Discrete Variational Calculus for B-spline Approximated Curves

TL;DR

This paper develops a discrete variational calculus framework for B-spline curves, enabling efficient solutions to aesthetic curve completion problems in 2D and 3D, with applications extending to other discrete variational problems.

Contribution

It introduces discrete Euler-Lagrange equations for B-spline approximated curves, simplifying complex continuous invariants for practical curve completion applications.

Findings

Discrete Euler-Lagrange equations for B-spline curves derived

Simple discrete Lagrangians effectively solve curve completion

Method applicable to various discrete variational problems

Abstract

We study variational problems for curves approximated by B-spline curves. We show that, one can obtain discrete Euler-Lagrange equations, for the data describing the approximated curves. Our main application is to the curve completion problem in 2D and 3D. In this case, the aim is to find various aesthetically pleasing solutions as opposed to a solution of a physical problem. The Lagrangians of interest are invariant under the special Euclidean group action for which B-spline approximated curves are well suited. Smooth Lagrangians with special Euclidean symmetries involve curvature, torsion, and arc length. Expressions in these, in the original coordinates, are highly complex. We show that, by contrast, relatively simple discrete Lagrangians offer excellent results for the curve completion problem. The methods we develop for the discrete curve completion problem are general and can be used to solve other discrete variational problems for B-spline curves.