quat2mat4: Quaternion to Homogeneous Matrix Conversion

Homogeneous Coordinate System Integration

The quat2mat4 function extends 3D quaternion rotations into 4×4 homogeneous transformation matrices. This conversion enables seamless integration of rotation operations with translation, scaling, and projection transformations within unified homogeneous coordinate frameworks.

The mathematical transformation:

$$M_{4×4} = \text{toMat4}(\text{quat2mat3}(q))$$

Where the 3×3 rotation matrix is embedded within the homogeneous coordinate structure:

$$M = \begin{bmatrix} R_{3×3} & \mathbf{0} \\ \mathbf{0}^T & 1 \end{bmatrix}$$

This representation preserves all rotational properties while enabling matrix composition with translation and scaling operations.

Temporal Quaternion Field Dynamics

This visualization demonstrates how quaternion-to-matrix conversion enables complex spatial transformations through time-varying rotation fields. The homogeneous matrix form allows for efficient composition with other geometric transformations.

Live Editor

const fragment = () => {
      const grid = uv.mul(3).fract().sub(0.5)
      const cellId = uv.mul(3).floor()
      const hash = cellId.x.add(cellId.y.mul(31.7)).fract().mul(43758.5)

      const phase = iTime.add(hash.mul(6.28))
      const q = vec4(
              phase.sin(),
              phase.mul(1.618).cos(),
              phase.mul(0.707).sin(),
              phase.mul(2.414).cos()
      ).normalize()

      const matrix = quat2mat4(q)
      const point = vec4(grid, 0, 1)
      const transformed = matrix.mul(point)

      const distance = transformed.xy.length()
      const rings = distance.mul(12).sub(iTime.mul(3)).sin().abs()
      const color = rings.mul(vec3(0.5, 0.8, 1.0))

      return vec4(color, 1)

}

The quat2mat4 function bridges quaternion mathematics with homogeneous coordinate systems, enabling sophisticated 3D transformations while maintaining computational efficiency and numerical stability inherent to quaternion representations.