学数学学院 徐小湛更多的更多的图形形16http:/ July 4, 2012四川大学数学学院 徐小湛17http:/ July 4, 2012四川大学数学学院 徐小湛18http:/ July 4, 2012四川大学数学学院 徐小湛with(plots):xzhou:=spacecurve(x,0,0, x=-2.2, thickness=1,color=black):yzhou:=spacecurve(0,y,0, y=-2.2, thickness=1,color=black):zzhou:=spacecurve(0,0,z, z=-2.4, thickness
11、=1,color=black):a:=0:b:=3:R:=1:r:=0.1:yuan:=spacecurve(0,a+R*cos(t),b+R*sin(t), t=0.2*Pi, thickness=3,color=red):a0:=0:a2:=0.2:a4:=0.4:a6:=0.6:a8:=0.8:a_2:=-0.2:a_4:=-0.4:a_6:=-0.6:a_8:=-0.8:b0:=3:b2:=3.2:b4:=3.4:b6:=3.6:b8:=3.8:b_2:=3-0.2:b_4:=3-0.4:b_6:=3-0.6:b_8:=3-0.8:y00:=spacecurve(0,a0+r*cos(
12、t),b0+r*sin(t), t=0.2*Pi,color=blue):y02:=spacecurve(0,a0+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y04:=spacecurve(0,a0+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y06:=spacecurve(0,a0+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y08:=spacecurve(0,a0+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y0_2:=s
13、pacecurve(0,a0+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y0_4:=spacecurve(0,a0+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y0_6:=spacecurve(0,a0+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y0_8:=spacecurve(0,a0+r*cos(t),b_8+r*sin(t), t=0.2*Pi,color=blue):y20:=spacecurve(0,a2+r*cos(t),b0+r*sin(t),
14、t=0.2*Pi,color=blue):y22:=spacecurve(0,a2+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y24:=spacecurve(0,a2+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y26:=spacecurve(0,a2+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y28:=spacecurve(0,a2+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y2_2:=spacecurve(0,a2+r
15、*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y2_4:=spacecurve(0,a2+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y2_6:=spacecurve(0,a2+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y2_8:=spacecurve(0,a2+r*cos(t),b_8+r*sin(t), t=0.2*Pi,color=blue):y40:=spacecurve(0,a4+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=b
16、lue):y42:=spacecurve(0,a4+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y44:=spacecurve(0,a4+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y46:=spacecurve(0,a4+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y48:=spacecurve(0,a4+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y4_2:=spacecurve(0,a4+r*cos(t),b_2+r*si
17、n(t), t=0.2*Pi,color=blue):y4_4:=spacecurve(0,a4+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y4_6:=spacecurve(0,a4+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y4_8:=spacecurve(0,a4+r*cos(t),b_8+r*sin(t), t=0.2*Pi,color=blue):y60:=spacecurve(0,a6+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y62:=spacec
18、urve(0,a6+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y64:=spacecurve(0,a6+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y66:=spacecurve(0,a6+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y6_2:=spacecurve(0,a6+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y6_4:=spacecurve(0,a6+r*cos(t),b_4+r*sin(t), t=0.2*Pi
19、,color=blue):y6_6:=spacecurve(0,a6+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y80:=spacecurve(0,a8+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y82:=spacecurve(0,a8+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y84:=spacecurve(0,a8+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y8_2:=spacecurve(0,a8+r*cos(t
20、),b_2+r*sin(t), t=0.2*Pi,color=blue):y8_4:=spacecurve(0,a8+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y8_6:=spacecurve(0,a8+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y_20:=spacecurve(0,a_2+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y_22:=spacecurve(0,a_2+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue)
21、:y_24:=spacecurve(0,a_2+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y_26:=spacecurve(0,a_2+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y_28:=spacecurve(0,a_2+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y_2_2:=spacecurve(0,a_2+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y_2_4:=spacecurve(0,a_2+r*cos(t),
