Geodesic Problem Research Articles

Logspace computations in graph products

We consider three important and well-studied algorithmic problems in group theory: the word, geodesic, and conjugacy problem. We show transfer results from individual groups to graph products. We concentrate on logspace complexity because the challenge is actually in small complexity classes, only. The most difficult transfer result is for the conjugacy problem. We have a general result for graph products, but even in the special case of a graph group the result is new. Graph groups are closely linked to the theory of Mazurkiewicz traces which form an algebraic model for concurrent processes. Our proofs are combinatorial and based on well-known concepts in trace theory. We also use rewriting techniques over traces. For the group-theoretical part we apply Bass–Serre theory. But as we need explicit formulae and as we design concrete algorithms all our group-theoretical calculations are completely explicit and accessible to non-specialists.

An n Center Problem as an Example of a Geodesic Billiard Map with Singularities: Poincaré Maps and Dynamics

This paper deals with the motion of a particle of infinitesimal mass and moving under the gravitational attraction of n particles of equal mass and fixed in the vertices of a regular polygon. The complete reduction of this problem is done using its finite group of symmetries. We show that it is the lift of a flow which is topologically conjugate to a geodesic billiard problem with the Jacobi metric and also study numerically the corresponding billiard maps.

Connectivity by geodesics in open subsets of globally hyperbolic spacetimes

We discuss the geodesic connectedness problem in open subsets with convex boundary of globally hyperbolic spacetimes endowed with a complete, timelike or lightlike, Killing vector field. Furthermore, we furnish applications to generalized plane waves.

Solving the initial value problem of discrete geodesics

Computing geodesic paths and distances is a common operation in computer graphics and computer-aided geometric design. The existing discrete geodesic algorithms are mainly designed to solve the boundary value problem, i.e., to find the shortest path between two given points. In this paper, we focus on the initial value problem, i.e., finding a uniquely determined geodesic path from a given point in any direction. Since the shortest paths do not provide the unique solution on triangle meshes, we solve the initial value problem in an indirect manner: given a fixed point and an initial tangent direction on a triangle mesh M, we first compute a geodesic curve γ̂ on a piecewise smooth surface M̂, which well approximates the input mesh M and can be constructed at little cost. Then, we solve a first-order ODE of the tangent vector using the fourth-order Runge–Kutta method, and parallel transport it along γ̂. When the geodesic curve reaches the boundary of the current patch, its tangent can be directly transported to the neighboring patch, thanks to the G1-continuity along the common boundary of two adjacent patches. Finally, once the geodesic curve γ̂ is available, we project it onto the underlying mesh M, producing the discrete geodesic path γ, which is guaranteed to be unique on M. It is worth noting that our method is different from the conventional methods of directly solving the geodesic equation (i.e., a second-order ODE of the position) on piecewise smooth surfaces, which are difficult to implement due to the complicated representation of the geodesic equation involving Christoffel symbols. The proposed method, based on the first-order ODE of the tangent vector, is intuitive and easy for implementation. Our method is particularly useful for computing geodesic paths on low-resolution meshes which may have large and/or skinny triangles, since the conventional straightest geodesic paths are usually far from the ground truth.

Near-Horizon Celestial Phenomena, a Study in Geometric Optics

Near horizon phenomena, like blank strips in the setting sun or Fata Morganas, can be largely understood in terms of geometric optics. Here the existence of a warm layer above the observer in which light rays can be either refracted of reflected, already suffices to explain many of the phenomena; a reasoning which goes back to Alfred Wegener in the 1920’s. We review this theory, where first the atmosphere is divided into discrete, optically homogeneous layers, and where later the atmosphere is considered as a continuous, isotropic medium. Thus the theory gets embedded in differential geometry, where light rays are geodesics. A simplifying assumption is that the refraction index only depends on the elevation above the surface of the earth. Here a connection is made with the geodesic problem on a surface of revolution. In the background of this we touch the variational calculus as this runs from Bernoulli to Hamilton, all the way maintaining a completely anachronistic viewpoint.

