01/11/2023

Folha: Bézier, Casteljau and the industrial design revolution

Reproduction of Marcelo Viana's column in Folha de S. Paulo.

We mark some points on the screen and the computer draws a smooth curve passing close to each of them. If we 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 more where convenient. 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 design surfaces in three-dimensional space, based on an adjustable grid of control points. And the computer would be connected directly to the turning machines, so that the part would be produced exactly according to the selected specifications. This is the vision brought by the engineer Pierre Bézier (1910–1999), starting in 1960, to the design and production of Renault cars.

The key idea for realizing this vision was discovered in 1912 by the 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.

Bézier had more freedom to disseminate his ideas. In the 1960s, he developed and popularized the Unisurf software, a precursor to CAD, which became fully operational in the automotive industry in 1975, the same year 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".

To read the full text, visit the newspaper's website.