22、b_4+r*sin(t), t=0.2*Pi,color=blue):y_2_6:=spacecurve(0,a_2+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y_2_8:=spacecurve(0,a_2+r*cos(t),b_8+r*sin(t), t=0.2*Pi,color=blue):y_40:=spacecurve(0,a_4+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y_42:=spacecurve(0,a_4+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blu
23、e):y_44:=spacecurve(0,a_4+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y_46:=spacecurve(0,a_4+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y_48:=spacecurve(0,a_4+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y_4_2:=spacecurve(0,a_4+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y_4_4:=spacecurve(0,a_4+r*cos(t
24、),b_4+r*sin(t), t=0.2*Pi,color=blue):y_4_6:=spacecurve(0,a_4+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y_4_8:=spacecurve(0,a_4+r*cos(t),b_8+r*sin(t), t=0.2*Pi,color=blue):y_60:=spacecurve(0,a_6+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y_62:=spacecurve(0,a_6+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=b
25、lue):y_64:=spacecurve(0,a_6+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y_66:=spacecurve(0,a_6+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y_6_2:=spacecurve(0,a_6+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y_6_4:=spacecurve(0,a_6+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y_6_6:=spacecurve(0,a_6+r*c
26、os(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y_80:=spacecurve(0,a_8+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y_82:=spacecurve(0,a_8+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y_84:=spacecurve(0,a_8+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y_8_2:=spacecurve(0,a_8+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color
27、=blue):y_8_4:=spacecurve(0,a_8+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y_8_6:=spacecurve(0,a_8+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):h00:=plot3d(b0+r*cos(t)*sin(u),a0+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h02:=plot3d(b2+r*cos(t)*sin(u),a0+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0
28、.2*Pi):h04:=plot3d(b4+r*cos(t)*sin(u),a0+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h06:=plot3d(b6+r*cos(t)*sin(u),a0+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h08:=plot3d(b8+r*cos(t)*sin(u),a0+r*sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h0_2:=plot3d(b_2+r*cos(t)*sin(u),a0+r*sin(t),(b
29、_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h0_4:=plot3d(b_4+r*cos(t)*sin(u),a0+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h0_6:=plot3d(b_6+r*cos(t)*sin(u),a0+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h0_8:=plot3d(b_8+r*cos(t)*sin(u),a0+r*sin(t),(b_8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h20:=pl
30、ot3d(b0+r*cos(t)*sin(u),a2+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h22:=plot3d(b2+r*cos(t)*sin(u),a2+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h24:=plot3d(b4+r*cos(t)*sin(u),a2+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h26:=plot3d(b6+r*cos(t)*sin(u),a2+r*sin(t),(b6+r*cos(t)*cos(u
31、),t=0.2*Pi,u=0.2*Pi):h28:=plot3d(b8+r*cos(t)*sin(u),a2+r*sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h2_2:=plot3d(b_2+r*cos(t)*sin(u),a2+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h2_4:=plot3d(b_4+r*cos(t)*sin(u),a2+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h2_6:=plot3d(b_6+r*cos(t)*s
32、in(u),a2+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h2_8:=plot3d(b_8+r*cos(t)*sin(u),a2+r*sin(t),(b_8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h40:=plot3d(b0+r*cos(t)*sin(u),a4+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h42:=plot3d(b2+r*cos(t)*sin(u),a4+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0
33、.2*Pi):h44:=plot3d(b4+r*cos(t)*sin(u),a4+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h46:=plot3d(b6+r*cos(t)*sin(u),a4+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h48:=plot3d(b8+r*cos(t)*sin(u),a4+r*sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h4_2:=plot3d(b_2+r*cos(t)*sin(u),a4+r*sin(t),(b
