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
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
v = from 0 to (2*PI) D2
b = rand2(0.5, a)
y = b*sin(v)
z = (a + b*cos(v))*cos(u)
u
= from 0 to (2*PI) D1v = from 0 to (2*PI) D2
a
= 10.0b = 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
b = rand2(1, 10)
z = (a + b*cos(v))*sin(u)
y = b*sin(v)
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
v = from 0 to (PI*2) D2
y = sin(v)
z = (1 + cos(v))*sin(u)
y = y*a
z = z*a
vertices = D1:100 D2:100
u
= from 0 to (PI*2) D1v = from 0 to (PI*2) D2
x
= (1 + cos(v))*cos(u)y = sin(v)
z = (1 + cos(v))*sin(u)
a
= 10x
= x*ay = y*a
z = z*a
(4)环桶
vertices = D1:72 D2:72
v = from 0 to (2*PI) D2
b = rand2(0.5, a)
y = b*sin(v) + if(sin(v) > 0, 10, –10)
z = (a + b*cos(v))*cos(u)
u
= from 0 to (2*PI) D1v = from 0 to (2*PI) D2
a
= 10.0b = 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
v = from 0 to (2*PI) D2
b = rand2(0.5, a)
y = b*sin(2*v)
z = (a + b*cos(v))*cos(u)
u
= from 0 to (2*PI) D1v = from 0 to (2*PI) D2
a
= 10.0b = 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
b = rand2(1, 10)
k = k / 2
z = (a+b*cos(v))*sin(u)
y = b*sin(v)*cos(k*u)
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)
e = rand2(-2,2)
z = t*sin(v)*sin(u) + e*(b + a*cos(v))*cos(u)
y = b*sin(v)
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
b = cos(u)
d = cos(v)
o = 2 * v
y = a + 2*d
z = r*cos(o)
y = y*5
z = z*5
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 + bo = 2 * v
x
= r*sin(o)y = a + 2*d
z = r*cos(o)
x
= x*5y = 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
c = rand2(1, a/2)
b = sqrt(a*a – c*c)
d = rand2(1, 10)
y = b*sin(u)*(a – d*cos(v))
z = b*sin(v)*(c*cos(u) – d)
y = y/w
z = z/w
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/wy = y/w
z = z/w
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