Geodesic Triangle Research Articles

Planar subspaces are intrinsically CAT(0)

Let $M^2_{\kappa}$ be the complete, simply connected, Riemannian 2-manifold of constant curvature $\kappa \leq 0$. Let $E$ be a closed, simply connected subspace of $M^2_{\kappa}$ with the property that every pair of points in $E$ is connected by a rectifiable path in $E$. We show that under the induced path metric, $E$ is a complete CAT($\kappa$) space. We also show that the natural notions of angle coming from the intrinsic and extrinsic metrics coincide for all simple geodesic triangles.

Использование поисковых методов для уравнивания и оценки точности элементарных геодезических построений

The author provides information about the search methods of adjustment and assessment of the elementary geodetic constructions accuracy on the example of a geodesic triangle under various initial conditions. An algorithm for combining Powell and DSC search methods used to solve the problem posed in the article is given. To equalize the geodesic constructions under consideration by the search method, a program for finding the minimum of the objective function is made, it implements the algorithm described in the article. This program works in the Microsoft Excel format as a macro recorded in the VBA language. Examples of equalization calculations are given. Accuracy assessment of the adjustment results by the search method of the geodesic triangle under different initial conditions is performed. The obtained adjustment results and accuracy estimates are compared with the calculations performed in the NW program by Professor Kougiya V. A. confirmed the correctness of the search method application.

Gromov hyperbolicity in lexicographic product graphs

If X is a geodesic metric space and $$x_1,x_2,x_3\in X$$ , a geodesic triangle $$T=\{x_1,x_2,x_3\}$$ is the union of the three geodesics $$[x_1x_2]$$ , $$[x_2x_3]$$ and $$[x_3x_1]$$ in X. The space X is $$\delta $$ -hyperbolic (in the Gromov sense) if any side of T is contained in a $$\delta $$ -neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by $$\delta (X)$$ the sharp hyperbolicity constant of X, i.e. $$\delta (X)=\inf \{\delta \ge 0: \, X \, \text { is }\delta \text {-hyperbolic}\}$$ . In this paper, we characterize the lexicographic product of two graphs $$G_1\circ G_2$$ which are hyperbolic, in terms of $$G_1$$ and $$G_2$$ : the lexicographic product graph $$G_1\circ G_2$$ is hyperbolic if and only if $$G_1$$ is hyperbolic, unless if $$G_1$$ is a trivial graph (the graph with a single vertex); if $$G_1$$ is trivial, then $$G_1\circ G_2$$ is hyperbolic if and only if $$G_2$$ is hyperbolic. In particular, we obtain the sharp inequalities $$\delta (G_1)\le \delta (G_1\circ G_2) \le \delta (G_1) + 3/2$$ if $$G_1$$ is not a trivial graph, and we characterize the graphs for which the second inequality is attained.

The symplectic area of a geodesic triangle in a Hermitian symmetric space of compact type

Let M be an irreducible Hermitian symmetric space of compact type, and let $$\omega $$ be its Kahler form. For a triplet $$(p_1,p_2,p_3)$$ of points in M we study conditions under which a geodesic triangle $$\mathcal T(p_1,p_2,p_3)$$ with vertices $$p_1,p_2,p_3$$ can be unambiguously defined. We consider the integral $$A(p_1,p_2,p_3)=\int _\Sigma \omega $$ , where $$\Sigma $$ is a surface filling the triangle $$\mathcal T(p_1,p_2,p_3)$$ and discuss the dependence of $$A(p_1,p_2,p_3)$$ on the surface $$\Sigma $$ . Under mild conditions on the three points, we prove an explicit formula for $$A(p_1,p_2,p_3)$$ analogous to the known formula for the symplectic area of a geodesic triangle in a non-compact Hermitian symmetric space.

The dimension of the Hitchin component for triangle groups

Let \( p,q,r \) be positive integers satisfying \( 1/p + 1/q + 1/r < 1 \), and let \( \Delta (p,q,r) \) be a geodesic triangle in the hyperbolic plane with angles \( \pi /p \,,\,\pi /q \,,\,\pi /r \). Then there exists a tiling of the hyperbolic plane by triangles congruent to \( \Delta (p,q,r) \), and we define the triangle group\( T(p,q,r) \) to be the group of orientation preserving isometries of this tiling. Representation varieties of closed surface groups into \( \mathrm {SL(n,{\mathbb {R}})}\) have been studied extensively by Hitchin and Labourie, and the dimension of a certain distinguished component of the variety was obtained by Hitchin using Higgs bundles. Here we determine the corresponding dimension for representations of triangle groups into \( \mathrm {SL(n,{\mathbb {R}})}\), generalising some earlier work of Choi and Goldman in the case \(n = 3\).

$$\mathbf {Nil}$$ Nil Geodesic Triangles and Their Interior Angle Sums

In this paper we study the interior angle sums of geodesic triangles in \(\mathbf {Nil}\) geometry and prove that these can be larger, equal or less than \(\pi \). We use for the computations the projective model of \(\mathbf {Nil}\) introduced by Molnar (Beitr. Algebra Geom. 38(2):261–288, 1997).