Explicit methods in extended phase space for inseparable Hamiltonian problems

We present a method for explicit leapfrog integration of inseparable Hamiltonian systems by means of an extended phase space. A suitably defined new Hamiltonian on the extended phase space leads to equations of motion that can be numerically integrated by standard symplectic leapfrog (splitting) methods. When the leapfrog is combined with coordinate mixing transformations, the resulting algorithm shows good long term stability and error behaviour. We extend the method to non-Hamiltonian problems as well, and investigate optimal methods of projecting the extended phase space back to original dimension. Finally, we apply the methods to a Hamiltonian problem of geodesics in a curved space, and a non-Hamiltonian problem of a forced non-linear oscillator. We compare the performance of the methods to a general purpose differential equation solver LSODE, and the implicit midpoint method, a symplectic one-step method. We find the extended phase space methods to compare favorably to both for the Hamiltonian problem, and to the implicit midpoint method in the case of the non-linear oscillator.

Isometry-invariant geodesics and the fundamental group

We prove that on closed Riemannian manifolds with infinite abelian, but not cyclic, fundamental group, any isometry that is homotopic to the identity possesses infinitely many invariant geodesics. We conjecture that the result remains true if the fundamental group is infinite cyclic. We also formulate a generalization of the isometry-invariant geodesics problem, and a generalization of the celebrated Weinstein conjecture: on a closed contact manifold with a selected contact form, any strict contactomorphism that is contact-isotopic to the identity possesses an invariant Reeb orbit.

A unified framework for computing geodesic distances on nonorientable manifold polyhedral surfaces

Nonorientable manifold surfaces not only play an important role in topology, but also have numerous applications in many topics such as visualization and computation of minimal surfaces. From the topological point of view, a 2-manifold surface is locally homeomorphic to an open disk. This property is independent of the global orientability. However, as far as the discrete representation is concerned, orientable manifold surfaces are usually discretized with halfedge data structure, while nonorientable surfaces are discretized into polygon soups, which is inconvenient for digital geometry processing that often takes orientable meshes as input. In this paper, we propose a uni ed framework for transforming geodesic distance problems de ned on nonorientable 2-manifold meshes to the counter-parts on orientable surfaces, and thereby bridging up nonorientable 2-manifold meshes and conventional geometric algorithms. In order to illustrate the universal adaptability, we apply this new approach to study three problems on nonorientable meshes, including computing exact geodesic paths, discrete exponential mapping and farthest point sampling.

Geodesics on Shape Spaces with Bounded Variation and Sobolev Metrics

This paper studies the space of $BV^2$ planar curves endowed with the $BV^2$ Finsler metric over its tangent space of displacement vector fields. Such a space is of interest for applications in image processing and computer vision because it enables piecewise regular curves that undergo piecewise regular deformations, such as articulations. The main contribution of this paper is the proof of the existence of the shortest path between any two $BV^2$-curves for this Finsler metric. Such a result is proved by applying the direct method of calculus of variation to minimize the geodesic energy. This method applies more generally to similar cases such as the space of curves with $H^k$ metrics for $k\geq 2$ integer. This space has a strong Riemannian structure and is geodesically complete. Thus, our result shows that the exponential map is surjective, which is complementary to geodesic completeness in infinite dimensions. We propose a finite element discretization of the minimal geodesic problem, and use a gradient descent method to compute a stationary point of the energy. Numerical illustrations show the qualitative difference between $BV^2$ and $H^2$ geodesics.

The Time Scale Calculus Approach to the Geodesic Problem in 3D Dynamic Data Sets

Geodesics have a fundamental role in the geometry of curved surfaces, as well as in discrete geometry. We present the time scale analogy of the dynamic data sets parameterized by a tensor product of two times scales. The goal of our study is the find the shortest and straightest path between two points on a point cloud like data sets which also involves continuous data.

