Abstract This article presents geometric optimal control techniques for analyzing geodesics in time-optimal Zermelo navigation problems on 2-spheres of revolution. We classify the problem by analyzing the pair ( F 0 , g ) , which represents the current (or wind) and the Riemannian metric. Using the maximum principle, the dynamics of geodesics are described by a Hamiltonian vector field on the cotangent bundle T ∗ S 2 . Our primary motivation is the application to micromagnetism, specifically spin magnetization reversal in ferromagnetic ellipsoidal samples. This model depends on four parameters and the amplitude of the applied magnetic field. The problem is formulated as a Zermelo navigation on the 2-sphere, where geodesics are classified as elliptic, hyperbolic, or abnormal. We demonstrate that the transition set | F 0 | g = 1 , which separates weak and strong current domains, is critical for understanding optimality. A key result shows that abnormal geodesics intersect this set with semi-cubical cusp singularities, a phenomenon we term the Landau–Lifshitz billiard. The analysis of the transition set’s connected components is complex and complemented by algebraic geometry and symbolic computations. We further reveal that hyperbolic geodesics lose optimality at their second intersection with the abnormal arc. Our numerical simulations complement this analysis by computing conjugate and cut loci, wavefronts, and accessibility sets, providing new insights into optimal magnetization switching under bounded control.

In this paper, we will study the relation between the caustics produced by the wavefronts of rays reflected in plane curves and the cut locus, as the self-intersection points of the wavefronts will generate the cut locus of the caustic. This is related to the study of caustics in general and their cusps, as other authors have made in [1, 2, 3]. Firstly, we will define the wavefronts and other related elements to analyse this statement, then we will use two different methods for evaluating the self-intersection points of the wavefronts, which are useful for determining self-intersections of curves in general. In the end, we will use the results obtained and see their relationship with the confocal conics when the reflection curve is an ellipse.

Abstract This work presents the Affine Heat Method for computing logarithmic maps. These maps are local surface parameterizations defined by the direction and distance along shortest geodesic paths from a given source point, and arise in many geometric tasks from local texture mapping to geodesic distance‐based optimization. Our main insight is to define a connection Laplacian with a homogeneous coordinate accounting for the translation between tangent coordinate frames; the action of short‐time heat flow under this Laplacian gives both the direction and distance from the source, along shortest geodesics. The resulting numerical method is straightforward to implement, fast, and improves accuracy compared to past approaches. We present two variants of the method, one of which enables pre‐computation for fast repeated solves, while the other resolves the map even near the cut locus in high detail. As with prior heat methods, our approach can be applied in any dimension and to any spatial discretization, including polygonal meshes and point clouds, which we demonstrate along with applications of the method.

Topological and control theoretic properties of Hamilton–Jacobi equations via Lax-Oleinik commutators

This paper presents a new method for the computation of the generalized Voronoi diagram of planar polygons. First, we show that the vertices of the cut locus can be computed efficiently. This is achieved by enumerating the tripoints of the polygon, a superset of the cut locus vertices. This is the set of all points that are of equal distance to three distinct topological entities. Then our algorithm identifies and connects the appropriate tripoints to form the cut locus vertex connectivity graph, where edges define linear or parabolic boundary segments between the Voronoi regions, resulting in the generalized Voronoi diagram. Our proposed method is validated on complex polygon soups. We apply the algorithm to represent the exact signed distance function of the polygon by augmenting the Voronoi regions with linear and radial functions, calculating the cut locus both inside and outside.

In this article, we investigate the cut locus of closed (not necessarily compact) submanifolds in a forward complete Finsler manifold. We explore the deformation and characterization of the cut locus, extending the results of Basu and Prasad (Algebr Geom Topol 23(9):4185–4233, 2023). Given a submanifold N, we consider an N-geodesic loop as an N-geodesic starting and ending in N, possibly at different points. This class of geodesics were studied by Omori (J Differ Geom 2:233–252, 1968). We obtain a generalization of Klingenberg’s lemma for closed geodesics (Klingenberg in: Ann Math 2(69):654–666, 1959). for N-geodesic loops in the reversible Finsler setting.

We prove a central limit theorem (CLT) for the Fréchet mean of independent and identically distributed observations in a compact Riemannian manifold assuming that the population Fréchet mean is unique. Previous general CLT results in this setting have assumed that the cut locus of the Fréchet mean lies outside the support of the population distribution. In this paper we present a CLT under some mild technical conditions on the manifold plus the following assumption on the population distribution: in a neighbourhood of the cut locus of the population Fréchet mean, the population distribution is absolutely continuous with respect to the volume measure on the manifold and in this neighhbourhood the Radon–Nikodym derivative has a version that is continuous. So far as we are aware, the CLT given here is the first which allows the cut locus to have co-dimension one or two when it is included in the support of the distribution. A key part of the proof is establishing an asymptotic approximation for the parallel transport of a certain vector field. Whether or not a non-standard term arises in the CLT depends on whether the co-dimension of the cut locus is one or greater than one: in the former case a non-standard term appears but not in the latter case. This is the first paper to give a general and explicit expression for the non-standard term which arises when the co-dimension of the cut locus is one.

