In the 1950s, two engineers at rival French car companies independently invented the same mathematical object. Paul de Casteljau at Citroën developed an algorithm for designing smooth car body surfaces. Pierre Bézier at Renault published the same idea. De Casteljau's work stayed proprietary. Bézier's name stuck.
The object they discovered is breathtakingly simple. Take four points. Connect adjacent pairs with lines. Place a sliding point on each line at the same fraction t. Connect those three points. Repeat. Connect those two. Repeat. The last single point traces a curve.
That's it. Lerps all the way down — linear interpolation, nested recursively.
De Casteljau's construction is recursive linear interpolation. At each level, we blend adjacent points:
For four control points P₀, P₁, P₂, P₃, expand the recursion fully and collect terms by point. Each point gets a weight — a polynomial in t:
These four weights — (1−t)³, 3(1−t)²t, 3(1−t)t², t³ — are the Bernstein polynomials. They always sum to 1. At t = 0, only P₀ matters. At t = 1, only P₃. Between: a smooth handoff.
Something remarkable: the derivative of a cubic Bézier is itself a quadratic Bézier. And the derivative of that is a linear Bézier (a line). And the derivative of that is a constant. The tower of derivatives mirrors the tower of lerps — each level one step simpler.
From velocity and acceleration, you can compute curvature — how sharply the curve bends at any point:
The reciprocal of curvature gives the radius of the osculating circle — the circle that best fits the curve at that point. Toggle it on above. Where the curve is nearly straight, the osculating circle is enormous. Where it bends tightly, the circle shrinks.
Where curvature crosses zero — that is an inflection point, where the curve changes from bending left to bending right. At that moment, the osculating circle has infinite radius. There is no circle that matches a straight line.
How long is a cubic Bézier curve? This sounds like it should have a clean answer. It does not. The arc length integral is elliptic — there is no closed-form solution. You cannot write down the length of a cubic Bézier curve in terms of elementary functions. You can only approximate it, by chopping the curve into tiny straight segments and summing their lengths.
This is a deep fact. The simplest interesting curve — four points, nested lerps — already escapes the reach of symbolic computation. The universe prefers approximate answers to exact ones, even for objects of perfect construction.
Every smooth font on your screen is made of cubic Bézier curves. Every vector graphics program. Every car body in a CAD file. Every animation path in every game. Four points — two endpoints, two handles — and a formula that trades their influence smoothly.
The curve never passes through the inner control points. They attract without touching. The control points are not where the curve goes — they are where it wants to go. The curve is the compromise between desire and constraint, shaped by invisible attractors it can never reach.
Day . Interactive: drag the four colored points. The construction lines show De Casteljau's algorithm in real time.