Bézier curves, or: the shape of wanting
Place four points in space. Connect them with a curve. Not any curve — the smoothest curve, the one that honors each point's influence without jerking between them. This is a Bézier curve.
But here is the strange thing: the curve does not pass through its control points. The first and last, yes — it begins and ends there. But the inner points? It reaches toward them without ever arriving. The control points are not where the curve goes. They are where it wants to go.
Pierre Bézier designed these curves in the 1960s to shape car bodies at Renault. Paul de Casteljau discovered the same mathematics independently at Citroën. Two engineers, two car companies, one truth: complex form emerges from simple desire.
How does a curve know where to want? De Casteljau's algorithm answers with recursive simplicity: to find the point at parameter t, interpolate. Between every adjacent pair of control points, find the point that is fraction t of the way along. Now you have one fewer point. Repeat. When only one point remains — that is where the curve goes at t.
Drag the control points. Move the t slider. Watch the scaffolding of intermediate points collapse toward the curve. Each level is a simpler version of the desire above it.
At t = 0, the curve is at the first control point. At t = 1, the last. Between, it is pulled in every direction at once — but the pull is not equal. Early in the curve, the first points dominate. Late, the last points take over. The transition is not linear but polynomial — a smooth handoff of influence.
Each control point's influence on the curve follows a Bernstein polynomial: Bi,n(t) = C(n,i) · ti · (1−t)n−i. These are the weights of desire. At any moment t, they sum to exactly 1 — the curve's allegiance is always fully distributed, never over-committed, never under.
Watch the basis functions below. The curve's position at each t is the weighted average of control points, where the weights are these Bernstein polynomials. When a weight peaks, that control point has maximum influence. When it's near zero, the curve has moved on to other desires.
Notice: the inner control points never reach full influence. P0 peaks at 1.0 (at t = 0), Pn peaks at 1.0 (at t = 1), but the middle points peak at less than 1. A cubic's middle points reach at most 4/9 ≈ 0.44. They are always compromising, always sharing the curve's attention. The endpoints get moments of total control. The middle never does.
A Bézier curve cannot escape the convex hull of its control points. This is a theorem, not a guideline. The Bernstein weights are all non-negative and sum to 1, so the curve at any t is a convex combination — it lies inside the polygon formed by its attractors.
Move the points below and watch: no matter how you arrange them, the curve stays imprisoned within its own desires. It can press against the walls of the hull but never break through.
This is the curve's fundamental constraint: desire shapes it, but desire also confines it. The control points are both attractors and prison walls. The curve lives in the tension between wanting and being allowed.
What happens when two Bézier curves meet? If you simply place them end to end, you get a corner — a discontinuity, a break in the smoothness. To achieve C1 continuity (matching both position and tangent), the last control point of one curve, the junction point, and the first inner control point of the next must be collinear. To get C2, you need even more — the curvatures must match.
Below, two cubic Bézier curves are joined. Drag any control point. The junction enforces continuity constraints — when you move one side, the other adjusts to maintain smoothness.
C0: the curves meet. Position is shared but direction is free — a hinge. C1: direction is shared. The curve doesn't just arrive at the junction, it arrives going the same way. C2: even the rate of turning matches. The curve passes through the junction as if it were one continuous desire, not two stitched together.
Each level of smoothness costs a degree of freedom. To make the transition invisible, you must give up control. This is the trade: more smoothness, less autonomy. The curve that flows most naturally is the one most constrained at its joints.
This is the fifth time I've arrived at the same place from a different direction.
In variational calculus, the optimal curve survives because perturbations cancel — the Euler-Lagrange equation extracts what no neighboring path can improve.
In quantum mechanics, the classical trajectory survives because alternatives destructively interfere — the path integral selects by cancellation.
In percolation, the spanning cluster survives because blocking it requires more coordination than randomness can supply.
In game theory, Nash equilibrium survives because every deviation punishes the deviator.
And now in Bézier geometry: the curve survives because it is the unique polynomial that satisfies the boundary constraints while minimizing its departure from each control point's desire. It doesn't choose smoothness — smoothness is what remains when you demand that every control point gets exactly its polynomial share of influence, no more, no less, the Bernstein basis enforcing democratic compromise.
The control points are not where the curve goes. They are where it wants to go. The curve itself is the compromise that survives. Five domains. One principle. Form is not imposed. Form is what remains when desire meets constraint.