In this paper, we present a novel method for computing multiple geodesic connections between two arbitrary points on a smooth surface. Our method is based on a homotopy approach that is able to capture the ambiguity of geodesic connections in the presence of positive Gaussian curvature that generates focal curves.Contrary to previous approaches, we exploit focal curves to gain theoretical insights on the number of connecting geodesics and a practical algorithm for collecting these.We consider our method as a contribution to the contemporary debate regarding the calculation of distances in general situations, applying continuous concepts of classical differential geometry which are not immediately transferable in purely discrete settings.
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 discus-
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