4D Gradient Vector Generation

Mathematical Foundation: 4D Gradient Vector Generation

The grad4 function generates pseudo-random 4D gradient vectors using deterministic mathematical operations to ensure consistent spatial coherence in noise algorithms.

Mathematical Definition:

$$\begin{align} \mathbf{p}_{xyz} &= \lfloor \text{frac}(j \cdot \mathbf{ip}_{xyz}) \cdot 7 \rfloor \cdot ip_z - 1 \\ p_w &= 1.5 - |\mathbf{p}_{xyz}| \cdot \mathbf{1}_{xyz} \\ \mathbf{s} &= \text{lessThan}(\mathbf{p}, \mathbf{0}) \\ \mathbf{p}_{xyz} &= \mathbf{p}_{xyz} + (\mathbf{s}_{xyz} \cdot 2 - 1) \cdot s_w \end{align}$$

Algorithm Components:

• $j$: Scalar seed value for deterministic generation
• $\mathbf{ip} = (ip_x, ip_y, ip_z, ip_w)$: Parameter vector controlling distribution
• $\mathbf{p}$: Resulting 4D gradient vector
• $\mathbf{s}$: Sign correction vector for proper gradient distribution

Properties:

• Deterministic generation ensures spatial coherence
• Vectors maintain unit-sphere distribution characteristics
• Consistent mathematical relationships between neighboring coordinates
• Optimal for Perlin noise and procedural texture generation

Multiple Gradient Fields

ライブエディター

const fragment = () => {
      const coord = uv.mul(20)
      const oscillation = iTime.mul(0.5)

      const base = coord.x.mul(13.0).add(coord.y.mul(71.0)).add(oscillation.mul(23.0))
      const params = vec4(0.08, 0.12, 8.0, 1.2)

      const field1 = grad4(base, params)
      const field2 = grad4(base.add(500.0), params.mul(vec4(1.5, 0.8, 0.9, 1.1)))
      const field3 = grad4(base.add(1000.0), params.mul(vec4(0.6, 1.4, 1.1, 0.9)))

      const combined = field1.xyz.add(field2.xyz.mul(0.6)).add(field3.xyz.mul(0.4))
      const normalized = combined.normalize().mul(0.5).add(0.5)

      const pulse = combined.length().mul(10).add(oscillation.mul(2)).sin().mul(0.3).add(0.7)

      return vec4(normalized.mul(pulse), 1)

}