Saddle vertex graph (SVG)

This paper presents the Saddle Vertex Graph (SVG), a novel solution to the discrete geodesic problem. The SVG is a sparse undirected graph that encodes complete geodesic distance information: a geodesic path on the mesh is equivalent to a shortest path on the SVG, which can be solved efficiently using the shortest path algorithm (e.g., Dijkstra algorithm). The SVG method solves the discrete geodesic problem from a local perspective. We have observed that the polyhedral surface has some interesting and unique properties, such as the fact that the discrete geodesic exhibits a strong local structure, which is not available on the smooth surfaces. The richer the details and complicated geometry of the mesh, the stronger such local structure will be. Taking advantage of the local nature, the SVG algorithm breaks down the discrete geodesic problem into significantly smaller sub-problems, and elegantly enables information reuse. It does not require any numerical solver, and is numerically stable and insensitive to the mesh resolution and tessellation. Users can intuitively specify a model-independent parameter K , which effectively balances the SVG complexity and the accuracy of the computed geodesic distance. More importantly, the computed distance is guaranteed to be a metric. The experimental results on real-world models demonstrate significant improvement to the existing approximate geodesic methods in terms of both performance and accuracy.

Is General Relativity a simplified theory?

Gravity is understood as a geometrization of spacetime. But spacetime is also the manifold of the boundary values of the spinless point particle in a variational approach. Since all known matter, baryons, leptons and gauge bosons are spinning objects, it means that the manifold, which we call the kinematical space, where we play the game of the variational formalism of an elementary particle is greater than spacetime. This manifold for any mechanical system is a Finsler metric space such that the variational formalism can always be interpreted as a geodesic problem on this space. This manifold is just the flat Minkowski space for the free spinless particle. Any interaction modifies its flat Finsler metric as gravitation does. The same thing happens for the spinning objects but now the Finsler metric space has more dimensions and its metric is modified by any interaction, so that to reduce gravity to the modification only of the spacetime metric is to make a simpler theory, the gravitational theory of spinless matter. Even the usual assumption that the modification of the metric only involves dependence on the metric coefficients of the spacetime variables is also a restriction because in general these coefficients are dependent on the velocities. In the spirit of unification of all forces, gravity cannot produce, in principle, a different and simpler geometrization than any other interaction.

Affine-quadratic problems on Lie groups

This paper focuses on a class of left invariant variational problems on a Lie group$ G $whose Lie algebra $\mathfrak{g}$ admits Cartan decomposition $ \mathfrak{g}=\mathfrak{p}+\mathfrak{k}$ with the usual Lie algebraic conditions \begin{equation*} \lbrack \mathfrak{p},\mathfrak{p]\subseteq k }, \mathfrak{[p},\mathfrak{ k]\subseteq p},\mathfrak{ [k},\mathfrak{k]\subseteq k.} \end{equation*} &nbsp The Maximum Principle of optimal control leads to the Hamiltonians $H$ on $ \mathfrak{g }$that admit spectral parameter representations with important contributions to the theory of integrable Hamiltonian systems. Particular cases will be singled out that provides natural explanations for the classical results of Fock and Moser linking Kepler's problem to the geodesics on spaces of constant curvature, C.L. Jacobi's geodesic problem on an ellipsoid and J.Moser's work on integrability based on isospectral methods. The paper also shows the relevance of this class of Hamiltonians to the elastic curves on spaces of constant curvature.

The geodesic boundary value problem and its solution on a triaxial ellipsoid

The geodesic boundary value problem and its solution on a triaxial ellipsoid

On the dynamic Markov-Dubins problem: From path planning in robotics and biolocomotion to computational anatomy