The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems, dynamic low-rank decompositions and model reduction. With this mostly expository work, we aim to provide a collection of the essential facts and formulae on the geometry of the Grassmann manifold in a fashion that is fit for tackling the aforementioned problems with matrix-based algorithms. Moreover, we expose the Grassmann geometry both from the approach of representing subspaces with orthogonal projectors and when viewed as a quotient space of the orthogonal group, where subspaces are identified as equivalence classes of (orthogonal) bases. This bridges the associated research tracks and allows for an easy transition between these two approaches. Original contributions include a modified algorithm for computing the Riemannian logarithm map on the Grassmannian that is advantageous numerically but also allows for a more elementary, yet more complete description of the cut locus and the conjugate points. We also derive a formula for parallel transport along geodesics in the orthogonal projector perspective, formulae for the derivative of the exponential map, as well as a formula for Jacobi fields vanishing at one point.

In this paper, we prove the conjecture stating that, on any closed convex surface, the cut locus of a finite set $M$ with more than two points has length at least half the diameter of the surface.

In this paper, we prove the conjecture stating that, on any closed convex surface, the cut locus of a finite set $M$ with more than two points has length at least half the diameter of the surface.

We study the sub-Riemannian cut time and cut locus of a given point in a class of step-2 Carnot groups of Reiter–Heisenberg type. Following the Hamiltonian point of view, we write and analyze extremal curves, getting the cut time of any of them, and a precise description of the set of cut points.

We consider a left-invariant sub-Riemannian problem on the Lie group of rotations of a three-dimensional space. We find the cut locus numerically, in fact we construct the optimal synthesis numerically, i. e., the shortest arcs. The software package nutopy designed for the numerical solution of optimal control problems is used. With the help of this package we investigate sub-Riemannian geodesics, conjugate points, Maxwell points and diffeomorphic domains of the exponential map. We describe some operating features of this software package.

Associated to every closed, embedded submanifold N in a connected Riemannian manifold M, there is the distance function d N which measures the distance of a point in M from N. We analyze the square of this function and show that it is Morse-Bott on the complement of the cut locus Cu.N / of N provided M is complete. Moreover, the gradient flow lines provide a deformation retraction of M Cu.N / to N. If M is a closed manifold, then we prove that the Thom space of the normal bundle of N is homeomorphic to M=Cu.N /. We also discuss several interesting results which are either applications of these or related observations regarding the theory of cut locus. These results include, but are not limited to, a computation of the local homology of singular matrices, a classification of the homotopy type of the cut locus of a homology sphere inside a sphere, a deformation of the indefinite unitary group U.p; q/ to U.p/ U.q/ and a geometric deformation of GL.n; R/ to O.n; R/ which is different from the Gram-Schmidt retraction.

Isomorphism classes of cut loci for a cube

There are not so many kinds of surface of revolution whose cut locus structure have been determined, although the cut locus structures of very familiar surfaces of revolution (in Euclidean space) such as ellipsoids, 2-sheeted hyperboloids, paraboloids and tori are now known. Except for tori, the known cut locus structures are very simple, i.e., a single point or an arc. In this article, a new family $\{M_n\}_n$ of 2-spheres of revolution with simple cut locus structure is introduced. This family is also new in the sense that the number of points on each meridian which assume a local minimum or maximum of the Gaussian curvature function on the meridian goes to infinity as $n$ tends to infinity. Thus, our family includes surfaces which have arbitrarily many bands of alternately increasing or decreasing Gaussian curvature, although each member of this family has a simple cut locus structure.

Cut locus realizations on convex polyhedra

We study the variational problem of finding the fastest path between two points that belong to different anisotropic media, each with a prescribed speed profile and a common interface. The optimal curves are Finsler geodesics that are refracted — broken — as they pass through the interface, due to the discontinuity of their velocities. This “breaking” must satisfy a specific condition in terms of the Finsler metrics defined by the speed profiles, thus establishing the generalized Snell’s law. In the same way, optimal paths bouncing off the interface — without crossing into the second domain — provide the generalized law of reflection. The classical Snell’s and reflection laws are recovered in this setting when the velocities are isotropic. If one considers a wave that propagates in all directions from a given ignition point, the trajectories that globally minimize the traveltime generate the wavefront at each instant of time. We study in detail the global properties of such wavefronts in the Euclidean plane with anisotropic speed profiles. Like the individual rays, they break when they encounter the discontinuity interface. But they are also broken due to the formation of cut loci — stemming from the self-intersection of the wavefronts — which typically appear when they approach a high-speed profile domain from a low-speed profile.

<abstract><p>In the present paper, we study the Randers metric on two-spheres of revolution in order to obtain new families of Finsler of Randers type metrics with simple cut locus. We determine the geodesics behavior, conjugate and cut loci of some families of Finsler metrics of Randers type whose navigation data is not a Killing field and without sectional or flag curvature restrictions. Several examples of Randers metrics whose cut locus is simple are shown.</p></abstract>

In this article, the historical study from Carathéodory-Zermelo about computing the quickest nautical path is generalized to Zermelo navigation problems on surfaces of revolution, in the frame of geometric optimal control. Using the Maximum Principle, we present two methods dedicated to analyzing the geodesic flow and to compute the conjugate and cut loci. We apply these calculations to investigate case studies related to applications in hydrodynamics, space mechanics and geometry.

Left-invariant optimal control problems on Lie groups are an important class of problems with a large symmetry group. They are theoretically interesting because they can often be investigated in full and general laws can be studied by using these model problems. In particular, problems on nilpotent Lie groups provide a fundamental nilpotent approximation to general problems. Also, left-invariant problems often arise in applications such as classical and quantum mechanics, geometry, robotics, visual perception models, and image processing. The aim of this paper is to present a survey of the main concepts, methods, and results pertaining to left-invariant optimal control problems on Lie groups that can be integrated by elliptic functions. The focus is on describing extremal trajectories and their optimality, the cut time and cut locus, and optimal synthesis. Bibliography: 162 titles.