quat2mat3: Quaternion to Matrix Transformation

Mathematical Foundation of Spatial Orientation

Quaternions represent rotations in 4D space using unit vectors on the complex hypersphere. This function converts the quaternion representation $q = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} + w$ into its equivalent 3×3 rotation matrix form.

The transformation follows the mathematical relationship:

$$R = \begin{bmatrix} 1-2(q_y^2+q_z^2) & 2(q_xq_y-q_zq_w) & 2(q_xq_z+q_yq_w) \\ 2(q_xq_y+q_zq_w) & 1-2(q_x^2+q_z^2) & 2(q_yq_z-q_xq_w) \\ 2(q_xq_z-q_yq_w) & 2(q_yq_z+q_xq_w) & 1-2(q_x^2+q_y^2) \end{bmatrix}$$

This conversion preserves orthogonality and determinant properties essential for rigid body transformations.

Hypersphere Navigation Through Quaternion Space

The quaternion-to-matrix transformation reveals the geometric relationship between 4D unit sphere coordinates and 3D rotational manifolds. Each point on the quaternion hypersphere maps to a unique rotation in 3D space.

ライブエディター

const fragment = () => {
      const angle = uv.y.atan2(uv.x)
      const radius = uv.length()
      const q = vec4(
              angle.add(iTime).sin(),
              angle.mul(2).cos(),
              radius.mul(3).add(iTime.mul(0.7)).sin(),
              radius.mul(2).sub(iTime.mul(0.3)).cos()
      ).normalize()
      const matrix = quat2mat3(q)
      const basePattern = vec3(0, 0, 1)
      const rotated = matrix.mul(basePattern)
      const color = rotated.mul(0.5).add(0.5)
      return vec4(color, 1)
}

The mathematical beauty lies in how quaternion multiplication preserves rotation composition, while the matrix form enables direct geometric transformations in Euclidean space.