quatLength: Quaternion Magnitude Calculation

Euclidean Norm Computation

The quatLength function calculates the Euclidean magnitude of a quaternion in 4D space. This operation computes the square root of the sum of squared components, providing the absolute length measurement needed for normalization and distance calculations.

Mathematical Definition: For quaternion $q = (x, y, z, w)$:

$$||q|| = \sqrt{x^2 + y^2 + z^2 + w^2}$$

Properties:

• Always returns non-negative values
• Essential for quaternion normalization: $\hat{q} = \frac{q}{||q||}$
• Preserves rotational magnitude information
• Used in distance calculations between quaternions

This function provides fundamental magnitude measurement for quaternion mathematics and normalization operations.

Magnitude-based Spatial Deformation

This example demonstrates how quaternion magnitude creates non-uniform spatial deformation fields. The length measurement drives geometric distortion patterns that vary based on quaternion magnitude distribution.

Live Editor

const fragment = () => {
      const center = vec2(0.5, 0.5)
      const radius = uv.sub(center).length()
      const angle = atan2(uv.y.sub(0.5), uv.x.sub(0.5))

      const timePhase = iTime.mul(0.3)
      const q1 = vec4(angle.mul(3).add(timePhase).sin(), angle.mul(2).cos(), radius.mul(8).sin(), timePhase.cos())
      const q2 = vec4(angle.add(timePhase.mul(1.3)).cos(), radius.mul(5).sin(), angle.mul(4).cos(), radius.mul(6).sin())

      const len1 = quatLength(q1)
      const len2 = quatLength(q2)

      const magnitudeDifference = len1.sub(len2).abs()
      const deformationFactor = magnitudeDifference.mul(radius.mul(15))

      const warpedCoords = uv.sub(0.5).mul(float(1).add(deformationFactor.mul(0.1)))
      const distortedPattern = warpedCoords.x.mul(warpedCoords.y).mul(20).sin().abs()

      const intensityMap = distortedPattern.mul(magnitudeDifference).mul(2)
      const color = vec3(intensityMap.mul(0.8), intensityMap, intensityMap.mul(0.5))

      return vec4(color, 1)

}

The quatLength function enables precise quaternion magnitude calculations for normalization operations and magnitude-based geometric transformations in 4D space.