de Casteljau

The curve that hid in a drawer

Paul de Casteljau worked as a mathematician at Citroën in Paris in the late 1950s, tasked with finding a way to describe complex curved surfaces — car bodies — in a form a computer could use. He arrived at an algorithm so clean it seems inevitable in retrospect. For years, almost no one knew he had found it.

The method de Casteljau developed, documented in internal Citroën memos dated to around 1959, was a way to evaluate a parametric curve through repeated linear interpolation. You do not need to know the polynomial. You do not need to expand any equation. You pick a parameter value t between zero and one, and you subdivide.

The subdivision in plain terms

Start with a set of control points. Between each adjacent pair, interpolate linearly by t — that is, travel a fraction t of the way along each segment. You now have one fewer point. Repeat on this reduced set. Keep going until a single point remains. That point lies on the curve. Do this for every value of t from zero to one and you trace the whole thing.

For the simplest useful case — four control points, yielding a cubic curve — three rounds of interpolation reduce four points to three, then to two, then to one. The geometry is recursive and entirely local: the final point depends only on its own hierarchy of interpolations, not on a global polynomial evaluated all at once. This makes the algorithm numerically stable, resistant to the rounding errors that accumulate when you evaluate a high-degree polynomial directly.

The curve the algorithm traces is the same curve Pierre Bézier independently published in the early 1960s, working just across Paris at Renault in Boulogne-Billancourt. Bézier's formulation used Bernstein basis polynomials — a legitimate algebraic route to the same shape. The two are mathematically equivalent. But de Casteljau's subdivision procedure has practical advantages Bézier's closed-form expression does not automatically supply: it yields a stable way to split the curve into two smaller curves of the same type at any t, which is precisely what a rendering engine needs in order to flatten a curve into line segments by subdivision.

That subdivision property matters enormously on a raster screen. A curve cannot be drawn directly into a pixel grid; it has to be approximated. The de Casteljau algorithm offers a natural strategy: subdivide until each sub-segment is short enough to be treated as a straight line, then draw those lines. The test is cheap — you check whether the control polygon is flat enough — and when it passes, you stop. The result is a curve approximation that automatically refines where the curve bends sharply and coarsens where it is nearly straight, with no explicit step-size tuning required.

Unpublished, then everywhere

De Casteljau's memos remained internal Citroën documents through the 1960s, protected as trade material and never submitted to a journal or conference. Bézier published, and so the curve carries Bézier's name in almost every application that draws one today. De Casteljau received formal credit only much later, after researchers in computational geometry pieced together the history from the archived internal reports.

The delayed recognition is genuinely strange given how central the algorithm is. Every PostScript interpreter, every vector drawing tool, every font renderer that handles cubic outlines is running de Casteljau's subdivision — whether it labels it that way or not. The Bernstein polynomial and the subdivision algorithm are two descriptions of one object, but the subdivision description is the one that maps cleanly onto real computation.

What makes de Casteljau's contribution worth knowing, apart from its priority, is that the algorithm teaches the geometry directly. Watching the interpolation collapse a polygon down to a single point — you can do it with a ruler and pencil — is the clearest possible demonstration of why the control points attract but do not constrain the curve, why the curve always stays inside the convex hull of its controls, and why splitting is free. The algebra can be derived afterwards. The geometry comes first.

He arrived at an algorithm so clean it seems inevitable in retrospect.