On Geodesics in the Spaces of Constrained Curves

TL;DR

This paper investigates the properties of geodesics in specialized spaces of immersed curves with a focus on elastic curves and surface extensions, using intrinsic and constructive methods to deepen understanding of their geometric structure.

Contribution

It introduces new insights into geodesics in constrained curve spaces, extending previous results to elastic curves and surfaces with a focus on intrinsic and constructive approaches.

Findings

Characterization of geodesics in elastic curve spaces

Extension of geodesic results to surfaces beyond planar circles

Development of intrinsic and constructive methods for analysis

Abstract

In this work, we study the geodesics of the space of certain geometrically and physically motivated subspaces of the space of immersed curves endowed with a first order Sobolev metric. This includes elastic curves and also an extension of some results on planar concentric circles to surfaces. The work focuses on intrinsic and constructive approaches.