cubicIn: Cubic Polynomial Acceleration

Mathematical Foundation of Cubic Motion

The cubicIn function implements acceleration through cubic polynomial progression. The mathematical expression $f(t) = t^3$ represents the simplest form of cubic easing, creating smooth acceleration with increasing rate throughout the motion.

The cubic function provides $C^2$ continuity with zero initial velocity and zero initial acceleration, making it ideal for natural motion that starts gently and builds momentum progressively.

The derivative $f'(t) = 3t^2$ shows quadratic velocity growth, while the second derivative $f''(t) = 6t$ indicates linear acceleration increase, characteristic of polynomial motion curves.

Polynomial Properties and Smoothness

Cubic functions maintain excellent smoothness properties due to their polynomial nature. The motion begins with near-zero velocity, gradually increasing acceleration that builds to maximum rate at completion.

This acceleration profile creates motion that feels natural and predictable, without the jarring transitions that can occur with linear or discontinuous easing functions.

Computational Efficiency

The TSL implementation uses direct multiplication: t.mul(t).mul(t), providing computational efficiency while maintaining mathematical precision. This approach avoids more expensive power operations while achieving identical results.

The auto-typed parameters enable compatibility with both scalar and vector inputs, maintaining consistent cubic behavior across different data types.

Live Editor

const fragment = () => {
  const w = 0.01
  const t = iTime.fract()
  const y = cubicIn(t)
  const Y = cubicIn(uv.x)
  const a = vec3(0.1, 0.9, 0.3)
  const b = vec3(0.8, 0.9, 0.7)
  const c = a.mix(b, t).mul(uv.x.step(t))
  const lines = mmin2(smoothstep(0, w, uv.mod(0.1).min(uv.sub(vec2(t, y)).abs())))
  const curve = stroke(uv.y.sub(Y), 0, w).mul(c)
  const color = lines.oneMinus().mul(0.2).add(curve)
  return vec4(color, 1)
}