The Bezier curves are polynomial curves with Bernstein basis.
The history on this topic is quite complicated. My professor likes to tell these stories as he has met the two guys (Bézier and de Casteljau) personally.
As mentioned earlier, Bézier was working for Renault while de Casteljau was working for Citroën (both French car companies). Both invented Bézier curves independently of each other, de Casteljau probably a little bit earlier than Bézier. But, he wasn't allowed to publish his research. That is why Bézier was the first to publish this type of curves. Since then the two have become enemies and can't stand being in a room together. As the curves are already named after Bézier the science community also wanted to acknowledge the work of de Casteljau. So, they named the algorithm for the computation of a point on the curve after him. I'm not sure, though, if this algorithm should be attributed to him.
But, as the Bernstein polynomials are older than Bézier curves, maybe somebody else did define them before. Probably there wasn't a use case for these kinds of curves before CAD. Only there they became useful for the first time.
To the best of my knowledge, that would depend on what you mean by "defined". If you mean its use in Graphics, it would be Pierre Bézier. Casteljau did study these curves at an earlier time (late 1950's), and provided the algorithm for drawing Bezier Curves (as we know them today). The study of Bernstein polynomials predates this research by a number of decades.
The first description using control points and so on is due to Pierre Bézier who was a french engineer for car design.
When Bezier was working with French Renault car company, developed the procedure to make these curves and surfaces to design car body shapes. See a good introduction here:
http://kowon.dongseo.ac.kr/~lbg/cagd/history1.pdf
http://fei.edu.br/~psergio/CG_arquivos/IntroSplines.pdf
The history on this topic is quite complicated. My professor likes to tell these stories as he has met the two guys (Bézier and de Casteljau) personally.
As mentioned earlier, Bézier was working for Renault while de Casteljau was working for Citroën (both French car companies). Both invented Bézier curves independently of each other, de Casteljau probably a little bit earlier than Bézier. But, he wasn't allowed to publish his research. That is why Bézier was the first to publish this type of curves. Since then the two have become enemies and can't stand being in a room together. As the curves are already named after Bézier the science community also wanted to acknowledge the work of de Casteljau. So, they named the algorithm for the computation of a point on the curve after him. I'm not sure, though, if this algorithm should be attributed to him.
But, as the Bernstein polynomials are older than Bézier curves, maybe somebody else did define them before. Probably there wasn't a use case for these kinds of curves before CAD. Only there they became useful for the first time.
Bernstein introduced the so called Bernstein polynomial in one variable for the purpose of interpolation and approximation (and he gave a new proof of Weierstrass theorem on the density of polynomials on the set of continuous functions). So, Bézier curves already existed but only for graph of functions. Pierre Bézier was the first to publish their use for the design of plane curves using the actual formalism (control polygons and so on) but he did not describe the De Casteljau algorithm.
The De Casteljau interpolation scheme was really introduced by De Casteljau later (very close time from the publication of Bézier).
But Bernstein polynomial was of common use for approximation of function and was a lot of other application (Bernstein-Sato polynomials for D-module and so on).
The so-called Bezier curves were first introduced by Pierre Bezier in 1968. See more in the reference: Bezier, P.E., How Renault Uses Numerical Control for Car Body Design and Tooling, SAE Paper 680010, Society of Automotive Engineers Congress, Detroit, MI, 1968.
The Bezier curve is a parametric curve, which is defined by a minimum of three points consisting of an origin, endpoint and at least one control point. Unlike straight lines and circles where one of the variables can be set to discover a point on the line, with Bezier curves you sample as many times as required from t in [0,1] to obtain that many points. Each of these points returned is a function of the control points on the origin and endpoint.
The mathematical basis for Bezier curves — the Bernstein polynomial — has been known since 1912, but its applicability to graphics was understood half a century later. Bezier curves were widely publicized in 1962 by the French engineer Pierre Bezier, who used them to design automobile bodies at Renault. The study of these curves was however first developed in 1959 by mathematician Paul de Casteljau using de Casteljau’s algorithm, a numerically stable method to evaluate Bezier curves, at Citroen, another French automaker.
I understand that Paul de Faget de Casteljau, was working on the same question. Paradoxically, was an engineer who worked at Citroën. But by not publishing anything, only reference was Bézier (Renault).
Dear @Federico, The name of De Casteljau was given to a practical algorithm to compute the Bezier curve.
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