Hermite subdivision on manifolds via parallel transport
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Hermite subdivision on manifolds via parallel transport 1 ¨ Caroline Moosmuller
Received: 27 April 2016 / Accepted: 18 January 2017 © The Author(s) 2017. This article is an open access publication
Abstract We propose a new adaption of linear Hermite subdivision schemes to the manifold setting. Our construction is intrinsic, as it is based solely on geodesics and on the parallel transport operator of the manifold. The resulting nonlinear Hermite subdivision schemes are analyzed with respect to convergence and C 1 smoothness. Similar to previous work on manifold-valued subdivision, this analysis is carried out by proving that a so-called proximity condition is fulfilled. This condition allows to conclude convergence and smoothness properties of the manifold-valued scheme from its linear counterpart, provided that the input data are dense enough. Therefore the main part of this paper is concerned with showing that our nonlinear Hermite scheme is “close enough”, i.e., in proximity, to the linear scheme it is derived from. Keywords Hermite subdivision · Manifolds subdivision · C 1 analysis · Proximity Mathematics Subject Classification (2010) 41A25 · 65D17 · 53A99
1 Introduction Hermite subdivision is an iterative method for constructing a curve together with its derivatives from discrete point-vector data. It has mainly been studied in the linear Communicated by: Tomas Sauer The author gratefully acknowledges support by the doctoral program “Discrete Mathematics”, funded by the Austrian Science Fund FWF under grant agreement W1230. Caroline Moosm¨uller
[email protected] 1
Institut f¨ur Geometrie, TU Graz, Kopernikusgasse 24, 8010 Graz, Austria
C. Moosm¨uller
setting, where many results concerning convergence and smoothness are available, such as [3, 4, 6–8, 14] and others. In a recent paper [15] we propose an analogue of linear Hermite schemes in manifolds which are equipped with an exponential map. This construction works via conversion of vector data to point data, and makes use of the well-established methods of non-Hermite subdivision in manifold, see [9] for an overview. The present paper investigates manifold analogues of Hermite subdivision rules which work directly with vectors and employ the parallel transport operators available in Riemannian manifolds and also in Lie groups. Our motivation for doing subdivision in this way is to use only such operations which are intrinsic to the underlying geometry and which therefore commute with isomorphisms of the respective geometric structures. The C 1 convergence analysis of the nonlinear schemes we obtain by the parallel transport approach is provided from their linear counterparts by means of a proximity condition for Hermite schemes introduced by [15]. This condition allows to conclude C 1 convergence of the manifold-valued scheme if it is “close enough” to a C 1 convergent linear one. Similar to most previous results on manifold subdivision, C 1 convergence can only be deduced if the input data are dense enough. The paper is organized as follows: In
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