The sum of the interior angles in geodesic and translation triangles of Sl2(R)~ geometry

We study the interior angle sums of translation and geodesic triangles in the universal cover of real 2 x 2 matrices with unit determinant, as, a Thurston geometry denoted by P of SL2(R)~ geometry. We prove that the angle sum ?3i =1(?i) ? ? for translation triangles and for geodesic triangles the angle sum can be larger, equal or less than ?.

Decomposition of closed orientable geometric surfaces into acute geodesic triangles

We prove that for every $$g\ge 2$$ , a differentiable closed orientable geometric surface of genus g may be decomposed into $$16g-16$$ acute geodesic triangles. We also determine the number of acute geodesic triangles needed for the sphere and the torus.

Generalized Chordality, Vertex Separators and Hyperbolicity on Graphs

A graph is chordal if every induced cycle has exactly three edges. A vertex separator set in a graph is a set of vertices that disconnects two vertices. A graph is δ -hyperbolic if every geodesic triangle is δ -thin. In this paper, we study the relation between vertex separator sets, certain chordality properties that generalize being chordal and the hyperbolicity of the graph. We also give a characterization of being quasi-isometric to a tree in terms of chordality and prove that this condition also characterizes being hyperbolic, when restricted to triangles, and having stable geodesics, when restricted to bigons.

The hyperbolicity constant of infinite circulant graphs

Abstract If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. Deciding whether or not a graph is hyperbolic is usually very difficult; therefore, it is interesting to find classes of graphs which are hyperbolic. A graph is circulant if it has a cyclic group of automorphisms that includes an automorphism taking any vertex to any other vertex. In this paper we prove that infinite circulant graphs and their complements are hyperbolic. Furthermore, we obtain several sharp inequalities for the hyperbolicity constant of a large class of infinite circulant graphs and the precise value of the hyperbolicity constant of many circulant graphs. Besides, we give sharp bounds for the hyperbolicity constant of the complement of every infinite circulant graph.

Replacing the lower curvature bound in Toponogov's comparison theorem by a weaker hypothesis

Toponogov's triangle comparison theorem and its generalizations are important tools for studying the topology of Riemannian manifolds. In these theorems, one assumes that the curvature of a given manifold is bounded from below by the curvature of a model surface. The models are either of constant curvature, or, in the generalizations, rotationally symmetric about some point. One concludes that geodesic triangles in the manifold correspond to geodesic triangles in the model surface which have the same corresponding side lengths, but smaller corresponding angles. In addition, a certain rigidity holds: Whenever there is equality in one of the corresponding angles, the geodesic triangle in the surface embeds totally geodesically and isometrically in the manifold. In this paper, we discuss a condition relating the geometry of a Riemannian manifold to that of a model surface which is weaker than the usual curvature hypothesis in the generalized Toponogov theorems, but yet is strong enough to ensure that a geodesic triangle in the manifold has a corresponding triangle in the model with the same corresponding side lengths, but smaller corresponding angles. In contrast, it is interesting that rigidity fails in this setting.

Convexity in Tree Spaces

We study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all \ ultrametrics. The ${\rm CAT}(0)$-metric of Billera-Holmes-Vogtman arises from the theory of orthant spaces. While its geodesics can be computed by the Owen-Provan algorithm, geodesic triangles are complicated. We show that the dimension of such a triangle can be arbitrarily high. Tropical convexity and the tropical metric behave better. They exhibit properties desirable for geometric statistics, such as geodesics of small depth.

Angle geometry in asymptotic Teichmüller spaces

The angular geometry of asymptotic Teichmüller spaces is studied. Although it has been proved that the angles between any two geodesic rays from the base point of AT(X) always exist, it is shown in this paper that there are infinitely many pairs of intersecting geodesic segments such that the angles between them do not exist. The sums of inner angles of geodesic triangles are also studied. It is proved that for any number from 0 to 3π there exists a geodesic triangle in AT(X) such that the sum of its inner angles is equal to such a number.

Chordality Properties and Hyperbolicity on Graphs

Let $G$ be a graph with the usual shortest-path metric. A graph is $\delta$-hyperbolic if for every geodesic triangle $T$, any side of $T$ is contained in a $\delta$-neighborhood of the union of the other two sides. A graph is chordal if every induced cycle has at most three edges. In this paper we study the relation between the hyperbolicity of the graph and some chordality properties which are natural generalizations of being chordal. We find chordality properties that are weaker and stronger than being $\delta$-hyperbolic. Moreover, we obtain a characterization of being hyperbolic on terms of a chordality property on the triangles.

Small values of the hyperbolicity constant in graphs

If X is a geodesic metric space and x1,x2,x3∈X, a geodesic triangleT={x1,x2,x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharpest hyperbolicity constant of X, i.e., δ(X):=inf{δ≥0:X is δ-hyperbolic}. In the study of any parameter on graphs it is natural to study the graphs for which this parameter has small values. In this paper we study the graphs with small hyperbolicity constant, i.e., the graphs which are like trees (in the Gromov sense). We obtain simple characterizations of the graphs G with δ(G)=1 and δ(G)=54 (the case δ(G)<1 is known). Also, we give a necessary condition in order to have δ(G)=32 (we know that δ(G) is a multiple of 14). Although it is not possible to obtain bounds for the diameter of graphs with small hyperbolicity constant, we obtain such bounds for the effective diameter if δ(G)<32. This is the best possible result, since we prove that it is not possible to obtain similar bounds if δ(G)≥32.

