Geodesic Triangle Research Articles

On Geodesic Triangles in Non-Euclidean Geometry

On Geodesic Triangles in Non-Euclidean Geometry

A Steward inequality in the Hyperbolic Geometry

The main aim of this study is to show that the steward inequality holds for any hyperbolic domain and distinct points , where denote the hyperbolic metric in and be a hyperbolic geodesic triangle with vertices and sides and be a point on .

On Menelaus’ and Ceva’s theorems in Nil geometry

Abstract In this paper we deal with Nil geometry, which is one of the homogeneous Thurston 3-geometries. We define the “surface of a geodesic triangle” using generalized Apollonius surfaces. Moreover, we show that the “lines” on the surface of a geodesic triangle can be defined by the famous Menelaus’ condition and prove that Ceva’s theorem for geodesic trianglesistruein Nil space. In our work we will use the projective model of Nil geometry described by E. Molnár in [6].

Classical Notions and Problems in Thurston Geometries

Of the Thurston geometries, those with constant curvature geometries (Euclidean $ E^3$, hyperbolic $ H^3$, spherical $ S^3$) have been extensively studied, but the other five geometries, $ H^2\times R$, $ S^2\times R$, $Nil$, $\widetilde{SL_2 R}$, $Sol$ have been thoroughly studied only from a differential geometry and topological point of view. However, classical concepts highlighting the beauty and underlying structure of these geometries -- such as geodesic curves and spheres, the lattices, the geodesic triangles and their surfaces, their interior sum of angles and similar statements to those known in constant curvature geometries -- can be formulated. These have not been the focus of attention. In this survey, we summarize our results on this topic and pose additional open questions.

Cat(0) polygonal complexes are 2-median

Median spaces are spaces in which for every three points the three intervals between them intersect at a single point. It is well known that rank-1 affine buildings are median spaces, but by a result of Haettel, higher rank buildings are not even coarse median. We define the notion of “2-median space”, which roughly says that for every four points the minimal discs filling the four geodesic triangles they span intersect in a point or a geodesic segment. We show that CAT(0) Euclidean polygonal complexes, and in particular rank-2 affine buildings, are 2-median. In the appendix, we recover a special case of a result of Stadler of a Fary–Milnor type theorem and show in elementary tools that a minimal disc filling a geodesic triangle is injective.

Translation-Like Isoptic Surfaces and Angle Sums of Translation Triangles in textbf{Nil} Geometry

After having investigated the geodesic and translation triangles and their angle sums in textbf{Sol} and widetilde{{textbf{S}}{textbf{L}}_2{textbf{R}}} geometries we consider the analogous problem in textbf{Nil} space that is one of the eight 3-dimensional Thurston geometries. We analyze the interior angle sums of translation triangles in textbf{Nil} geometry and we provide a new approach to prove that it can be larger than or equal to pi . Moreover, for the first time in non-constant curvature Thurston geometries we have developed a procedure for determining the equations of textbf{Nil} isoptic surfaces of translation-like segments and as a special case of this we examine the textbf{Nil} translation-like Thales sphere, which we call Thaloid. In our work we will use the projective model of textbf{Nil} described by Molnár (Beitr Algebra Geom 38(2):261–288, 1997).

Iterated Medial Triangle Subdivision in Surfaces of Constant Curvature

Consider a geodesic triangle on a surface of constant curvature and subdivide it recursively into four triangles by joining the midpoints of its edges. We show the existence of a uniform delta >0 such that, at any step of the subdivision, all the triangle angles lie in the interval (delta ,pi -delta ). Additionally, we exhibit stabilising behaviours for both angles and lengths as this subdivision progresses.

On groups presented by inverse-closed finite confluent length-reducing rewriting systems

We show that groups presented by inverse-closed finite confluent length-reducing rewriting systems are characterised by a striking geometric property: their Cayley graphs are geodetic and side-lengths of non-degenerate geodesic triangles are uniformly bounded. This leads to a new algebraic result: the group is plain (isomorphic to the free product of finitely many finite groups and copies of Z) if and only if a certain relation on the set of non-trivial finite-order elements of the group is transitive on a bounded set. We use this to prove that deciding if a group presented by an inverse-closed finite confluent length-reducing rewriting system is not plain is in NP. A ‘yes’ answer to an instance of this problem would disprove a longstanding conjecture of Madlener and Otto from 1987. We also prove that the isomorphism problem for plain groups presented by inverse-closed finite confluent length-reducing rewriting systems is in PSPACE.