34、_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h4_4:=plot3d(b_4+r*cos(t)*sin(u),a4+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h4_6:=plot3d(b_6+r*cos(t)*sin(u),a4+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h4_8:=plot3d(b_8+r*cos(t)*sin(u),a4+r*sin(t),(b_8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h60:=pl
35、ot3d(b0+r*cos(t)*sin(u),a6+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h62:=plot3d(b2+r*cos(t)*sin(u),a6+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h64:=plot3d(b4+r*cos(t)*sin(u),a6+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h66:=plot3d(b6+r*cos(t)*sin(u),a6+r*sin(t),(b6+r*cos(t)*cos(u
36、),t=0.2*Pi,u=0.2*Pi):h6_2:=plot3d(b_2+r*cos(t)*sin(u),a6+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h6_4:=plot3d(b_4+r*cos(t)*sin(u),a6+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h6_6:=plot3d(b_6+r*cos(t)*sin(u),a6+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h80:=plot3d(b0+r*cos(t)*
37、sin(u),a8+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h82:=plot3d(b2+r*cos(t)*sin(u),a8+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h84:=plot3d(b4+r*cos(t)*sin(u),a8+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h8_2:=plot3d(b_2+r*cos(t)*sin(u),a8+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0
38、.2*Pi):h8_4:=plot3d(b_4+r*cos(t)*sin(u),a8+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_20:=plot3d(b0+r*cos(t)*sin(u),a_2+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_22:=plot3d(b2+r*cos(t)*sin(u),a_2+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_24:=plot3d(b4+r*cos(t)*sin(u),a_2+r*si
39、n(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_26:=plot3d(b6+r*cos(t)*sin(u),a_2+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_28:=plot3d(b8+r*cos(t)*sin(u),a_2+r*sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_2_2:=plot3d(b_2+r*cos(t)*sin(u),a_2+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h
40、_2_4:=plot3d(b_4+r*cos(t)*sin(u),a_2+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_2_6:=plot3d(b_6+r*cos(t)*sin(u),a_2+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_2_8:=plot3d(b_8+r*cos(t)*sin(u),a_2+r*sin(t),(b_8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_40:=plot3d(b0+r*cos(t)*sin(u),a_4+r*si
41、n(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_42:=plot3d(b2+r*cos(t)*sin(u),a_4+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_44:=plot3d(b4+r*cos(t)*sin(u),a_4+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_46:=plot3d(b6+r*cos(t)*sin(u),a_4+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_48
42、:=plot3d(b8+r*cos(t)*sin(u),a_4+r*sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_4_2:=plot3d(b_2+r*cos(t)*sin(u),a_4+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_4_4:=plot3d(b_4+r*cos(t)*sin(u),a_4+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_4_6:=plot3d(b_6+r*cos(t)*sin(u),a_4+r*sin(t)
43、,(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_4_8:=plot3d(b_8+r*cos(t)*sin(u),a_4+r*sin(t),(b_8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_60:=plot3d(b0+r*cos(t)*sin(u),a_6+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_62:=plot3d(b2+r*cos(t)*sin(u),a_6+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_64
44、:=plot3d(b4+r*cos(t)*sin(u),a_6+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_66:=plot3d(b6+r*cos(t)*sin(u),a_6+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_6_2:=plot3d(b_2+r*cos(t)*sin(u),a_6+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_6_4:=plot3d(b_4+r*cos(t)*sin(u),a_6+r*sin(t),(b
45、_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_6_6:=plot3d(b_6+r*cos(t)*sin(u),a_6+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_80:=plot3d(b0+r*cos(t)*sin(u),a_8+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_82:=plot3d(b2+r*cos(t)*sin(u),a_8+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_84:=p
46、lot3d(b4+r*cos(t)*sin(u),a_8+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_8_2:=plot3d(b_2+r*cos(t)*sin(u),a_8+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_8_4:=plot3d(b_4+r*cos(t)*sin(u),a_8+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y0
47、8,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82
