Hyperball in mathematical graphics

There are many fruits in nature that belong to the shape of the Superball. I have previously written an article about the hyper-circle: Mathematical graphics (1.44) hyper-circle. This article will extend it from the two-dimensional curve of the hyper-circle to the three-dimensional surface of the hyper-sphere.

A supercircle is a graph generated by x ^ A + y ^ B = C. It is a circle when a = B = 2.

Superelliptic is an equation: M * x ^ A + N * y ^ B = C. When a = B = 2, it is an elliptic.

Then the hyperball is defined as follows:

Hyperball equation: x ^ A + y ^ B + Z ^ c = d

Super elliptical equation: M * x ^ A + N * y ^ B + K * Z ^ c = d

I use my own custom syntax script code to generate a hyper-ball image. For more information about the software, see: Mathematical graphics visualization tool. This software is free open source. QQ chat group: 367752815

(1) The image generated by rotating the supercircle along the X or Y axis is also a type of SuperBall.

vertices = D1:100 D2:100u = from (-PI/2) to (PI/2)  D1v = from 0 to (2*PI) D2a = rand2(0.1, 4)b = rand2(0.1, 4)r = 10.0x = r*pow_sign(sin(u), a)n = r*pow_sign(cos(u), b)y = n*cos(v)z = n*sin(v)

vertices = D1:100 D2:100u = from 0 to (PI) D1v = from 0 to (2*PI) D2a = rand2(0.1, 4)b = rand2(0.1, 4)r = 10.0n = r*pow_sign(sin(u), a)y = r*pow_sign(cos(u), b)x = n*cos(v)z = n*sin(v)

(2) hyper-sphere (thin)

vertices = D1:100 D2:100u = from 0 to (2*PI) D1v = from (-PI*0.5) to (PI*0.5) D2a = 10m = 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) hyper-sphere (FAT)

vertices = D1:100 D2:100u = from 0 to (2*PI) D1v = from (-PI*0.5) to (PI*0.5) D2a = 10m = 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) hyper-sphere (double parameter)

vertices = D1:100 D2:100u = from 0 to (2*PI) D1v = from (-PI*0.5) to (PI*0.5) D2a = 10m = 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) hyper-sphere (three parameters)

vertices = D1:100 D2:100u = from 0 to (2*PI) D1v = from (-PI*0.5) to (PI*0.5) D2r = 10a = 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) hyper-elliptical sphere

vertices = D1:100 D2:100u = from (-PI*0.5) to (PI*0.5) D1v = from (-PI) to (PI) D2a = rand2(1, 5)b = rand2(1, 5)c = rand2(1, 5)m = 5n = 3x = a*(cos(u)^m)*(cos(v)^n)z = b*(cos(u)^m)*(sin(v)^n)y = c*(sin(u)^m)

vertices = D1:100 D2:100u = from (-PI*0.5) to (PI*0.5) D1v = from (-PI) to (PI) D2a = rand2(1, 5)b = rand2(1, 5)c = rand2(1, 5)m = 0.6n = 0.3x = 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))