Semi-discrete linear hyperbolic polyharmonic flows of closed polygons

TL;DR

This paper studies a semi-discrete hyperbolic flow of polygons driven by a linear polyharmonic analogue, demonstrating the ability to morph any polygon into another and analyzing convergence properties of these flows.

Contribution

It introduces a novel semi-discrete hyperbolic flow model for polygons and proves its capability to transform any polygon into any other, with exponential convergence results.

Findings

Polygons can be transformed into any target polygon via the proposed flow.

The flow converges exponentially to a planar basis polygon after transformation.

The flow of the Yau curvature difference can reach any target polygon in infinite time.

Abstract

We consider the damped hyperbolic motion of polygons by a linear semi-discrete analogue of polyharmonic curve diffusion. We show that such flows may transition any polygon to any other polygon, reminiscent of the Yau problem of evolving one curve to another by a curvature flow, before converging exponentially to a point that, under appropriate rescaling, is a planar basis polygon. We also consider a hyperbolic linear semi-discrete flow of the Yau curvature difference flow, where a polygonal curve is able to flow to any other such that we get convergence to the target polygon in infinite time.