48、,y_84,y_8_2,y_8_4,h00,h02,h04,h06,h08,h0_2,h0_4,h0_6,h0_8,h20,h22,h24,h26,h28,h2_2,h2_4,h2_6,h2_8,h40,h42,h44,h46,h48,h4_2,h4_4,h4_6,h4_8,h60,h62,h64,h66,h6_2,h6_4,h6_6,h80,h82,h84,h8_2,h8_4,h_20,h_22,h_24,h_26,h_28,h_2_2,h_2_4,h_2_6,h_2_8,h_40,h_42,h_44,h_46,h_48,h_4_2,h_4_4,h_4_6,h_4_8,h_60,h_62,h
49、_64,h_66,h_6_2,h_6_4,h_6_6,h_80,h_82,h_84,h_8_2,h_8_4,scaling=constrained);19http:/ July 4, 2012四川大学数学学院 徐小湛 display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y
50、84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h04,h08,h0_4,h0_8,h22,h26,h2_2,h2_6,h44,h48,h4_4,h4_8,h62,h66,h6_2,h6_6,h80,h84,h8_4,h_20,h_24,h_28,h_2_4,h_2_8,h_44,h_4
51、8,h_4_4,h_4_8,h_62,h_66,h_6_2,h_6_6,h_80,h_84,h_8_4,scaling=constrained);20http:/ July 4, 2012四川大学数学学院 徐小湛 display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84
52、,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h08,h0_8,h26,h2_6,h44,h4_4,h62,h6_2,h80,h84,h8_4,h_20,h_28,h_2_8,h_44,h_4_4,h_66,h_6_6,h_80,h_84,h_8_4,scaling=constrained
53、);21http:/ July 4, 2012四川大学数学学院 徐小湛display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_
54、44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h_20,h_28,h_2_8,h_44,h_4_4,h_66,h_6_6,h_80,h_84,h_8_4,scaling=constrained,color=green);22http:/ July 4, 2012四川大学数学学院 徐小湛display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20
55、,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h_2
56、0,h_28,h_2_8,h_44,h_4_4,h_66,h_6_6,h_80,h_84,h_8_4,scaling=constrained,color=green);23http:/ July 4, 2012四川大学数学学院 徐小湛24http:/ July 4, 2012四川大学数学学院 徐小湛25http:/ July 4, 2012四川大学数学学院 徐小湛26http:/ July 4, 2012四川大学数学学院 徐小湛 with(plots):xzhou:=spacecurve(x,0,0, x=-2.2, thickness=1,color=black):yzhou:=spacec
57、urve(0,y,0, y=-2.2, thickness=1,color=black):zzhou:=spacecurve(0,0,z, z=-2.4, thickness=1,color=black):a:=0:b:=3:R:=1:r:=0.1:yuan:=spacecurve(0,a+R*cos(t),b+R*sin(t), t=0.2*Pi, thickness=3,color=red):a0:=0:a2:=0.2:a4:=0.4:a6:=0.6:a8:=0.8:a_2:=-0.2:a_4:=-0.4:a_6:=-0.6:a_8:=-0.8:b0:=3:b2:=3.2:b4:=3.4:
58、b6:=3.6:b8:=3.8:b_2:=3-0.2:b_4:=3-0.4:b_6:=3-0.6:b_8:=3-0.8:y00:=spacecurve(0,a0+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y02:=spacecurve(0,a0+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y04:=spacecurve(0,a0+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y06:=spacecurve(0,a0+r*cos(t),b6+r*sin(t), t=0.2
59、*Pi,color=blue):y08:=spacecurve(0,a0+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y0_2:=spacecurve(0,a0+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y0_4:=spacecurve(0,a0+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y0_6:=spacecurve(0,a0+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y0_8:=spacecurve(0,a0+
60、r*cos(t),b_8+r*sin(t), t=0.2*Pi,color=blue):y20:=spacecurve(0,a2+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y22:=spacecurve(0,a2+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y24:=spacecurve(0,a2+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y26:=spacecurve(0,a2+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):
61、y28:=spacecurve(0,a2+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y2_2:=spacecurve(0,a2+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y2_4:=spacecurve(0,a2+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y2_6:=spacecurve(0,a2+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y2_8:=spacecurve(0,a2+r*cos(t),b_8+r*s
62、in(t), t=0.2*Pi,color=blue):y40:=spacecurve(0,a4+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y42:=spacecurve(0,a4+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y44:=spacecurve(0,a4+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y46:=spacecurve(0,a4+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y48:=spacecurve(
63、0,a4+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y4_2:=spacecurve(0,a4+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y4_4:=spacecurve(0,a4+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y4_6:=spacecurve(0,a4+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y4_8:=spacecurve(0,a4+r*cos(t),b_8+r*sin(t), t=0.2*Pi,
