数学图形之圆环 – 叶飞影

这一节将为你展示如何生成圆环,以及各种与圆环相关的图形,有Cyclide surface,Horn Torus, tore de klein等.

相关软件参见:数学图形可视化工具,使用自己定义语法的脚本代码生成数学图形.

我之前写过生成圆环的C++程序,代码发布在圆环(Ring)图形的生成算法

 

(1)圆环

vertices = D1:72 D2:72
u = from 0 to (2*PI) D1
v = from 0 to (2*PI) D2

r = 3*cos(u) + 7

z = 3*sin(u)
y = r*sin(v)
x = r*cos(v)

y = y + 5

(2)随机半径的圆环

这里提供了两种写法:

vertices = D1:72 D2:72

u = from 0 to (2*PI) D1
v
= from 0 to (2*PI) D2

a = 10.0
b
= rand2(0.5, a)

x = (a + b*cos(v))*sin(u)
y
= b*sin(v)
z
= (a + b*cos(v))*cos(u)

#http://www.mathcurve.com/surfaces/tore/tore.shtml
vertices = D1:100 D2:100


u
= from 0 to (PI*2) D1
v
= from 0 to (PI*2) D2

a = rand2(1, 10)
b
= rand2(1, 10)

x = (a + b*cos(v))*cos(u)
z
= (a + b*cos(v))*sin(u)
y
= b*sin(v)

(3)Horn Torus

其特点是小圈半径等于大圈的一半

#http://mathworld.wolfram.com/HornTorus.html

vertices
= D1:100 D2:100

u = from 0 to (PI*2) D1
v
= from 0 to (PI*2) D2

x = (1 + cos(v))*cos(u)
y
= sin(v)
z
= (1 + cos(v))*sin(u)

a = 10

x = x*a
y
= y*a
z
= z*a

(4)环桶

vertices = D1:72 D2:72

u = from 0 to (2*PI) D1
v
= from 0 to (2*PI) D2

a = 10.0
b
= rand2(0.5, a)

x = (a + b*cos(v))*sin(u)
y
= b*sin(v) + if(sin(v) > 0, 10, –10)
z
= (a + b*cos(v))*cos(u)

(5)轮子

vertices = D1:72 D2:72

u = from 0 to (2*PI) D1
v
= from 0 to (2*PI) D2

a = 10.0
b
= rand2(0.5, a)

x = (a + b*cos(v))*sin(u)
y
= b*sin(2*v)
z
= (a + b*cos(v))*cos(u)

(6)tore de klein

#http://www.mathcurve.com/surfaces/klein/toredeklein.shtml

vertices
= D1:100 D2:100
u
= from 0 to (PI*2) D1
v
= from 0 to (PI*2) D2

a = rand2(1, 10)
b
= rand2(1, 10)

k = rand_int2(1, 20)
k
= k / 2

x = (a+b*cos(v))*cos(u)
z
= (a+b*cos(v))*sin(u)
y
= b*sin(v)*cos(k*u)

(7)拧着的圆环

#http://www.mathcurve.com/surfaces/tore/tore.shtml
vertices = D1:100 D2:100
u
= from 0 to (PI*2) D1
v
= from 0 to (PI*2) D2
a
= rand2(1, 10)
b
= rand2(0.5, a)

t = sqrt(a*a – b*b)
e
= rand2(-2,2)

x = t*sin(v)*cos(u) – e*(b + a*cos(v))*sin(u)
z
= t*sin(v)*sin(u) + e*(b + a*cos(v))*cos(u)
y
= b*sin(v)

(8)多圈的环

vertices = D1:100 D2:100
u
= from 0 to (2*PI) D1
v
= from 0 to (2*PI) D2

a = sin(u)
b
= cos(u)

c = sin(v)
d
= cos(v)

r = 3 + c + b
o
= 2 * v

x = r*sin(o)
y
= a + 2*d
z
= r*cos(o)

x = x*5
y
= y*5
z
= z*5

(9)偏圆环

vertices = D1:100 D2:100
u
= from 0 to (2*PI) D1
v
= from 0 to (2*PI) D2

a = rand2(5, 10)
c
= rand2(1, a/2)
b
= sqrt(a*a – c*c)
d
= rand2(1, 10)

w = a – c*cos(u)*cos(v)

x = d*(c – a*cos(u)*cos(v)) + b*b*cos(u)
y
= b*sin(u)*(a – d*cos(v))
z
= b*sin(v)*(c*cos(u) – d)

x = x/w
y
= y/w
z
= z/w

 

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