Wednesday, 13 July 2011

Ease functions

Ease functions are important for interpolation of just about anything. Popular functions are "smoothstep" or "ease in" and "ease out". This images shows what they look like for x[0,1] and y[0,1]:

linear: y=x, ease out: y=(x-1)^3+1, ease in: y=x^3, smoothstep: y=3x^2-2x^3,
y=sin(x*PI/2), y=1-cos(x*PI/2)
There are other popular functions for ease in and ease out, e.g. y=sin(x*PI/2) resp. y=1-cos(x*PI/2). What is important about those functions is if they're C1- and/or C2-continuous when you want to combine functions. C1-continuity means that the first derivative or the tangent of the function is the same for x=0 and x=1, meaning you can piece together two of the curves and there are no sudden changes or jumps for y. C2-continuity means that the second derivative or how the tangent changes is the same.
y=sin(x*PI/2) and y=1-cos(x*PI/2) are not C1-continuous, but you can use them to start/end a linear interpolation for example.

The nice picture was plotted with

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