64、color=blue):y60:=spacecurve(0,a6+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y62:=spacecurve(0,a6+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y64:=spacecurve(0,a6+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y66:=spacecurve(0,a6+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y6_2:=spacecurve(0,a6+r*cos(t),b
65、_2+r*sin(t), t=0.2*Pi,color=blue):y6_4:=spacecurve(0,a6+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y6_6:=spacecurve(0,a6+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y80:=spacecurve(0,a8+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y82:=spacecurve(0,a8+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y84:=s
66、pacecurve(0,a8+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y8_2:=spacecurve(0,a8+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y8_4:=spacecurve(0,a8+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y8_6:=spacecurve(0,a8+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y_20:=spacecurve(0,a_2+r*cos(t),b0+r*sin(t),
67、 t=0.2*Pi,color=blue):y_22:=spacecurve(0,a_2+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y_24:=spacecurve(0,a_2+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y_26:=spacecurve(0,a_2+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y_28:=spacecurve(0,a_2+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y_2_2:=spacecu
68、rve(0,a_2+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y_2_4:=spacecurve(0,a_2+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y_2_6:=spacecurve(0,a_2+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y_2_8:=spacecurve(0,a_2+r*cos(t),b_8+r*sin(t), t=0.2*Pi,color=blue):y_40:=spacecurve(0,a_4+r*cos(t),b0+r*sin(t
69、), t=0.2*Pi,color=blue):y_42:=spacecurve(0,a_4+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y_44:=spacecurve(0,a_4+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y_46:=spacecurve(0,a_4+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y_48:=spacecurve(0,a_4+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y_4_2:=space
70、curve(0,a_4+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y_4_4:=spacecurve(0,a_4+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y_4_6:=spacecurve(0,a_4+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y_4_8:=spacecurve(0,a_4+r*cos(t),b_8+r*sin(t), t=0.2*Pi,color=blue):y_60:=spacecurve(0,a_6+r*cos(t),b0+r*sin
71、(t), t=0.2*Pi,color=blue):y_62:=spacecurve(0,a_6+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y_64:=spacecurve(0,a_6+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y_66:=spacecurve(0,a_6+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y_6_2:=spacecurve(0,a_6+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y_6_4:=s
72、pacecurve(0,a_6+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y_6_6:=spacecurve(0,a_6+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y_80:=spacecurve(0,a_8+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y_82:=spacecurve(0,a_8+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y_84:=spacecurve(0,a_8+r*cos(t),b4+r*sin
73、(t), t=0.2*Pi,color=blue):y_8_2:=spacecurve(0,a_8+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y_8_4:=spacecurve(0,a_8+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y_8_6:=spacecurve(0,a_8+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):h00:=plot3d(b0+r*cos(t)*sin(u),a0+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*P
74、i,u=1.5.6.6):h02:=plot3d(b2+r*cos(t)*sin(u),a0+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=1.4.6.3):h04:=plot3d(b4+r*cos(t)*sin(u),a0+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=1.56.6.4):h06:=plot3d(b6+r*cos(t)*sin(u),a0+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=1.3.6.31):h08:=plot3d(b8+r*cos(t)*sin(u),a0+r*
75、sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=1.2.6.29):h0_2:=plot3d(b_2+r*cos(t)*sin(u),a0+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=1.2*Pi):h0_4:=plot3d(b_4+r*cos(t)*sin(u),a0+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=1.59.6.33):h0_6:=plot3d(b_6+r*cos(t)*sin(u),a0+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=1.
