A Survey of Algorithms for Geodesic Paths and Distances

Abstract

Numerical computation of shortest paths or geodesics on curved domains, as well as the associated geodesic distance, arises in a broad range of applications across digital geometry processing, scientific computing, computer graphics, and computer vision. Relative to Euclidean distance computation, these tasks are complicated by the influence of curvature on the behavior of shortest paths, as well as the fact that the representation of the domain may itself be approximate. In spite of the difficulty of this problem, recent literature has developed a wide variety of sophisticated methods that enable rapid queries of geodesic information, even on relatively large models. This survey reviews the major categories of approaches to the computation of geodesic paths and distances, highlighting common themes and opportunities for future improvement.

Ray-Tracing Model for Generalized Geodesic-Lens Multiple-Beam Antennas

“…For the solution of the variational equation (1), we can resort to well-established numerical procedures for the calculation of geodesic paths on an arbitrary polyhedral surface given in terms of a mesh [55]- [58]. For example, in Fig.…”

A comprehensive analysis for classification and regression of surface points based on geodesics and machine learning algorithms

“…Patrikalakis and Ko obtained geodesics by solving the initial value problem (IVP) and the boundary value problem (BVP) with four boundary conditions (Patrikalakis and Ko, 2003). Numerical computation of geodesics or shortest paths has crucial applications in computer graphics, digital geometry processing, computer vision and scientific computing (Crane et al ., 2020). Geodesic paths are computed on Bezier and Nurbs patches in (Patrikalakis and Bardis, 1989; Patrikalakis and Maekawa, 2010).…”

“…In the literature, geodesics are studied as geodesic curves as in (Crane et al ., 2020; Arvanitakis et al ., 2013; Scholz and Maekawa, 2021) or as geodesic distances as in (Nassih et al ., 2021). The difference of our study from the others is the classification of the surface points with respect to the geodesic values of the given surface based on the differential geometric properties of the surface points using a variety of machine learning techniques.…”

A linear finite-difference scheme for approximating randers distances on cartesian grids

“…Remark 1.2 (Advantages of linear schemes for distance map computation). Distance maps are ubiquitous in mathematics and their applications, and a variety of approaches have been proposed for their numerical computation [22], including Randers distances [39,42]. The use of a linear PDE (1.3), is here largely motivated by its application to the optimal transport problem (1.7), but this approach has other advantages, see [23] for a more detailed discussion:…”