Bézier curves are named after Pierre Bézier, and primary used is geometry modeling. They are good at describing various shapes in 2D and 3D. A Bézier curve is a function x(t), y(t) - it gives points in space (x, y) for some parameter t = 0..1. But nowadays they are also used in computer graphics for animation, as easing functions. There, we need to evaluate y(x), because x is the time parameter and y is the evaluated variable.
How does the formula of a Bézier curve look like as y(x)? What constraints do the 4 control points need to meet for this function to be correct - to have only one value of y for each x, with no loops or arcs? Finally, how can this function be approximated to store it in computer memory and evaluate it efficiently in modern game engines? These sound like fundamental questions, but apparently no one researched this topic thoroughly before, so it became the subject of the Ph.D. thesis of my friend Łukasz Izdebski.
A part of his research has just been published as paper "Bézier Curve as a Generalization of the Easing Function in Computer Animation" in Advances in Computer Graphics, 37th Computer Graphics International Conference, CGI 2020, Geneva, Switzerland. We want to share an excerpt of his findings online as an article: Bezier Curve as Easing Function.