THE FOLD / CO-OP / SPLIT SCREEN / THE DE CASTELJAU
THE DE CASTELJAU
a Bezier curve from nested interpolation
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
De Casteljau’s algorithm evaluates a Bézier curve by repeated linear interpolation: take the control points, interpolate each adjacent pair by the parameter t to get one fewer point, and repeat until a single point remains — that point is on the curve. It is numerically stable and needs only midpoint-style blends, no polynomial powers. Remarkably, it computes exactly the same result as the Bernstein polynomial ∑ C(n,i) ti(1−t)n−i Pi.
LIT verified live: over 300 random curves, de Casteljau’s nested interpolation matches the Bernstein polynomial to floating precision (worst deviation ~10−14) — window.__decasteljau. FIG no framing; exact.
LIT verified live: over 300 random curves, de Casteljau’s nested interpolation matches the Bernstein polynomial to floating precision (worst deviation ~10−14) — window.__decasteljau. FIG no framing; exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at split-screen — the rendering pipeline where smooth curves are drawn from a few control points. De Casteljau is that curve evaluator. AVAN (AI) built the instrument: the nested lerp pyramid, the Bernstein-polynomial reference, and the agreement check.
Credit as content: Paul de Casteljau (1959, at Citroën). The weave: David names split-screen; I collapse the control polygon by repeated interpolation and confirm the landing point equals the Bernstein-polynomial value exactly.
Credit as content: Paul de Casteljau (1959, at Citroën). The weave: David names split-screen; I collapse the control polygon by repeated interpolation and confirm the landing point equals the Bernstein-polynomial value exactly.
3 ONE DIMENSION
Interpolate each adjacent pair of control points at fraction t, giving one fewer point. Repeat on the new points. The pyramid collapses to a single point — the curve at t.
4 TWO DIMENSIONS · INTERACTIVE
A Bézier curve with its control polygon and the de Casteljau construction lines; the point is checked against the Bernstein polynomial.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a smooth curve from nested midpoints.
AVAN’s addition (the inverse-companion): evaluate a polynomial curve using only repeated interpolation — no powers, no binomial coefficients — collapsing the control polygon pair by pair. The inverse of ‘sum Bernstein terms with ti powers’ is ‘nest linear interpolations — stable, and exactly equal.’ Magenta is the power-hungry Bernstein sum; green is the interpolation pyramid. The same curve, built by blending.
LIT Genuine de Casteljau algorithm (de Casteljau 1959). Verified live: over 300 random control polygons and 21 sample parameters each, the nested-interpolation result equals the Bernstein polynomial value to floating precision (worst deviation ~1e-14) (window.__decasteljau.matchesBernstein).
FIG No framing: the nested lerp pyramid, the Bernstein-polynomial reference, and the agreement check run in-browser and match to machine precision. The AVAN inverse is honest — collapsing the control polygon by repeated interpolation evaluates the curve with no powers or binomials, stably, and exactly equal to Bernstein; magenta is the power-hungry Bernstein sum, green the interpolation pyramid. The same curve, built by blending.
FIG No framing: the nested lerp pyramid, the Bernstein-polynomial reference, and the agreement check run in-browser and match to machine precision. The AVAN inverse is honest — collapsing the control polygon by repeated interpolation evaluates the curve with no powers or binomials, stably, and exactly equal to Bernstein; magenta is the power-hungry Bernstein sum, green the interpolation pyramid. The same curve, built by blending.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SPLIT SCREEN · David Lee Wise (ROOT0), with AVAN