Designing a Geodesic Faceted Acoustical Volumetric Array Using a Novel Analytical Method

We present a novel analytical method as an efficient approach to design a geodesic-faceted array (GFA) for achieving a beam performance equivalent to that of a typical spherical array (SA). GFA is a triangle-based quasi-spherical configuration, which is conventionally created using the icosahedron method imitated from the geodesic dome roof construction process. In this conventional approach, the geodesic triangles have nonuniform geometries due to some distortions that occur during the random icosahedron division process. In this study, we took a paradigm shift from this approach and adopt a new technique to design a GFA that is based on uniform triangles. The characteristic equations that relate the geodesic triangle with a spherical platform were first developed as functions of the operating frequency and geometric parameters of the array. Then, the directional factor was derived to calculate the beam pattern associated with the array. A sample design of GFA for a given underwater sonar imaging system was synthesized through an optimization process. The GFA design was compared with that of a typical SA, and a reduction of 16.5% in the number of array elements was recorded in the GFA at a nearly equivalent performance. Both arrays were modeled, simulated, and analyzed using the finite element method (FEM) to validate the theoretical designs. Comparison of the results showed a high degree of compliance between the FEM and the theoretical method for both arrays. The proposed novel approach is faster and requires fewer computer resources than the FEM. Moreover, this approach is more flexible than the traditional icosahedron method in adjusting geometrical parameters in response to desired performance outputs.

On the hyperbolicity of Delaunay triangulations

<abstract><p>If $ X $ is a geodesic metric space and $ x_1, x_2, x_3\in X $, a <italic>geodesic triangle</italic> $ 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 <italic>hyperbolic</italic> if there exists a constant $ \delta \ge 0 $ such that any side of any geodesic triangle in $ X $ is contained in the $ \delta $-neighborhood of the union of the two other sides. In this paper, we study the hyperbolicity of an important kind of Euclidean graphs called Delaunay triangulations. Furthermore, we characterize the Delaunay triangulations contained in the Euclidean plane that are hyperbolic.</p></abstract>

Acylindrical hyperbolicity of cubical small cancellation groups

We provide an analogue of Strebel's classification of geodesic triangles in classical $C'(\frac16)$ groups for groups given by Wise's cubical presentations satisfying sufficiently strong metric cubical small cancellation conditions. Using our classification, we prove that, except in specific degenerate cases, such groups are acylindrically hyperbolic.

Geodesic Triangles in H² with Short Sides

We prove that if M is an orientable hyperbolic surface without boundary (possibly compact, possibly with infinitely generated fundamental group) and γ is a closed geodesic in M then any side of any triangle formed by distinct lifts of γ in H2 is shorter than γ.

Discontinuous Normals in Non-Euclidean Geometries and Two-Dimensional Gravity

This paper builds two detailed examples of generalized normal in non-Euclidean spaces, i.e., the hyperbolic and elliptic geometries. In the hyperbolic plane we define a n-sided hyperbolic polygon P, which is the Euclidean closure of the hyperbolic plane H, bounded by n hyperbolic geodesic segments. The polygon P is built by considering the unique geodesic that connects the n+2 vertices z˜,z0,z1,…,zn−1,zn. The geodesics that link the vertices are Euclidean semicircles centred on the real axis. The vector normal to the geodesic linking two consecutive vertices is evaluated and turns out to be discontinuous. Within the framework of elliptic geometry, we solve the geodesic equation and construct a geodesic triangle. Additionally in this case, we obtain a discontinuous normal vector field. Last, the possible application to two-dimensional Euclidean quantum gravity is outlined.

Geometrical interpretation of the argument of weak values of general observables in N-level quantum systems

Observations in quantum weak measurements are determined by complex numbers called weak values. We present a geometrical interpretation of the argument of weak values of general Hermitian observables in N-dimensional quantum systems in terms of geometric phases. We formulate an arbitrary weak value in terms of three real vectors on the unit sphere in N 2 − 1 dimensions, . These vectors are linked to the initial and final states, and to the weakly measured observable, respectively. We express pure states in the complex projective space of N − 1 dimensions, , which has a non-trivial representation as a 2N − 2 dimensional submanifold of (a generalization of the Bloch sphere for qudits). The argument of the weak value of a projector on a pure state of an N-level quantum system describes a geometric phase associated to the symplectic area of the geodesic triangle spanned by the vectors representing the pre-selected state, the projector and the post-selected state in . We then proceed to show that the argument of the weak value of a general observable is equivalent to the argument of an effective Bargmann invariant. Hence, we extend the geometrical interpretation of projector weak values to weak values of general observables. In particular, we consider the generators of SU(N) given by the generalized Gell–Mann matrices. Finally, we study in detail the case of the argument of weak values of general observables in two-level systems and we illustrate weak measurements in larger dimensional systems by considering projectors on degenerate subspaces, as well as Hermitian quantum gates. To conclude, we discuss the interpretation and usefulness of geometric phases in weak values in connection to weak measurements.

