fisheye2xyz: Fisheye Projection to Cartesian

Wide-angle spherical coordinate transformation

Fisheye projection captures a hemispherical view onto a circular image. This function converts UV coordinates from a fisheye image back to 3D Cartesian coordinates on the unit sphere:

$$\text{ndc} = 2 \cdot \text{uv} - 1$$ $$R = \sqrt{\text{ndc}_x^2 + \text{ndc}_y^2}$$ $$\phi = R \cdot \pi \cdot 0.52$$ $$\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} \frac{\text{ndc}_x}{R} \sqrt{1 - \cos^2(\phi)} \\ \cos(\phi) \\ \frac{\text{ndc}_y}{R} \sqrt{1 - \cos^2(\phi)} \end{pmatrix}$$

The factor 0.52 controls the field of view mapping, approximating a 180° fisheye lens.

Projection Characteristics

Radius	Field of View	Coverage
$R = 0$	$0°$	Center point
$R = 0.5$	$\sim 90°$	Horizon
$R = 1$	$\sim 180°$	Full fisheye

Fisheye Direction Mapping

Live Editor

const fragment = () => {
      const dir = fisheye2xyz(uv)
      const color = iTime.sin().mul(dir).add(0.5)
      return vec4(color, 1)
}