voronoi: Voronoi Diagram Generation

Cellular Pattern Construction Through Distance Analysis

Voronoi diagrams partition space into regions based on proximity to seed points. Each region contains all points closer to its seed than to any other seed, creating natural cellular patterns found in biology, geology, and architecture.

Mathematical Foundation

The Voronoi diagram construction follows these principles:

1. Seed Point Distribution: Random points $p_i$ distributed across a grid
2. Distance Calculation: For any point $q$, compute distances $d_i = ||q - p_i||$
3. Region Assignment: Assign $q$ to region $i$ where $d_i = \min(d_1, d_2, ..., d_n)$
4. Boundary Formation: Boundaries occur where $d_i = d_j$ for distinct seeds

The distance metric typically uses Euclidean distance: $d(q, p_i) = \sqrt{(q_x - p_{i,x})^2 + (q_y - p_{i,y})^2}$

Function Variants

Function	Input	Output	Purpose
voronoi	vec2, float	vec3	Time-animated Voronoi
voronoiVec2	vec2	vec3	Static 2D Voronoi
voronoiVec3	vec3	vec3	3D coordinates as time

Return Value Structure

The functions return vec3(seed_x, seed_y, distance) where:

• x, y: Coordinates of the nearest seed point
• z: Distance to the nearest seed point

Implementation Examples

ライブエディター

const fragment = () => {
      const p = uv.mul(12)
      const cell = voronoiVec2(p)
      const borders = smoothstep(0, 0.05, cell.z)
      const cellId = cell.xy.dot(vec2(12.9898, 78.233))
      const cellColor = vec3(
              cellId.sin().mul(0.5).add(0.5),
              cellId.mul(2.1).sin().mul(0.5).add(0.5),
              cellId.mul(3.3).sin().mul(0.5).add(0.5)
      )
      return vec4(cellColor.mul(borders), 1)
}

ライブエディター

const fragment = () => {
      const baseScale = 8
      const p = uv.mul(baseScale)
      const time = iTime.mul(0.2)

      const cell1 = voronoi(p, time)
      const cell2 = voronoi(p.mul(2), time.mul(1.3))

      const distortion = cell1.z.sub(cell2.z).mul(2)
      const ripples = cell1.z.mul(40).sub(time.mul(8)).sin().mul(0.1)

      const intensity = distortion.add(ripples).clamp(0, 1)
      const hue = vec2(0.5, 0.8).dot(cell1.xy).add(time.mul(0.1))

      const r = hue.mul(6.28).sin().mul(0.5).add(0.5)
      const g = hue.mul(6.28).add(2.09).sin().mul(0.5).add(0.5)
      const b = hue.mul(6.28).add(4.19).sin().mul(0.5).add(0.5)

      return vec4(vec3(r, g, b).mul(intensity), 1)

}