Bézier

The curve that came from a car factory

Pierre Bézier was a mechanical engineer at Renault, and in the early 1960s his problem was not aesthetics — it was manufacturing. A car body panel must be smooth in a precise, measurable sense: any deviation in surface curvature shows up as a distortion in a reflected highlight, and those distortions telegraph poor craftsmanship at fifty metres. Describing that surface to a milling machine, in terms the machine could act on, required mathematics that could represent free-form curves exactly — not approximations sketched by hand and re-measured, but a compact, unambiguous definition that a computer could evaluate at any point along the path.

The system Bézier published — formally described in a 1970 paper and underlying his UNISURF CAD system, which Renault adopted through the late 1960s — used a polynomial parameterisation based on the work of the mathematician Sergei Bernstein. The Bernstein basis polynomials have a convenient property: they are all positive and they sum to one across the parameter interval, which means a weighted combination of control points defined by those polynomials stays inside the convex hull of those points. In plain terms, the curve never wanders outside the bounding shape formed by drawing lines between its control points. That is the property that makes Bézier curves tractable as a design tool: the designer has a guarantee about where the curve will go.

What Bézier did not know at the time — and what only became widely understood later — was that Paul de Casteljau, an engineer at Citroën working a few kilometres from Boulogne-Billancourt, had independently derived the same family of curves several years earlier. De Casteljau's internal Citroën reports, written in 1959, remained proprietary and unpublished for decades. His algorithm for evaluating the curve by recursive linear interpolation is now the standard method of computation: faster, more numerically stable, and geometrically transparent. The result carries Bézier's name because he published first; the evaluation method carries de Casteljau's.

Four points and why they are enough

The cubic Bézier — four control points, a degree-three polynomial — is the form that settled into virtually every drawing tool because four points are the minimum needed for a curve with independently adjustable direction at both ends. Two points define the endpoints, the curve passes through both. Two more are the "handles" — they do not lie on the curve; they pull it. The tangent to the curve at the start endpoint is the straight line from that endpoint to the first handle, and the tangent at the end is the straight line from the end endpoint to the second handle. Move a handle and the curve bends toward it; the endpoint stays fixed. This separation between the points the curve passes through and the points that govern its direction is what makes interactive manipulation feel coherent.

The parameter t runs from zero to one along the curve. At t = 0 you are at the first endpoint; at t = 1 you are at the last. For any value in between, the de Casteljau algorithm performs three rounds of linear interpolation between adjacent points, reducing four points to three, three to two, two to one, and that one point is on the curve. The geometry is always subdivision of straight lines — no trigonometry, no square roots, just ratios — which is why the computation is fast even on the hardware of the late 1960s and early 1970s.

Higher-degree curves exist: a quadratic Bézier uses three points, a quintic uses six. The quadratic is what the TrueType outline format uses internally for glyph outlines, partly for historical reasons around memory economy, and partly because the arithmetic is simpler still. But for general-purpose drawing the cubic is the natural unit. It is expressive enough to approximate a circular arc to within a fraction of a percent and compact enough that a complex path is just a chain of cubics joined at their endpoints.

How paths are stitched from segments

A single cubic covers limited ground. Real shapes — a letterform, a car silhouette, a mapmaker's coastline — are built from sequences of Bézier segments joined end to end. The join condition matters: if the two handles flanking a join point are collinear with it — that is, if they lie on the same straight line through the endpoint — the curve passes through smoothly, with matching tangent direction on both sides. If they share not just direction but also distance from the endpoint, the curvature matches too, and the join is visually indistinguishable from a continuous curve. Drawing tools expose this as a toggle: the locked, symmetric handle pair versus the broken pair that lets each side have its own angle.

The compactness of this representation is why PostScript, designed by John Warnock and Chuck Geschke at Adobe in 1982, built cubic Bézier curves directly into the language as a primitive — the curveto operator. Every scalable typeface, every Illustrator path, every PDF page description that followed is ultimately a list of cubic Bézier segments. The curve went from an engineering report in a car company to the substrate of modern print and screen graphics inside roughly twenty years.

Alvy Ray Smith, working at Xerox PARC in the 1970s before co-founding Pixar, was among those who helped move Bézier mathematics into graphics software, and the SuperPaint system at Xerox PARC represents part of that early bridge between the theoretical foundations and interactive tools. The broader community formalised, shared and extended the mathematics through venues like ACM SIGGRAPH, whose annual proceedings became the clearinghouse for computational geometry as it intersected with image-making. The de Casteljau algorithm as a formal object has been studied extensively in the numerical analysis literature, providing the theoretical grounding for the subdivision techniques that underpin modern curve rendering.

Pierre Bézier was a mechanical engineer at Renault, and in the early 1960s his problem was not aesthetics — it was manufacturing.

Four points. A parameter from zero to one. Repeated linear interpolation. The curve that solved a manufacturing problem for Renault turned out to describe the shape of nearly everything a designer draws — because it is the simplest closed form that gives smooth, steerable, computable shape.