超球,自然界有很多果实属于超球的形状.之前曾经写过关于超圆的文章:数学图形(1.44)超圆, 这篇文章将对其扩展一下,由超圆的二维曲线转化为超球三维曲面.
超圆就是方程式:x^a+y^b= c 生成的图形.当a==b==2时,为一个圆.
超椭圆是方程式:m*x^a+n*y^b= c 生成的图形.当a==b==2时,为一个椭圆.
那么超球则是如下定义:
超球的方程式:x^a+y^b+z^c= d
超椭球的方程式:m*x^a+n*y^b+k*z^c= d
我使用自己定义语法的脚本代码生成超球图形.相关软件参见:数学图形可视化工具,该软件免费开源.QQ交流群: 367752815
(1)将超圆沿X轴或Y轴旋转生成的图形也是超球的一种
vertices = D1:100 D2:100 u = from (-PI/2) to (PI/2) D1 v = from 0 to (2*PI) D2 a = rand2(0.1, 4) b = rand2(0.1, 4) r = 10.0 x = r*pow_sign(sin(u), a) n = r*pow_sign(cos(u), b) y = n*cos(v) z = n*sin(v)
vertices = D1:100 D2:100 u = from 0 to (PI) D1 v = from 0 to (2*PI) D2 a = rand2(0.1, 4) b = rand2(0.1, 4) r = 10.0 n = r*pow_sign(sin(u), a) y = r*pow_sign(cos(u), b) x = n*cos(v) z = n*sin(v)
(2)超球面(瘦)
vertices = D1:100 D2:100 u = from 0 to (2*PI) D1 v = from (-PI*0.5) to (PI*0.5) D2 a = 10 m = rand2(1, 5) x = a*pow_sign(cos(u)*cos(v), m) y = a*pow_sign(sin(v), m) z = a*pow_sign(sin(u)*cos(v), m)
(3)超球面(胖)
vertices = D1:100 D2:100 u = from 0 to (2*PI) D1 v = from (-PI*0.5) to (PI*0.5) D2 a = 10 m = rand2(0.1, 1) x = a*pow_sign(cos(u)*cos(v), m) y = a*pow_sign(sin(v), m) z = a*pow_sign(sin(u)*cos(v), m)
(4)超球面(双参)
vertices = D1:100 D2:100 u = from 0 to (2*PI) D1 v = from (-PI*0.5) to (PI*0.5) D2 a = 10 m = rand2(0.2, 5) n = rand2(0.2, 5) x = a*pow_sign(cos(u)*cos(v), m) y = a*pow_sign(sin(v), n) z = a*pow_sign(sin(u)*cos(v), m)
(5)超球面(三参)
vertices = D1:100 D2:100 u = from 0 to (2*PI) D1 v = from (-PI*0.5) to (PI*0.5) D2 r = 10 a = rand2(0.2, 5) b = rand2(0.2, 5) c = rand2(0.2, 5) x = r*pow_sign(cos(u)*cos(v), a) y = r*pow_sign(sin(v), b) z = r*pow_sign(sin(u)*cos(v), c)
(6)超椭球面
vertices = D1:100 D2:100 u = from (-PI*0.5) to (PI*0.5) D1 v = from (-PI) to (PI) D2 a = rand2(1, 5) b = rand2(1, 5) c = rand2(1, 5) m = 5 n = 3 x = a*(cos(u)^m)*(cos(v)^n) z = b*(cos(u)^m)*(sin(v)^n) y = c*(sin(u)^m)
vertices = D1:100 D2:100 u = from (-PI*0.5) to (PI*0.5) D1 v = from (-PI) to (PI) D2 a = rand2(1, 5) b = rand2(1, 5) c = rand2(1, 5) m = 0.6 n = 0.3 x = a*(pow_sign(cos(u), m))*(pow_sign(cos(v),n)) z = b*(pow_sign(cos(u),m))*(pow_sign(sin(v),n)) y = c*(pow_sign(sin(u),m))
