B-Splines

Source inspiration: (Mathew 2000-2019).

Animations

These animations show two complementary views of cubic B-splines: the smooth curve traced from control points, and the individual basis functions \(N_{i,4}(t)\) whose weighted sum defines the curve.

Julia source scripts that generated these animations are linked under each case.

Case 1 — Cubic B-spline curve traced from 7 control points

Behavior: A cubic B-spline curve with a clamped knot vector passes through the first and last control points and is smoothly pulled toward the interior control points. The curve is \(C^2\)-continuous everywhere. The animation traces the curve parameter \(t\) from start to end.

Julia source

Cubic B-spline curve traced through 7 control points using de Boor’s algorithm; the smooth curve lies within the convex hull of the control polygon

Case 2 — Basis functions \(N_{1,4}, \ldots, N_{7,4}\) and the partition of unity

Behavior: Each cubic B-spline basis function \(N_{i,4}(t)\) is non-negative and has local support over exactly 4 knot spans. Their sum equals 1 everywhere in the domain (partition of unity), which is why the B-spline curve lies within the convex hull of the control polygon.

Julia source

Seven cubic B-spline basis functions N_{i,4} revealed one at a time; running sum shown in black approaches 1 as all functions are added

Derivation Notes (Planned)

Short derivations will be added to explain the core equations and assumptions.

References

Mathew, John H. 2000-2019. Numerical Analysis - Numerical Methods Modules. https://web.archive.org/web/20190808102217/http://mathfaculty.fullerton.edu/mathews/n2003/NumericalUndergradMod.html.