TY - JOUR

T1 - Geodesic bifurcation on smooth surfaces

AU - Thielhelm, Hannes

AU - Vais, Alexander

AU - Wolter, Franz Erich

N1 - Publisher Copyright: © 2014, Springer-Verlag Berlin Heidelberg.

PY - 2015/2

Y1 - 2015/2

N2 - 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.

AB - 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.

KW - Connecting geodesics

KW - Cut locus

KW - Distance computation

KW - Focal curves

KW - Geodesic exponential map

KW - Voronoi diagram

UR - http://www.scopus.com/inward/record.url?scp=84925513453&partnerID=8YFLogxK

U2 - 10.1007/s00371-014-1041-3

DO - 10.1007/s00371-014-1041-3

M3 - Article

AN - SCOPUS:84925513453

VL - 31

SP - 187

EP - 204

JO - Visual Computer

JF - Visual Computer

SN - 0178-2789

IS - 2

ER -