parabola: Parabolic Curvature Metamorphosis Engine

Parabolic Curve Generation

The parabola function creates smooth parabolic curves based on the formula $4x(1-x)$ raised to a power. This generates bell-shaped curves that peak at the center (x=0.5) and fall off towards the edges, useful for creating smooth gradients and falloffs.

The fundamental transformation follows the curvature metamorphosis equation:

$$f(x, k) = [4x(1-x)]^k$$

Where $x \in [0,1]$ defines the field domain and $k > 0$ controls curvature intensity. The base expression $4x(1-x)$ generates the canonical parabolic arch peaking at $x = 0.5$, while the exponential parameter $k$ enables adaptive curvature modulation:

$$\lim_{k \to 0^+} f(x,k) = 1 \quad \text{(uniform field)}$$ $$\lim_{k \to \infty} f(x,k) = \begin{cases} 1 & \text{if } x = 0.5 \\ 0 & \text{otherwise} \end{cases}$$

This mathematical foundation enables seamless transitions between uniform distributions and highly focused field concentrations.

ライブエディター

const fragment = () => {
      const center = vec2(0.5, 0.5)
      const fieldPos = uv.sub(center).abs()

      const radialDistance = fieldPos.length().clamp(0, 0.5).mul(2)
      const temporalModulation = iTime.mul(0.4).sin().mul(0.5).add(1.5)

      const parabolicField = parabola(radialDistance, temporalModulation)
      const intensityMapping = parabola(parabolicField, iTime.mul(0.3).cos().abs().mul(3).add(1))

      const focusedEnergy = intensityMapping.pow(2.2)
      const energyDistribution = focusedEnergy.mul(fieldPos.x.add(fieldPos.y).mul(1.5))

      const r = focusedEnergy.mul(energyDistribution.pow(0.7))
      const g = energyDistribution.mul(parabolicField.pow(1.8))
      const b = parabolicField.mul(radialDistance.oneMinus().pow(0.9))

      return vec4(r, g, b, 1)

}