Bézier, Casteljau and the industrial design revolution – 10/31/2023 – Marcelo Viana

We mark some points on the screen and the computer draws a smooth curve passing close to each of them. If you don’t like it, we can modify it by dragging each of the control points here or there. If the number of points is insufficient, we add them where appropriate. If it is excessive, we eliminate the ones we don’t want. In each case, the computer recalculates the curve to adapt to the new set of control points, without us ever needing to know which equations it uses to do so.

The same idea would be used to draw surfaces in three-dimensional space, using an adjustable mesh of control points. And the computer would be connected directly to the turning machines, so that the part was produced exactly according to the selected specifications. This is the vision brought by engineer Pierre Bézier (1910–1999), from 1960 onwards, to the design and production of Renault cars.

The key idea for realizing this vision was discovered in 1912 by Russian-Ukrainian mathematician Sergei N. Bernstein: it is a sequence of polynomials that Bernstein used to prove the so-called Weierstrass Approximation Theorem. But for practical applications it was essential to find a fast and robust way to calculate the Bernstein polynomial corresponding to each set of control points.

This problem was solved in 1959 by another French engineer, Paul de Casteljau (1930–2022), who worked for Citroën’s competing industry. Casteljau’s algorithm is the most used in CAD (computer-aided design) to this day. But unfortunately for him, Citroën’s strict rules prevented him from publishing his work until his ideas had been rediscovered by others.

Bézier had more freedom to publicize his ideas. In the 1960s, he developed and popularized the Unisurf software, a precursor to CAD, which became fully operational in the automobile industry in 1975, the same year that Bézier retired from Renault. Two years later, at the age of 67, he defended his doctoral thesis in mathematics entitled “Essay on the numerical definition of curves and surfaces”.

Today, so-called Bézier curves are ubiquitous in industrial design. Just to give you an idea, dear reader, it is very likely that the types (letters) in which this article appears written to you – on paper or on the screen – were designed using Bézier curves. For his contribution, Bézier received numerous distinctions and even gave his name to the Solid Modeling Association’s most important award. In 2012, the winner of the Bézier prize was… Paul de Casteljau.

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