On the hyperbolicity of edge-chordal and path-chordal graphs

If X is a geodesic metric space and x1, x2, x3 ( X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is ?-hyperbolic (in the Gromov sense) if any side of T is contained in a ?-neighborhood of the union of the other two sides, for every geodesic triangle T in X. An important problem in the study of hyperbolic graphs is to relate the hyperbolicity with some classical properties in graph theory. In this paper we find a very close connection between hyperbolicity and chordality: we extend the classical definition of chordality in two ways, edge-chordality and path-chordality, in order to relate this propertywith Gromov hyperbolicity. In fact, we prove that every edge-chordal graph is hyperbolic and that every hyperbolic graph is path-chordal. Furthermore, we prove that every path-chordal cubic graph with small path-chordality constant is hyperbolic.

Path Planning and Replanning for Mobile Robot Navigation on 3D Terrain: An Approach Based on Geodesic

In this paper, mobile robot navigation on a 3D terrain with a single obstacle is addressed. The terrain is modelled as a smooth, complete manifold with well-defined tangent planes and the hazardous region is modelled as an enclosing circle with a hazard grade tuned radius representing the obstacle projected onto the terrain to allow efficient path-obstacle intersection checking. To resolve the intersections along the initial geodesic, by resorting to the geodesic ideas from differential geometry on surfaces and manifolds, we present a geodesic-based planning and replanning algorithm as a new method for obstacle avoidance on a 3D terrain without using boundary following on the obstacle surface. The replanning algorithm generates two new paths, each a composition of two geodesics, connected via critical points whose locations are found to be heavily relying on the exploration of the terrain via directional scanning on the tangent plane at the first intersection point of the initial geodesic with the circle. An advantage of this geodesic path replanning procedure is that traversability of terrain on which the detour path traverses could be explored based on the local Gauss-Bonnet Theorem of the geodesic triangle at the planning stage. A simulation demonstrates the practicality of the analytical geodesic replanning procedure for navigating a constant speed point robot on a 3D hill-like terrain.

Bounds on Gromov hyperbolicity constant

If X is a geodesic metric space and $$x_{1},x_{2},x_{3} \in X$$ , a geodesic triangle $$T=\{x_{1},x_{2},x_{3}\}$$ is the union of the three geodesics $$[x_{1}x_{2}]$$ , $$[x_{2}x_{3}]$$ and $$[x_{3}x_{1}]$$ in X. The space X is $$\delta $$ -hyperbolic in the Gromov sense if any side of T is contained in a $$\delta $$ -neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by $$\delta (X)$$ the sharp hyperbolicity constant of X, i.e., $$\delta (X) =\inf \{ \delta \ge 0: X ~\text {is}~ \delta \text {-hyperbolic} \}.$$ To compute the hyperbolicity constant is a very hard problem. Then it is natural to try to bound the hyperbolicity constant in terms of some parameters of the graph. Denote by $$\mathcal {G}(n,m)$$ the set of (simple) graphs G with n vertices and m edges, and such that every edge has length 1. In this work we estimate $$A(n,m):=\min \{\delta (G)\mid G \in \mathcal {G}(n,m) \}$$ and $$B(n,m):=\max \{\delta (G)\mid G \in \mathcal {G}(n,m) \}$$ . In particular, we obtain good bounds for B(n, m), and we compute the precise value of A(n, m) for all values of n and m. We also study this problem for non-simple and weighted graphs.

An analytical solution of the weighted Fermat–Torricelli problem on a unit sphere

We obtain an analytical solution of the weighted Fermat–Torricelli (wFT) problem for a specific equilateral geodesic triangle. This approach is a generalization of Cockayne’s solution given in Cockayne (Math Mag 45:216–219, 1972) for three equal weights. Furthermore, we derive a necessary condition for the wFT point in the form of three transcendental equations with respect to some specific length of geodesic arcs. This condition locates the wFT point at the interior of a geodesic triangle with side lengths less than $$\frac{\pi }{2}$$ on a unit sphere.

Hyperbolicity in the corona and join of graphs

If X is a geodesic metric space and $${x_1, x_2, x_3 \in X}$$ , a geodesic triangle T = {x 1, x 2, x 3} is the union of the three geodesics [x 1 x 2], [x 2 x 3] and [x 3 x 1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e. δ(X) = inf{δ ≥ 0: X is δ-hyperbolic}. In this paper we characterize the hyperbolic product graphs for graph join G + H and the corona $${G\odot\mathcal H: G + H}$$ is always hyperbolic, and $${G\odot\mathcal H}$$ is hyperbolic if and only if G is hyperbolic. Furthermore, we obtain simple formulae for the hyperbolicity constant of the graph join G + H and the corona $${G \odot \mathcal H}$$ .