76、4.5.2):h0_8:=plot3d(b_8+r*cos(t)*sin(u),a0+r*sin(t),(b_8+r*cos(t)*cos(u),t=0.2*Pi,u=1.6.3):h20:=plot3d(b0+r*cos(t)*sin(u),a2+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=1.4.6.5):h22:=plot3d(b2+r*cos(t)*sin(u),a2+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=1.3.6.26):h24:=plot3d(b4+r*cos(t)*sin(u),a2+r*sin(t)
77、,(b4+r*cos(t)*cos(u),t=0.2*Pi,u=1.25.6.4):h26:=plot3d(b6+r*cos(t)*sin(u),a2+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=1.2*Pi):h28:=plot3d(b8+r*cos(t)*sin(u),a2+r*sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=1.8.6.26):h2_2:=plot3d(b_2+r*cos(t)*sin(u),a2+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=1.6.6.5):h2_4:=
78、plot3d(b_4+r*cos(t)*sin(u),a2+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=1.5.6.31):h2_6:=plot3d(b_6+r*cos(t)*sin(u),a2+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=1.6.6.34):h2_8:=plot3d(b_8+r*cos(t)*sin(u),a2+r*sin(t),(b_8+r*cos(t)*cos(u),t=0.2*Pi,u=1.57.6.25):h40:=plot3d(b0+r*cos(t)*sin(u),a4+r*sin(t),(
79、b0+r*cos(t)*cos(u),t=0.2*Pi,u=1.25.6.4):h42:=plot3d(b2+r*cos(t)*sin(u),a4+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=1.4.6.2):h44:=plot3d(b4+r*cos(t)*sin(u),a4+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=1.2.2*Pi):h46:=plot3d(b6+r*cos(t)*sin(u),a4+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=1.25.6.4):h48:=plot
80、3d(b8+r*cos(t)*sin(u),a4+r*sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=1.1.2*Pi):h4_2:=plot3d(b_2+r*cos(t)*sin(u),a4+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=1.4.2*Pi):h4_4:=plot3d(b_4+r*cos(t)*sin(u),a4+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=1.25.6.4):h4_6:=plot3d(b_6+r*cos(t)*sin(u),a4+r*sin(t),(b_6+r
81、*cos(t)*cos(u),t=0.2*Pi,u=1.6.2*Pi):h4_8:=plot3d(b_8+r*cos(t)*sin(u),a4+r*sin(t),(b_8+r*cos(t)*cos(u),t=0.2*Pi,u=1.4.6.2):h60:=plot3d(b0+r*cos(t)*sin(u),a6+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=1.7.6.4):h62:=plot3d(b2+r*cos(t)*sin(u),a6+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=1.25.6.4):h64:=plot3d
82、(b4+r*cos(t)*sin(u),a6+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=1.4.6.2):h66:=plot3d(b6+r*cos(t)*sin(u),a6+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=1.6.2*Pi):h6_2:=plot3d(b_2+r*cos(t)*sin(u),a6+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=1.25.6.4):h6_4:=plot3d(b_4+r*cos(t)*sin(u),a6+r*sin(t),(b_4+r*cos(t
83、)*cos(u),t=0.2*Pi,u=1.5.2*Pi):h6_6:=plot3d(b_6+r*cos(t)*sin(u),a6+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=1.7.2*Pi):h80:=plot3d(b0+r*cos(t)*sin(u),a8+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=1.25.6.6):h82:=plot3d(b2+r*cos(t)*sin(u),a8+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=1.4.2*Pi):h84:=plot3d(b4+
84、r*cos(t)*sin(u),a8+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=1.25.6.4):h8_2:=plot3d(b_2+r*cos(t)*sin(u),a8+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=1.3.6.5):h8_4:=plot3d(b_4+r*cos(t)*sin(u),a8+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=1.6.2*Pi):h_20:=plot3d(b0+r*cos(t)*sin(u),a_2+r*sin(t),(b0+r*cos(t)*
85、cos(u),t=0.2*Pi,u=1.6.2*Pi):h_22:=plot3d(b2+r*cos(t)*sin(u),a_2+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=1.25.6.4):h_24:=plot3d(b4+r*cos(t)*sin(u),a_2+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=1.2*Pi):h_26:=plot3d(b6+r*cos(t)*sin(u),a_2+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=1.25.6.4):h_28:=plot3d(b8+
