Function Repository Resource:

# SmoothStep

A sigmoidal interpolation function

Contributed by: Flip Phillips (FlipPhillips.com)
 ResourceFunction["SmoothStep"][x] returns a sigmoidal, Hermite interpolation between [0,1] for x on [0,1]. ResourceFunction["SmoothStep"][x,{min,max}] returns a sigmoidal, Hermite interpolation between [0,1] for x on [min,max].

## Details and Options

ResourceFunction["SmoothStep"] is an interpolation function commonly used in computer graphics.
The ResourceFunction["SmoothStep"] implementation is consistent with the GL and RenderMan (RSL) shading languages.
ResourceFunction["SmoothStep"][x] clamps to 0 or 1 for x less than or greater than 0 or 1, respectively.
ResourceFunction["SmoothStep"][x,{min,max}] is the same as the RenderMan smoothstep(min,max,x) invocation.
ResourceFunction["SmoothStep"] is not second-order continuous at either x=0 or x=1. This creates discontinuities. The resource function SmootherStep solves this problem.

## Examples

### Basic Examples (2)

The function is sigmoidal and inputs smaller than 0.5 underestimate the input:

 In[1]:=
 Out[1]=

Values larger than 0.5 do the opposite:

 In[2]:=
 Out[2]=

Plot the function:

 In[3]:=
 Out[3]=

### Scope (2)

The single-argument form of SmoothStep passes through the point {0.5,0.5}:

 In[4]:=
 Out[4]=

Specify an input domain by giving a second argument:

 In[5]:=
 Out[5]=

### Applications (2)

Smoothly interpolate with more gradual changes at the start and stop of the interpolation domain:

 In[6]:=
 Out[6]=

Interpolate colors:

 In[7]:=
 Out[7]=
 In[8]:=
 Out[8]=

### Properties and Relations (1)

The derivative of SmoothStep approaches 0 as x approaches 0 or 1 from either side:

 In[9]:=
 Out[9]=
 In[10]:=
 Out[10]=

### Possible Issues (1)

SmoothStep is second-order discontinuous at x=0 and x=1:

 In[11]:=
 Out[11]=

### Neat Examples (1)

SmoothStep approximates the animation technique of "ease-in / ease-out":

 In[12]:=
 Out[12]=

Flip Phillips

## Version History

• 1.0.0 – 08 August 2019