Details
| Original language | English |
|---|---|
| Pages (from-to) | 187-204 |
| Number of pages | 18 |
| Journal | Visual Computer |
| Volume | 31 |
| Issue number | 2 |
| Early online date | 4 Nov 2014 |
| Publication status | Published - Feb 2015 |
Abstract
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.
Keywords
- Connecting geodesics, Cut locus, Distance computation, Focal curves, Geodesic exponential map, Voronoi diagram
ASJC Scopus subject areas
- Computer Science(all)
- Software
- Computer Science(all)
- Computer Vision and Pattern Recognition
- Computer Science(all)
- Computer Graphics and Computer-Aided Design
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In: Visual Computer, Vol. 31, No. 2, 02.2015, p. 187-204.
Research output: Contribution to journal › Article › Research › peer review
}
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 -