cart2polar: Cartesian to Polar Coordinate Transformation

Converting rectangular coordinates to polar representation

The cart2polar functions transform Cartesian coordinates into polar coordinate systems, providing radius and angular components for both 2D and 3D spaces.

2D Polar Transformation

For 2D Cartesian coordinates $(x, y)$, the polar transformation produces $(\theta, r)$:

$\theta = \text{atan2}(y, x)$ $r = \sqrt{x^2 + y^2}$

3D Spherical Transformation

For 3D Cartesian coordinates $(x, y, z)$, the spherical transformation produces $(r, \phi, \theta)$:

$r = \sqrt{x^2 + y^2 + z^2}$ $\phi = \arccos\left(\frac{z}{r}\right)$ $\theta = \text{atan2}(y, x)$

Function Variants

Function	Input	Output	Description
cart2polar(st)	vec2	vec2	2D polar coordinates $(\theta, r)$
cart2polar3D(st)	vec3	vec3	3D spherical coordinates $(r, \phi, \theta)$

Live Editor

const fragment = () => {
      const center = uv.sub(0.5)
      const polar = cart2polar(center)
      const angle = polar.x
      const radius = polar.y
      const pattern = angle.mul(8).sin().mul(0.5).add(0.5)
      const radialMask = smoothstep(0.02, 0.05, radius)
      return vec4(vec3(pattern.mul(radialMask)), 1)
}