equirect2xyz: Equirectangular to Cartesian Conversion

Spherical coordinate transformation

The equirectangular projection maps a spherical surface onto a rectangular plane. This function converts UV coordinates $(u, v) \in [0,1]^2$ back to 3D Cartesian coordinates on the unit sphere:

$$\phi = \pi - v \cdot \pi$$ $$\theta = u \cdot 2\pi$$ $$\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} \cos(\theta) \sqrt{1 - \cos^2(\phi)} \\ \cos(\phi) \\ \sin(\theta) \sqrt{1 - \cos^2(\phi)} \end{pmatrix}$$

Where $\phi$ represents elevation from the equator and $\theta$ represents azimuth around the sphere.

Coordinate System Mapping

UV Input	Angle Range	Description
$u = 0$	$\theta = 0$	Front center
$u = 0.5$	$\theta = \pi$	Back center
$v = 0$	$\phi = \pi$	South pole (bottom)
$v = 0.5$	$\phi = \pi/2$	Equator (middle)
$v = 1$	$\phi = 0$	North pole (top)

Directional Visualization

Live Editor

const fragment = () => {
      const dir = equirect2xyz(uv)
      const color = iTime.sin().mul(dir).add(0.5)
      return vec4(color, 1)
}