Andrei Andreyevich Markov proposed in 1889 the problem (solved by Dubins in 1957) of finding the twice continuously differentiable (arc length parameterized) curve with bounded curvature, of minimum length, connecting two unit vectors at two arbitrary points in the plane. In this note we consider the following variant, which we call the dynamic Markov-Dubins problem (dM-D): to find the time-optimal C 2 trajectory connecting two velocity vectors having possibly different norms. The control is given by a force whose norm is bounded. The acceleration may have a tangential component, and corners are allowed, provided the velocity vanishes there. We show that for almost all the two vectors boundary value conditions, the optimization problem has a smooth solution. We suggest some research directions for the dM-D problem on Riemannian manifolds, in particular we would like to know what happens if the underlying geodesic problem is completely integrable. Path planning in robotics and aviation should be the usual applications, and we suggest a pursuit problem in biolocomotion. Finally, we suggest a somewhat unexpected application to “dynamic imaging science”. Short time processes (in medicine and biology, in environment sciences, geophysics, even social sciences?) can be thought as tangent vectors. The time needed to connect two processes via a dynamic Markov-Dubins problem provides a notion of distance. Statistical methods could then be employed for classification purposes using a training set.

Active Contours on Graphs: Multiscale Morphology and Graphcuts

In this paper we propose two novel methods for formulating and implementing the methodology of geodesic active contours on arbitrary graphs, as applied to multiscale morphology and segmentation. Firstly, we propose approximations to the calculation of the gradient and the divergence of vector functions defined on graphs and use these approximations to apply the technique of geodesic active contours for object detection on graphs. To this end, we extend existing work on graph morphology to multiscale dilation and erosion and implement them recursively usinglevel sets of functions defined on the graph. Second, we propose a graphcut based solution to the geodesic active contour problem on graphs. Appropriate weights are calculated for each edge for which the Riemannian length of a contour can be approximated by the weighted sum of intersections of the contour with the edges of the graph. Finding the minimum Riemannian length contour then becomes equivalent to solving a max flow problem for which efficient solutions have been proposed in the literature.

Algorithms for geodesics

Algorithms for the computation of geodesics on an ellipsoid of revolution are given. These provide accurate, robust, and fast solutions to the direct and inverse geodesic problems and they allow differential and integral properties of geodesics to be computed.

Remarks on the minimizing geodesic problem in inviscid incompressible fluid mechanics

We consider L2 minimizing geodesics along the group of volume preserving maps SDiff(D) of a given 3-dimensional domain D. The corresponding curves describe the motion of an ideal incompressible fluid inside D and are (formally) solutions of the Euler equations. It is known that there is a unique possible pressure gradient for these curves whenever their end points are fixed. In addition, this pressure field has a limited but unconditional (internal) regularity. The present paper completes these results by showing: (1) the uniqueness property can be viewed as an infinite dimensional phenomenon (related to the possibility of relaxing the corresponding minimization problem by convex optimization), which is false for finite dimensional configuration spaces such as O(3) for the motion of rigid bodies; (2) the unconditional partial regularity is necessarily limited.

APPROXIMATION OF GEODESICS IN METABELIAN GROUPS

It is known that the bounded Geodesic Length Problem in free metabelian groups is NP-complete [A. Myasnikov, V. Roman'kov, A. Ushakov and A. Vershik, The word and geodesic problems in free solvable groups, Trans. Amer. Math. Soc.362(9) (2010) 4655–4682] (in particular, the Geodesic Problem is NP-hard). We construct a 2-approximation polynomial time deterministic algorithm for the Geodesic Problem. We show that the Geodesic Problem in the restricted wreath product of a finitely generated non-trivial group with a finitely generated abelian group containing ℤ2 is NP-hard and there exists a Polynomial Time Approximation Scheme for this problem. We also show that the Geodesic Problem in the restricted wreath product of two finitely generated non-trivial abelian groups is NP-hard if and only if the second abelian group contains ℤ2.

Geodesic pancyclicity of twisted cubes

The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement. An n-dimensional twisted cube, TQ n , is an important variation of the hypercube Q n and preserves many of its desirable properties. The problem of embedding linear arrays and cycles into a host graph has attracted substantial attention in recent years. The geodesic cycle embedding problem is for any two distinct vertices, to find all the possible lengths of cycles that include a shortest path joining them. In this paper, we prove that TQ n is geodesic 2-pancyclic for each odd integer n ⩾ 3. This result implies that TQ n is edge-pancyclic for each odd integer n ⩾ 3. Moreover, TQ n × K 2 is also demonstrated to be geodesic 4-pancyclic.