In trying to understand basics of CAGD I stumbled upon Bézier curves. Their presentation usually starts with the definition using Bernstein polynomials, and then the determination of their properties.
I still wonder how that definition came to be. How did Bézier or de Casteljau think of using Bernstein polynomials as basis polynomials for polynomial curves?
Could one derive the form of the Bézier curve in reverse, starting from some desirable properties and then finding the Bézier curve definition?
Best Answer
Bézier developed his curves by performing an affine deformation of the basic curve $C(t) = (t, t^2, t^3)$. This kind of deformation is common in drafting, so that’s probably where he got the idea. This actually leads to a different set of basis functions, not the Bernstein polynomials. A few years later, Robin Forrest noticed the connection with the Bernstein polynomials. Both sets of basis functions are described in Bezier’s book.
It’s less clear where de Casteljau’s idea came from. From his original notes, it looks like he was making curves by repeatedly cutting off corners, as a carpenter would. Also, he was strongly motivated by a desire to make the arithmetic simple, to get good performance. This was in 1959, so every arithmetic operation counted. But I'm just guessing, and we'll probably never know. Even de Casteljau himself thought his algorithm was a clever trick (and therefore, perhaps, a stroke of luck). In 1959, he wrote:
"Les courbes unicursales de degré $n$, òu $x,y,z$ sont réprésentées par des polynômes de la variable $t$, peuvent être construites par construction géométrique assez ingénieuse."