Necessary and sufficient conditions for a triangle comparison theorem

We prove a version of Topogonov's triangle comparison theorem with surfaces of revolution as model spaces. Given a model surface and a Riemannian manifold with a fixed base point, we give necessary and sufficient conditions under which every geodesic triangle in the manifold with a vertex at the base point has a corresponding Alexandrov triangle in the model. Under these conditions we also prove a version of the Maximal Radius Theorem and a Grove--Shiohama type Sphere Theorem.

Geometric definition of emission coordinates

We investigate a relativistic positioning system where the coordinates of the users are determined by the proper times broadcasted by clocks in motion in spacetime: these are the so-called emission coordinates. In particular, we focus on emission coordinates in flat spacetime: in this case, we show that these coordinates can be defined using an approach based on simple geometrical properties of geodesic triangles. We analyze the 2-dimensional case and then we show how the whole procedure can be applied to 4 dimensions, also in terms of ordinary three-dimensional spheres. An analytic solution for the coordinates of the receiver is obtained. The solution remains valid at almost any place, in particular when redundancy in the number of emitters can be exploited.

Theory, strict formula derivation and algorithm development for the computation of a geodesic polygon area

The paper presents a coherent, original theory and an effective algorithm to calculate the surface area of a geodesic polygon, i.e. the area on an ellipsoid of revolution limited by geodesics, connecting pairs of subsequent vertices of known geographical coordinates. Using the cylindrical coordinate system and introducing an equatorial geodesic triangle, the novel strict recursive formulas were derived for the longitude on the ellipsoid and the distance along the geodesic and the area of a geodesic polygon. The area is determined by constant coefficients, not the values of finite series. The orthodromic properties of geodesics, which led to the detection of bifurcation lines composed of the double solutions of the inverse geodetic problem were analysed. The novel algorithm presented, integrating the problem with geodesic polygon area calculation, is based on the newly proposed parameterisation with two alternative working unknowns and a new formula to obtain the initial value of the geodesic C parameter. It allows solutions for all cases, including near-antipodal or near-vertices, and offers two solutions if bifurcation occurs. The algorithm presented was tested and verified using numerical examples from known publications. The numerical tests proved the accuracy of 1m2 of the computed geodesic area using standard IEEE 754 Double Floating Point PC arithmetic, for geodesic lengths up to ten thousand km.

Triangle angle sums related to translation curves in Sol geometry

"After having investigated the geodesic and translation triangles and their angle sums in Nil and SL_2R geometries we consider the analogous problem in Sol space that is one of the eight 3-dimensional Thurston geometries. We analyse the interior angle sums of translation triangles in Sol geometry and prove that it can be larger or equal than pi . In our work we will use the projective model of Sol described by E. Molnar in [9]."

Apollonius Surfaces, Circumscribed Spheres of Tetrahedra, Menelaus’s and Ceva’s Theorems in S2 × R and H2 × R Geometries

Abstract In the present paper we study $\mathbf{S}^2\!\times\!\mathbf{R}$ and $\mathbf{H}^2\!\times\!\mathbf{R}$ geometries, which are homogeneous Thurston 3-geometries. We define and determine the generalized Apollonius surfaces and with them define the ‘surface of a geodesic triangle’. Using the above Apollonius surfaces we develop a procedure to determine the centre and the radius of the circumscribed geodesic sphere of an arbitrary $\mathbf{S}^2\!\times\!\mathbf{R}$ and $\mathbf{H}^2\!\times\!\mathbf{R}$ tetrahedron. Moreover, we generalize the famous Menelaus’s and Ceva’s theorems for geodesic triangles in both spaces. In our work we will use the projective model of $\mathbf{S}^2\!\times\!\mathbf{R}$ and $\mathbf{H}^2\!\times\!\mathbf{R}$ geometries described by E. Molnár in [6].

Sharp Reverse Isoperimetric Inequalities in Nonpositively Curved Cones

We prove a pair of sharp reverse isoperimetric inequalities for domains in nonpositively curved surfaces: (1) metric disks centered at the vertex of a Euclidean cone of angle at least \(2\pi \) have minimal area among all nonpositively curved disks of the same perimeter and the same total curvature; (2) geodesic triangles in a Euclidean (resp. hyperbolic) cone of angle at least \(2\pi \) have minimal area among all nonpositively curved geodesic triangles (resp. all geodesic triangles of curvature at most \(-1\)) with the same side lengths and angles.