wavelet: Turbulent Fractal Noise

Fluid-like turbulence with controlled vorticity

Wavelet noise generates turbulent patterns through iterative rotation and sampling. Unlike standard fractal noise, wavelet introduces rotation matrices at each octave, creating fluid-like turbulence with controllable vorticity.

The function combines multiple scales through a rotation-based accumulation process:

$$W(p, \phi, k) = \frac{1}{M} \sum_{i=0}^{n} \frac{\sin(10 \cdot q_i.x + \phi)}{s_i} \cdot S(\|q_i\|^2)$$

where:

• $p$ represents input coordinates
• $\phi$ controls phase offset for animation
• $k$ determines frequency scaling between octaves
• $q_i = R_i \cdot (fract(p \cdot s_i) - 0.5)$ applies rotation $R_i$
• $S(d) = smoothstep(0.25, 0, d)$ provides radial falloff
• $M = \sum_{i=0}^{n} \frac{1}{s_i}$ normalizes the result

The rotation angle at each octave: $\theta_i = random(floor(p \cdot s_i)) \cdot 1000 + \phi \cdot random_{factor}$

Function Variations

Function	Purpose	Parameters
wavelet(p, phase, k)	Full control	vec2 position, phase offset, frequency ratio
waveletVec2(p)	Basic 2D	vec2 position only
waveletVec2Phase(p, phase)	Animated 2D	vec2 position, phase for time
waveletVec3(p)	3D input	vec3 with z as phase
waveletVec3K(p, k)	3D with control	vec3 input, frequency control

Live Editor

const fragment = () => {
      const pos = uv.mul(8)
      const turbulence = waveletVec2Phase(pos, iTime.mul(0.5))
      const contours = turbulence.mul(12).sin().pow(2)
      return vec4(contours.mul(vec3(0.3, 0.6, 0.9)), 1)
}

Live Editor

const fragment = () => {
      const grid = uv.mul(4)
      const w1 = wavelet(grid, iTime, 1.8)
      const w2 = wavelet(grid.add(vec2(10, 3)), iTime.mul(0.7), 2.1)
      const interference = w1.add(w2).abs()
      const bands = interference.mul(15).fract().step(0.3)
      const heat = vec3(bands.pow(3), bands.pow(1.5), bands)
      return vec4(heat, 1)
}

Mathematical Properties

The wavelet function exhibits several key characteristics:

Rotation Matrix Application: Each octave applies a 2x2 rotation matrix:

$$R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$

Coordinate Transformation: The fractal iteration applies:

$$p_{i+1} = p_i \cdot \begin{pmatrix} 0.54 & -0.84 \\ 0.84 & 0.54 \end{pmatrix} + i$$

Frequency Progression: Scale factor evolves as $s_{i+1} = s_i \cdot k$ where $k \approx 1.24$ provides balanced octave spacing.

Radial Falloff: The smoothstep function $S(d) = smoothstep(0.25, 0, d)$ creates circular falloff for each sample point, preventing harsh discontinuities.

Implementation Characteristics

Wavelet noise differs from Perlin or simplex noise through:

1. Rotational Coupling: Each octave introduces rotation based on local random values
2. Sinusoidal Basis: Uses sine waves instead of gradient interpolation
3. Radial Weighting: Applies distance-based falloff at each sample
4. Phase Control: Supports temporal animation through phase parameter
5. Vorticity Generation: Rotation accumulation creates swirl patterns

The algorithm maintains bounded output range $[-1, 1]$ through normalization factor $M$, ensuring consistent amplitude regardless of octave count.