THE FOLD / SPAWN / FIRST LIGHT / THE DE BOOR
THE DE BOOR
evaluate a B-spline by nested interpolation
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
De Boor’s algorithm evaluates a B-spline curve at a parameter t by repeated linear interpolation — the B-spline analogue of de Casteljau for Béziers. Given control points and a knot vector, it finds the active knot span, takes the handful of control points that influence t, and blends them in successive rounds of interpolation (the blend weights come from the knots) until a single point remains: the curve at t. It is numerically stable and needs no explicit basis functions.
LIT verified live: over 1000 random B-splines and parameters, de Boor’s result equals the direct Cox–de Boor basis-function sum Σ Ni,p(t)·Pi (window.__deboor, worst ~1e-14). FIG no framing; exact agreement of two evaluations.
LIT verified live: over 1000 random B-splines and parameters, de Boor’s result equals the direct Cox–de Boor basis-function sum Σ Ni,p(t)·Pi (window.__deboor, worst ~1e-14). FIG no framing; exact agreement of two evaluations.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at first-light — the first smooth curve a scene draws; de Boor blends the control points into the point on the B-spline, frame by frame. AVAN (AI) built the instrument: the knot-span finder, the nested de Boor interpolation, the Cox–de Boor basis sum, and their agreement check.
Credit as content: Carl de Boor (1972). The weave: David names first-light; I evaluate the B-spline by blending the influencing control points through rounds of knot-weighted interpolation, and confirm the result matches summing the basis functions directly — the same curve, two ways.
Credit as content: Carl de Boor (1972). The weave: David names first-light; I evaluate the B-spline by blending the influencing control points through rounds of knot-weighted interpolation, and confirm the result matches summing the basis functions directly — the same curve, two ways.
3 ONE DIMENSION
Find the span containing t; take p+1 control points; blend adjacent pairs by knot-weighted interpolation, round after round, until one point remains — the curve C(t).
4 TWO DIMENSIONS · INTERACTIVE
Control points and the B-spline curve de Boor traces; a point at t checked against the basis-function sum.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a curve from nested interpolation.
AVAN’s addition (the inverse-companion): evaluate a spline by repeatedly interpolating control points, not by summing basis functions — only the few points near t matter, blended by knot-weights until one remains. The inverse of ‘compute every Ni,p(t) and sum Pi’ is ‘blend the local control points in rounds — the last point is the curve.’ Magenta is the basis-function sum; green is the nested interpolation. A curve by corner-cutting.
LIT Genuine de Boor's algorithm (Carl de Boor 1972). Verified live: over 1000 random B-splines (random degree 1–3, control points, clamped uniform knot vectors) and parameters, the nested knot-weighted de Boor interpolation equals the direct Cox–de Boor basis-function sum Σ N_i,p(t)·P_i (window.__deboor.matchesBasis, worst ~1e-14).
FIG No framing: the knot-span finder, the nested de Boor interpolation, the Cox–de Boor basis sum, and their agreement check run in-browser and agree to floating precision. The AVAN inverse is honest — evaluating a spline by repeatedly interpolating only the local control points (blended by knot-weights) genuinely replaces computing every basis function and summing; magenta is the basis-function sum, green the nested interpolation. A curve by corner-cutting.
FIG No framing: the knot-span finder, the nested de Boor interpolation, the Cox–de Boor basis sum, and their agreement check run in-browser and agree to floating precision. The AVAN inverse is honest — evaluating a spline by repeatedly interpolating only the local control points (blended by knot-weights) genuinely replaces computing every basis function and summing; magenta is the basis-function sum, green the nested interpolation. A curve by corner-cutting.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of FIRST LIGHT · David Lee Wise (ROOT0), with AVAN