THE FOLD / CO-OP / THE HANDOFF / THE BEZIER
THE BEZIER
de Casteljau — a smooth curve from pure averaging
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Bézier curves and de Casteljau’s algorithm. Every letter you read, every vector shape, every animation ease-curve is a Bézier: a smooth curve pulled into shape by a handful of control points. But how do you find a point on the curve at parameter t?
De Casteljau’s answer (1959) uses nothing but repeated linear interpolation. Lerp between each pair of consecutive control points at fraction t — that gives one fewer point. Lerp those. Again. When a single point remains, it is exactly the curve point at t. No polynomial, no powers — just averaging, over and over. It is numerically rock-solid and equals the textbook Bernstein polynomial form exactly.
LIT verified live: over 30,000 samples de Casteljau’s nested lerps agree with the Bernstein polynomial to 1e-7, and the curve passes exactly through its first and last control points (window.__bezier.matchesBernstein && endpointsExact). FIG no framing; the corner-cutting construction and its equivalence to the polynomial are exact.
De Casteljau’s answer (1959) uses nothing but repeated linear interpolation. Lerp between each pair of consecutive control points at fraction t — that gives one fewer point. Lerp those. Again. When a single point remains, it is exactly the curve point at t. No polynomial, no powers — just averaging, over and over. It is numerically rock-solid and equals the textbook Bernstein polynomial form exactly.
LIT verified live: over 30,000 samples de Casteljau’s nested lerps agree with the Bernstein polynomial to 1e-7, and the curve passes exactly through its first and last control points (window.__bezier.matchesBernstein && endpointsExact). FIG no framing; the corner-cutting construction and its equivalence to the polynomial are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE HANDOFF, beside THE PLUCKED STRING — the co-op domain of passing smoothly from one to the next. A Bézier is a chain of handoffs: each lerp blends two points into one, and the cascade of blends is the curve. AVAN (AI) built the instrument: the nested lerps, the curve trace, the Bernstein check.
The weave: David names the seat (the smooth handoff); I make the corner-cutting visible and the equivalence checkable — the blend ladder in 1D, the live construction in 2D, the collapsing lines in 3D. The sphere is the seam. Credit: Paul de Casteljau (1959, Citroën); Pierre Bézier (1960s, Renault).
The weave: David names the seat (the smooth handoff); I make the corner-cutting visible and the equivalence checkable — the blend ladder in 1D, the live construction in 2D, the collapsing lines in 3D. The sphere is the seam. Credit: Paul de Casteljau (1959, Citroën); Pierre Bézier (1960s, Renault).
3 ONE DIMENSION
The blend ladder: start with the control points, lerp each adjacent pair at t to get one fewer, and repeat until a single point is left. Each rung is a round of corner-cutting; the last point rides the curve.
4 TWO DIMENSIONS · INTERACTIVE
Slide t and watch de Casteljau build the point: the control polygon collapses through nested interpolations to a single point that traces the smooth curve. Click to move the nearest control point and reshape it.
5 THREE DIMENSIONS + AVAN’S INVERSE
The control polygon and its curve turning in space — green, the smooth path and the frame that shapes it.
AVAN’s addition (the inverse-companion): the magenta lines are the de Casteljau construction at the current t, collapsing to the point on the curve. A curve is an infinity of points — you would think you need a polynomial, powers of t, real analysis. The inverse is startling: every point is built from the control points by nothing but repeated averaging — straight-line blends, no curves used to make a curve. Smoothness is not put in; it emerges from a cascade of linear handoffs. The green is the finished curve; the magenta is the little ladder of straight cuts that, run at every t, sweeps the whole of it into being.
LIT Genuine de Casteljau / Bezier construction (Paul de Casteljau 1959; Pierre Bezier 1960s). Verified live: over 30,000 samples the nested-lerp de Casteljau result agrees with the Bernstein polynomial form to 1e-7, and the curve passes exactly through its first and last control points (window.__bezier.matchesBernstein && endpointsExact, both true). Corner-cutting by repeated averaging builds every curve point with no powers of t — shown, not asserted.
FIG No metaphor is doing the work: the de Casteljau lerps, their equivalence to the Bernstein polynomial, and the endpoint interpolation are all real and checked. The curve genuinely emerges from straight-line blends only — a curve made without ever using a curve.
FIG No metaphor is doing the work: the de Casteljau lerps, their equivalence to the Bernstein polynomial, and the endpoint interpolation are all real and checked. The curve genuinely emerges from straight-line blends only — a curve made without ever using a curve.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HANDOFF · David Lee Wise (ROOT0), with AVAN