數學圖形之圓環

這一節將爲你展現如何生成圓環,以及各類與圓環相關的圖形,有Cyclide surface,Horn Torus, tore de klein等.

相關軟件參見:數學圖形可視化工具,使用本身定義語法的腳本代碼生成數學圖形.

我以前寫過生成圓環的C++程序,代碼發佈在圓環(Ring)圖形的生成算法

 

(1)圓環

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)隨機半徑的圓環

這裏提供了兩種寫法:

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

其特色是小圈半徑等於大圈的一半

#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)環桶

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)輪子

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)擰着的圓環

#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)多圈的環

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)偏圓環

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

 

(10)最後再補充下,圓環能夠看作是一個圓圈繞一個軸旋轉生成的,因此能夠有如下腳本代碼

vertices = D1:100 D2:100

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

r = 2
m = rand2(r, r*10)

n = r*cos(u) + m
y = r*sin(u)

x = n*cos(v)
z = n*sin(v)

 

 (11)補充一種環曲面:Bohemian

#http://http://www.mathcurve.com/surfaces/boheme/boheme.shtml

vertices = D1:100 D2:100

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

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

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