scale3d: Hyperbolic Metamorphosis Engine

Constructing Dimensional Reality Through Multiplicative Topology

The scale3d function generates a 3×3 transformation matrix that encodes scale relationships across three-dimensional space. This matrix represents a linear transformation that preserves the origin while multiplying coordinate values by specified scaling factors.

Mathematical Definition:

$$S(\mathbf{s}) = \begin{pmatrix} s_x & 0 & 0 \\ 0 & s_y & 0 \\ 0 & 0 & s_z \end{pmatrix}$$

For uniform scaling with factor $s$:

$$S(s) = s \cdot I_3 = \begin{pmatrix} s & 0 & 0 \\ 0 & s & 0 \\ 0 & 0 & s \end{pmatrix}$$

Where $I_3$ represents the 3×3 identity matrix. The transformation applies as: $\mathbf{p'} = S(\mathbf{s}) \cdot \mathbf{p}$, where $\mathbf{p}$ is the original position vector.

Volumetric Resonance Chambers

Shows how scale3d transformations affect wave frequencies in 3D space, demonstrating the relationship between geometric scaling and wave propagation characteristics.

ライブエディター

const fragment = () => {
      const p = uv.mul(5).sub(2.5)
      const time = iTime.mul(0.5)

      // Scale factor affects wave frequencies
      const waveScale = time.mul(0.8).sin().mul(0.4).add(1.1)
      const scaleMatrix = scale3d(waveScale)

      // 3D wave coordinates
      const z = p.length().mul(2).add(time).cos()
      const wavePoint = vec3(p, z)

      // Apply scaling to wave coordinates
      const scaledWave = vec3(
              scaleMatrix[0][0].mul(wavePoint.x),
              scaleMatrix[1][1].mul(wavePoint.y),
              scaleMatrix[2][2].mul(wavePoint.z)
      )

      // Multiple wave frequencies affected by scaling
      const wave1 = scaledWave.x.mul(4).add(time.mul(2)).sin()
      const wave2 = scaledWave.y.mul(3).add(time.mul(1.5)).cos()
      const wave3 = scaledWave.z.mul(5).add(time.mul(2.5)).sin()

      // Wave interference patterns
      const interference = wave1.mul(wave2).add(wave2.mul(wave3)).add(wave1.mul(wave3))
      const resonance = smoothstep(-0.6, 0.6, interference)

      // Visualize scaling effect on wave amplitude
      const amplitude = scaledWave.x.abs().add(scaledWave.y.abs()).add(scaledWave.z.abs()).div(6)
      const frequency = waveScale.mul(0.5)

      const color = vec3(
              resonance.mul(amplitude.add(0.3)),
              resonance.mul(frequency),
              resonance.mul(float(2).sub(amplitude).mul(0.7))
      )

      return vec4(color, 1)

}