In this paper we wish to endow the manifold M of smooth curves in R^n (either closed curves or open curves with fixed endpoints) with a Riemannian structure that allows us to treat homotopies between two curves C0 and C1 as trajectories with computable lengths. If we regard a curve as all possible reparameterizations of a C^1 mapping of the unit interval into R^n, then the tangent space at a point on M corresponds to all possible non-tangential flows of the underlying curve. We note that a Riemannian metric corresponds to a choice of inner-product on this tangent space. Once this is defined, we can compute distances between curves and consider the natural problem of finding geodesics which will yield minimal length homotopies between C0 and C1. We begin by noting that a seemingly natural inner-product corresponding to a geometric version of 2 does not yield useful geodesics and will instead introduce a sequence of conformally modfied innerproducts which has interesting limit properties.
GEODESIC HOMOTOPIES
MENNUCCI, Andrea Carlo Giuseppe
2004
Abstract
In this paper we wish to endow the manifold M of smooth curves in R^n (either closed curves or open curves with fixed endpoints) with a Riemannian structure that allows us to treat homotopies between two curves C0 and C1 as trajectories with computable lengths. If we regard a curve as all possible reparameterizations of a C^1 mapping of the unit interval into R^n, then the tangent space at a point on M corresponds to all possible non-tangential flows of the underlying curve. We note that a Riemannian metric corresponds to a choice of inner-product on this tangent space. Once this is defined, we can compute distances between curves and consider the natural problem of finding geodesics which will yield minimal length homotopies between C0 and C1. We begin by noting that a seemingly natural inner-product corresponding to a geometric version of 2 does not yield useful geodesics and will instead introduce a sequence of conformally modfied innerproducts which has interesting limit properties.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
