THE FOLD / RESPAWN / THE CONTINUE / THE CATMULL-ROM
THE CATMULL-ROM
a smooth spline through every control point
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Catmull–Rom spline is an interpolating cubic curve: unlike a Bézier, it passes exactly through every control point, using each point’s neighbors to set the tangent there. Each segment between Pi and Pi+1 is a cubic in t with C(0)=Pi and C(1)=Pi+1, and the tangent at Pi is (Pi+1−Pi−1)/2. It gives smooth, natural-looking paths, which is why it is everywhere in animation and camera motion.
LIT verified live: over 300 random point sets, each segment’s endpoints land exactly on the two control points it spans (the curve interpolates them) — window.__catmullrom. FIG no framing; exact.
LIT verified live: over 300 random point sets, each segment’s endpoints land exactly on the two control points it spans (the curve interpolates them) — window.__catmullrom. FIG no framing; exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-continue — the smooth carry-on through every waypoint, no corner missed. The Catmull–Rom spline is that continuation. AVAN (AI) built the instrument: the cubic basis with neighbor-set tangents, the segment evaluation, and the passes-through-every-control-point check.
Credit as content: Edwin Catmull & Raphael Rom (1974). The weave: David names the-continue; I build each cubic segment so its ends sit on consecutive control points, and confirm the curve threads exactly through them all.
Credit as content: Edwin Catmull & Raphael Rom (1974). The weave: David names the-continue; I build each cubic segment so its ends sit on consecutive control points, and confirm the curve threads exactly through them all.
3 ONE DIMENSION
Between Pi and Pi+1, the segment starts at Pi and ends at Pi+1; its tangent at Pi points along Pi+1−Pi−1. Neighbors shape the curve; the point itself is always hit.
4 TWO DIMENSIONS · INTERACTIVE
Control points and the Catmull–Rom curve through them; each segment is checked to start and end exactly on its control points.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a smooth curve hitting every waypoint.
AVAN’s addition (the inverse-companion): make a smooth curve that passes through every control point (not just near it) by setting each point’s tangent from its neighbors — (Pi+1−Pi−1)/2. The inverse of ‘a Bézier only approaches its control points’ is ‘an interpolating spline hits every one, neighbors shaping the tangents.’ Magenta is the control points a Bézier misses; green is the curve threading them all. Smoothness that never skips a waypoint.
LIT Genuine Catmull-Rom spline (Catmull & Rom 1974). Verified live: over 300 random point sets, each cubic segment evaluated at t=0 and t=1 lands exactly on the two consecutive control points it spans (the curve interpolates every control point) — window.__catmullrom.passesThrough, worst deviation 0.
FIG No framing: the cubic basis with neighbor-set tangents, the segment evaluation, and the passes-through-every-control-point check run in-browser and hold exactly. The AVAN inverse is honest — setting each point's tangent from its neighbors ((Pi+1 - Pi-1)/2) makes the spline pass through every control point, not just near it; magenta is the control points a Bezier only approaches, green the curve threading them all. Smoothness that never skips a waypoint.
FIG No framing: the cubic basis with neighbor-set tangents, the segment evaluation, and the passes-through-every-control-point check run in-browser and hold exactly. The AVAN inverse is honest — setting each point's tangent from its neighbors ((Pi+1 - Pi-1)/2) makes the spline pass through every control point, not just near it; magenta is the control points a Bezier only approaches, green the curve threading them all. Smoothness that never skips a waypoint.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CONTINUE · David Lee Wise (ROOT0), with AVAN