elastica

Converts this polyline into an elastica through all points: a minimum-energy curve (MEC), the G2 curve that minimizes its bending energy ∫κ² ds, which is the shape of a thin elastic rod threaded through pins at the points. A pin applies a force but no bending moment, so the curvature is continuous where the rod passes it.

The bending energy shrinks as a curve grows, so where the curve loops it has no minimum: the loop would inflate without end. The scale-invariant form, the energy times the curve length, is minimized instead, after Moreton and Séquin, "Functional optimization for fair surface design" (SIGGRAPH 1992). By the Cauchy-Schwarz inequality that is at least (∫|κ| ds)², with equality only for constant curvature, so points on a circle give exactly that circle. Collinear points give a straight line, and the ends of an open polyline are free like the ends of a rod.

The curve is built and optimized like org.openrndr.extra.shapes.mvc.minimumVariationCurve: quintic Béziers that share the tangent and curvature at their joints, optimized with Levenberg-Marquardt, with edges that turn far split into several quintics. Compared to the minimum-variation curve, which spreads the change of curvature as evenly as it can, the elastica keeps the curvature itself low, so it bends more where it has to and runs straighter in between.

The optimization is local, so for erratic points the result is the nearest local minimum, which is still G2 and passes through all points.

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