Splines

The bent strip and the equation that replaced it

Before a draughtsman ruled a curve, he pinned a thin strip of wood or metal — the physical spline — to his drawing board and bent it through a set of fixed points called ducks. The strip's elasticity distributed the bend smoothly between each pair of points, minimising sudden changes of curvature. The curve you got was not arbitrary; it was the shape a flexible beam naturally takes under those constraints, and the mathematics of elastic beams determines it exactly.

That physical behaviour translates into a polynomial equation. Between any two adjacent control points, the curve is described by a cubic polynomial — degree three turns out to be the lowest degree that gives you positional, tangent and curvature continuity (the segments join, without a kink, and without a jump in bend). Patch enough cubics together and you get a spline in the mathematical sense: a piecewise polynomial that is smooth across every join. The smoothness conditions at each join, called knot conditions, are what bind the pieces into something that looks like one unbroken curve rather than several segments bolted end to end.

The most widely used family in computer graphics is the B-spline — B for basis — which rewrites the same idea so that each control point influences only a local span of the curve, not the whole thing. Move one point and only a nearby section shifts; the rest stays put. That local-support property is what makes interactive editing tractable. The NURBS (Non-Uniform Rational B-Spline) form extends this further, adding weights to each control point so that conic sections — circles, ellipses, parabolas — can be represented exactly rather than merely approximated, a requirement in industrial CAD that migrated into digital illustration.

Pierre Bézier arrived at a related but distinct construction while working on car bodies at Renault in Boulogne-Billancourt: his curve is defined entirely by its endpoint and handle positions, with no internal knots, which simplifies the algebra at the cost of global control. Both forms descend from the same physical ancestor, and both live in modern illustration software as different answers to the same question the draughtsman's bent strip first posed: how do you pass a smooth curve through or near a set of points you actually care about?

The de Casteljau algorithm shows how to evaluate any such curve efficiently — no trigonometry, only repeated linear interpolation — which is why renderers commonly use it.