Bézier curves are nowadays widely used in vector graphics, such as in Adobe Illustrator. Their useful property is that only a few points (traditionally 4) are needed to define a smooth curve.

I’ll explain how they work by an example. We start with 4 points and connect them to get a broken line. Not very smooth yet, eh? Now choose a parameter between 0 and 1, such as 0.25, and divide each segment of the broken line according to this parameter (treated as a proportion). Connect the 3 chosen points on the broken line to get a new broken line with one segment less. Repeat this procedure with the same parameter and finally you’ll arrive at a single point, which lays on the Bézier curve.

You can find more points by choosing different parameters and constructing their corresponding points.

Bézier curves were publicized by Pierre Bézier, who used them to design automobile bodies at Renault. His curves lay the foundation of modern CAD, and are also very important in fonts, as TTF and OTF fonts save their characters as splines (connected Bézier curves).