Smoothstep information

Smoothstep is a family of sigmoid-like interpolation and clamping functions commonly used in computer graphics,^[1][2] video game engines,^[3] and machine learning.^[4]

The function depends on three parameters, the input x, the "left edge" and the "right edge", with the left edge being assumed smaller than the right edge. The function receives a real number x as an argument and returns 0 if x is less than or equal to the left edge, 1 if x is greater than or equal to the right edge, and smoothly interpolates, using a Hermite polynomial, between 0 and 1 otherwise. The gradient of the smoothstep function is zero at both edges. This is convenient for creating a sequence of transitions using smoothstep to interpolate each segment as an alternative to using more sophisticated or expensive interpolation techniques.

In HLSL and GLSL, smoothstep implements the ${\displaystyle \operatorname {S} _{1}(x)}$, the cubic Hermite interpolation after clamping:

${\displaystyle \operatorname {smoothstep} (x)=S_{1}(x)={\begin{cases}0,&x\leq 0\\3x^{2}-2x^{3},&0\leq x\leq 1\\1,&1\leq x\\\end{cases}}}$

Assuming that the left edge is 0, the right edge is 1, with the transition between edges taking place where 0 ≤ x ≤ 1.

A modified C/C++ example implementation provided by AMD^[5] follows.

float smoothstep (float edge0, float edge1, float x) {
   // Scale, and clamp x to 0..1 range
   x = clamp((x - edge0) / (edge1 - edge0));

   return x * x * (3.0f - 2.0f * x);
}

float clamp(float x, float lowerlimit = 0.0f, float upperlimit = 1.0f) {
  if (x < lowerlimit) return lowerlimit;
  if (x > upperlimit) return upperlimit;
  return x;
}

The general form for smoothstep, again assuming the left edge is 0 and right edge is 1, is

${\displaystyle \operatorname {S} _{n}(x)={\begin{cases}0,&{\text{if }}x\leq 0\\x^{n+1}\sum _{k=0}^{n}{\binom {n+k}{k}}{\binom {2n+1}{n-k}}(-x)^{k},&{\text{if }}0\leq x\leq 1\\1,&{\text{if }}1\leq x\\\end{cases}}}$

${\displaystyle \operatorname {S} _{0}(x)}$ is identical to the clamping function:

${\displaystyle \operatorname {S} _{0}(x)={\begin{cases}0,&{\text{if }}x\leq 0\\x,&{\text{if }}0\leq x\leq 1\\1,&{\text{if }}1\leq x\\\end{cases}}}$

The characteristic S-shaped sigmoid curve is obtained with ${\displaystyle \operatorname {S} _{n}(x)}$ only for integers n ≥ 1. The order of the polynomial in the general smoothstep is 2n + 1. With n = 1, the slopes or first derivatives of the smoothstep are equal to zero at the left and right edge (x = 0 and x = 1), where the curve is appended to the constant or saturated levels. With higher integer n, the second and higher derivatives are zero at the edges, making the polynomial functions as flat as possible and the splice to the limit values of 0 or 1 more seamless.