数学图形之球面,椭球面,胶囊体,刺球

从这一节开始讲3D的数学图形,为此我准备了好久.这一节中将为你展示如何生成球面,椭球面,胶囊体,刺球,圆弧面.

相关软件参见:数学图形可视化工具,使用我自定义语法的脚本代码生成数学图形.

在我刚学计算机图形学的时候,就写过生成球面的程序,代码曾发布在球(Sphere)图形的生成算法.

球与圆很相关,一个是三维,一个是二维,可以参考下:圆,椭圆

 

(1)sphere的第一种写法

vertices = D1:100 D2:100



t = from 0 to (PI*2) D1

r = from 0 to 1 D2



x = 2*r*sin(t)*sqrt(1-r^2)

y = 2*r*cos(t)*sqrt(1-r^2)

z = 1-2*(r^2)

 

球的网格线:

(2)sphere的另两种写法

vertices = dimension1:36 dimension2:72

u = from 0 to (2*PI) dimension1

v = from (-PI*0.5) to (PI*0.5) dimension2

r = 10.0

x = r*cos(v)*sin(u)

y = r*sin(v)

z = r*cos(v)*cos(u)

vertices = dimension1:36 dimension2:72

u = from 0 to (2*PI) dimension1

v = from 0 to (PI) dimension2

r = 10.0

x = r*sin(v)*sin(u)

y = r*cos(v)

z = r*sin(v)*cos(u)

两种写法生成的图形是一样的

 

(3)彩色球

在脚本中给rgb变量设值,就能设置顶点色.

vertices = dimension1:72 dimension2:72



u = from 0 to (2*PI) dimension1

v = from (-PI*0.5) to (PI*0.5) dimension2



x = cos(v)*sin(u)

y = sin(v)

z = cos(v)*cos(u)



a = 10.0



r = (x+1.0)/2

g = (y+1.0)/2

b = (z+1.0)/2



x = a*x

y = a*y

z = a*z

(4)圆弧面

将球的第二维度范围减小,即得到圆弧面

vertices = dimension1:36 dimension2:72

u = from 0 to (2*PI) dimension1

v = from (PI*0.1) to (PI*0.5) dimension2

r = 10.0

x = r*cos(v)*sin(u)

y = r*sin(v)

z = r*cos(v)*cos(u)

 

(5)椭球面

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



vertices = D1:100 D2:100



u = from 0 to (2*PI) D1

v = from (-PI*0.5) to (PI*0.5) D2



a = rand2(1, 10)

b = rand2(1, 10)

c = rand2(1, 10)



x = a*cos(v)*sin(u)

y = b*sin(v)

z = c*cos(v)*cos(u)

 

(6)胶囊体

将球面向上下两头拉伸,即得到胶囊体,我也曾经写过胶囊体的生成算法胶囊体(Capsule)图形的生成算法

 

(7)刺球

将球面上顶点到球心的距离,有规律地变化,可以得到多变的球,如刺球

vertices = dimension1:129 dimension2:65



u = from 0 to (2*PI) dimension1

v = from (-PI*0.5) to (PI*0.5) dimension2



n =4



a = from 0 to 128 D1

b = from 0 to 64 D2



t = (mod(a, n) + mod(b, n))/n*4



r = 10.0 + t



x = r*cos(v)*sin(u)

y = r*sin(v)

z = r*cos(v)*cos(u)