A Sobolev gradient method for construction of elastic curves in regular surfaces

Abstract

Given a sequence of control points and tangent vectors in a regular surface, we treat the problem of constructing a curve that lies in the surface and has minimum total curvature subject to the constraint that it interpolates the control points and tangent vectors. Geodesics and parametric planar curves (nonlinear splines) are included as special cases. The surface is defined implicitly by a smooth function, and the curve is approximated by a discrete set of vertices along with first and second derivative vectors. The nonlinear optimization problem of minimizing curvature subject to the constraints is solved by a variable metric gradient descent method based on Neuberger's Sobolev gradient theory. We compare different methods for treating the nonlinear constraints.