caa a22 4500 605541337 CHVBK 20210128100916.0 cr unu---uuuuu 210128e20150201xx s 000 0 eng 10.1007/s00371-014-1041-3 doi (NATIONALLICENCE)springer-10.1007/s00371-014-1041-3 Geodesic bifurcation on smooth surfaces [Elektronische Daten] [Hannes Thielhelm, Alexander Vais, Franz-Erich Wolter] Within Riemannian geometry the geodesic exponential map is an essential tool for various distance-related investigations and computations. Several natural questions can be formulated in terms of its preimages, usually leading to quite challenging non-linear problems. In this context we recently proposed an approach for computing multiple geodesics connecting two arbitrary points on two-dimensional surfaces in situations where an ambiguity of these connecting geodesics is indicated by the presence of focal curves. The essence of the approach consists in exploiting the structure of the associated focal curve and using a suitable curve for a homotopy algorithm to collect the geodesic connections. In this follow-up discussion we extend those constructions to overcome a significant limitation inherent in the previous method, i.e. the necessity to construct homotopy curves artificially. We show that considering homotopy curves meeting a focal curve tangentially leads to a singularity that we investigate thoroughly. Solving this so-called geodesic bifurcation analytically and dealing with it numerically provides not only theoretical insights, but also allows geodesics to be used as homotopy curves. This yields a stable computational tool in the context of computing distances. This is applicable in common situations where there is a curvature induced non-injectivity of the exponential map. In particular we illustrate how applying geodesic bifurcation approaches the distance problem on compact manifolds with a single closed focal curve. Furthermore, the presented investigations provide natural initial values for computing cut loci using the medial differential equation which directly leads to a discussion on avoiding redundant computations by combining the presented concepts to determine branching points. Springer-Verlag Berlin Heidelberg, 2014 Geodesic exponential map nationallicence Focal curves nationallicence Connecting geodesics nationallicence Distance computation nationallicence Cut locus nationallicence Voronoi diagram nationallicence Thielhelm Hannes Welfenlab, Leibniz University of Hannover, Hannover, Germany aut Vais Alexander Welfenlab, Leibniz University of Hannover, Hannover, Germany aut Wolter Franz-Erich Welfenlab, Leibniz University of Hannover, Hannover, Germany aut The Visual Computer Springer Berlin Heidelberg 31/2(2015-02-01), 187-204 0178-2789 31:2<187 2015 31 371 https://doi.org/10.1007/s00371-014-1041-3 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00371-014-1041-3 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Thielhelm Hannes Welfenlab, Leibniz University of Hannover, Hannover, Germany aut NATIONALLICENCE 700 1- Vais Alexander Welfenlab, Leibniz University of Hannover, Hannover, Germany aut NATIONALLICENCE 700 1- Wolter Franz-Erich Welfenlab, Leibniz University of Hannover, Hannover, Germany aut NATIONALLICENCE 773 0- The Visual Computer Springer Berlin Heidelberg 31/2(2015-02-01), 187-204 0178-2789 31:2<187 2015 31 371