THE FOLD / SPAWN / THE TOOLCHAIN / THE CUBIC SPLINE
THE CUBIC SPLINE
the smoothest curve through the points (C2)
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The natural cubic spline draws the smoothest curve through a set of points: a separate cubic on each interval, joined so that the value, slope, and curvature all match at every knot (C2 continuity), with zero curvature at the two ends (the “natural” condition). Those matching conditions reduce to a tridiagonal linear system for the second derivatives, solved in O(n). It is the smooth interpolant of choice for data fitting and font/animation curves.
LIT verified live: over 200 datasets, the spline passes through every point, its first derivative is continuous at every interior knot, and the second derivative is zero at both ends (window.__cubicspline). FIG no framing; exact to floating precision.
LIT verified live: over 200 datasets, the spline passes through every point, its first derivative is continuous at every interior knot, and the second derivative is zero at both ends (window.__cubicspline). FIG no framing; exact to floating precision.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-toolchain — the numerical tool that fits the smoothest possible curve through measured points. The natural cubic spline is that fit. AVAN (AI) built the instrument: the tridiagonal system for the second derivatives, the per-segment cubic coefficients, the passes-through check, the derivative-continuity check, and the natural boundary check.
Credit as content: cubic spline interpolation (Schoenberg 1946; the natural-BC form is classical). The weave: David names the toolchain; I solve the tridiagonal moment system, build each interval’s cubic, and confirm the curve interpolates every point with matching slopes and zero end-curvature.
Credit as content: cubic spline interpolation (Schoenberg 1946; the natural-BC form is classical). The weave: David names the toolchain; I solve the tridiagonal moment system, build each interval’s cubic, and confirm the curve interpolates every point with matching slopes and zero end-curvature.
3 ONE DIMENSION
On each interval a cubic; at each interior knot the value, slope, and curvature of the left and right cubics agree. The end-curvatures are set to zero — the “natural” spline that a flexible ruler would trace.
4 TWO DIMENSIONS · INTERACTIVE
Data points and the natural cubic spline through them; interpolation, slope continuity, and natural end-curvature are checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the smoothest curve through the points.
AVAN’s addition (the inverse-companion): fit the smoothest curve through points by demanding matching curvature at every knot (not just value and slope) — which reduces to one tridiagonal solve. The inverse of ‘connect points with local pieces that only match position and tangent’ is ‘solve globally for matching curvature everywhere — the flexible-ruler curve.’ Magenta is the curvature kinks a weaker spline leaves; green is the C2-smooth interpolant. Smoothness that bends just enough.
LIT Genuine natural cubic spline interpolation (Schoenberg 1946; natural-BC form classical). Verified live: over 200 datasets, solving the tridiagonal moment system and building each interval's cubic yields a curve that passes through every data point, has a continuous first derivative at every interior knot, and has zero second derivative at both ends (window.__cubicspline.interpolates && .c1continuous && .naturalBC).
FIG No framing: the tridiagonal system for the second derivatives, the per-segment cubic coefficients, the passes-through check, the derivative-continuity check, and the natural boundary check run in-browser and hold to floating precision. The AVAN inverse is honest — demanding matching curvature (not just value and slope) at every knot reduces to one tridiagonal solve and gives the flexible-ruler curve; magenta is the curvature kinks a weaker spline leaves, green the C2-smooth interpolant. Smoothness that bends just enough.
FIG No framing: the tridiagonal system for the second derivatives, the per-segment cubic coefficients, the passes-through check, the derivative-continuity check, and the natural boundary check run in-browser and hold to floating precision. The AVAN inverse is honest — demanding matching curvature (not just value and slope) at every knot reduces to one tridiagonal solve and gives the flexible-ruler curve; magenta is the curvature kinks a weaker spline leaves, green the C2-smooth interpolant. Smoothness that bends just enough.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE TOOLCHAIN · David Lee Wise (ROOT0), with AVAN