86、r*cos(t)*sin(u),a_2+r*sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=1.1.2*Pi):h_2_2:=plot3d(b_2+r*cos(t)*sin(u),a_2+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=1.6.6.5):h_2_4:=plot3d(b_4+r*cos(t)*sin(u),a_2+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=1.4.6.2):h_2_6:=plot3d(b_6+r*cos(t)*sin(u),a_2+r*sin(t),(b_6+r*
87、cos(t)*cos(u),t=0.2*Pi,u=1.5.2*Pi):h_2_8:=plot3d(b_8+r*cos(t)*sin(u),a_2+r*sin(t),(b_8+r*cos(t)*cos(u),t=0.2*Pi,u=1.4.6.2):h_40:=plot3d(b0+r*cos(t)*sin(u),a_4+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=1.6.2*Pi):h_42:=plot3d(b2+r*cos(t)*sin(u),a_4+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=1.5.2*Pi):h_44:
88、=plot3d(b4+r*cos(t)*sin(u),a_4+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=1.7.2*Pi):h_46:=plot3d(b6+r*cos(t)*sin(u),a_4+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=1.58.2*Pi):h_48:=plot3d(b8+r*cos(t)*sin(u),a_4+r*sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=1.53.6.3):h_4_2:=plot3d(b_2+r*cos(t)*sin(u),a_4+r*sin(t)
89、,(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=1.3.6.2):h_4_4:=plot3d(b_4+r*cos(t)*sin(u),a_4+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=1.2*Pi):h_4_6:=plot3d(b_6+r*cos(t)*sin(u),a_4+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=1.3.6.31):h_4_8:=plot3d(b_8+r*cos(t)*sin(u),a_4+r*sin(t),(b_8+r*cos(t)*cos(u),t=0.2*Pi,u=1.5
90、.2*Pi):h_60:=plot3d(b0+r*cos(t)*sin(u),a_6+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=1.6.2*Pi):h_62:=plot3d(b2+r*cos(t)*sin(u),a_6+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=1.7.2*Pi):h_64:=plot3d(b4+r*cos(t)*sin(u),a_6+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=2.6.6):h_66:=plot3d(b6+r*cos(t)*sin(u),a_6+r*
91、sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=1.7.2*Pi):h_6_2:=plot3d(b_2+r*cos(t)*sin(u),a_6+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=1.5.2*Pi):h_6_4:=plot3d(b_4+r*cos(t)*sin(u),a_6+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=1.3.6.31):h_6_6:=plot3d(b_6+r*cos(t)*sin(u),a_6+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*
92、Pi,u=1.7.2*Pi):h_80:=plot3d(b0+r*cos(t)*sin(u),a_8+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=1.2.2*Pi):h_82:=plot3d(b2+r*cos(t)*sin(u),a_8+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=1.5.6.6):h_84:=plot3d(b4+r*cos(t)*sin(u),a_8+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=1.4.2*Pi):h_8_2:=plot3d(b_2+r*cos(t)*s
93、in(u),a_8+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=1.6.2*Pi):h_8_4:=plot3d(b_4+r*cos(t)*sin(u),a_8+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=1.3.6.31):display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8
94、,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h02,h04,h06,h08,h0_2,h0_4,h0_6,h0_8,h20,h22,h24,h26,h28,h2_2,h2_4,h2_6,h2_8,h40
95、,h42,h44,h46,h48,h4_2,h4_4,h4_6,h4_8,h60,h62,h64,h66,h6_2,h6_4,h6_6,h80,h82,h84,h8_2,h8_4,h_20,h_22,h_24,h_26,h_28,h_2_2,h_2_4,h_2_6,h_2_8,h_40,h_42,h_44,h_46,h_48,h_4_2,h_4_4,h_4_6,h_4_8,h_60,h_62,h_64,h_66,h_6_2,h_6_4,h_6_6,h_80,h_82,h_84,h_8_2,h_8_4,scaling=constrained,orientation=0,90); 27http:/ July 4, 2012四川大学数学学院 徐小湛更多的学习资料见：更多的学习资料见： http:/ 28
