Decomposition of closed orientable geometric surfaces into acute geodesic triangles

2017 ◽  
Vol 110 (1) ◽  
pp. 81-89
Author(s):  
Sheng Bau ◽  
Stephen M. Gagola
Keyword(s):  
Geodesic Triangles

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

2021 ◽  
Author(s):  
Jenő Szirmai
Keyword(s):  
Geodesic Sphere ◽  
Geodesic Triangle ◽  
Projective Model ◽  
Geodesic Triangles

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].

scholarly journals COMPUTATION IN WORD-HYPERBOLIC GROUPS

2001 ◽  
Vol 11 (04) ◽  
pp. 467-487 ◽  
Author(s):  
DAVID B. A. EPSTEIN ◽  
DEREK F. HOLT
Keyword(s):  
Cayley Graph ◽  
Difficult Problem ◽  
Hyperbolic Groups ◽  
Maximum Width ◽  
Finitely Presented Group ◽  
Practical Algorithms ◽  
Finite State ◽  
Geodesic Triangles ◽  
Word Hyperbolic Groups ◽  
Finitely Presented

We describe two practical algorithms for computing with word-hyperbolic groups, both of which we have implemented. The first is a method for estimating the maximum width, if it exists, of geodesic bigons in the Cayley graph of a finitely presented group G. Our procedure will terminate if and only this maximum width exists, and it has been proved by Papasoglu that this is the case if and only if G is word-hyperbolic. So the algorithm amounts to a method of verifying the property of word-hyperbolicity of G. The aim of the second algorithm is to compute the thinness constant for geodesic triangles in the Cayley graph of G. This seems to be a much more difficult problem, but our implementation does succeed with straightforward examples. Both algorithms involve substantial computations with finite state automata.

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

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 5023-5036
Author(s):  
Géza Csima ◽  
Jenő Szirmai
Keyword(s):  
Universal Cover ◽  
Interior Angle ◽  
Geodesic Triangles

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 ?.

scholarly journals A criterion for hyperbolicity

1999 ◽  
Vol 42 (3) ◽  
pp. 445-454
Author(s):  
Michael Batty
Keyword(s):  
Cayley Graph ◽  
Isoperimetric Inequality ◽  
Usual Definition ◽  
Finitely Presented Groups ◽  
Geodesic Triangles ◽  
Definition Of ◽  
New Criterion ◽  
Finitely Presented

The usual definition of hyperbolicity of a group G demands that all geodesic triangles in the Cayley graph of G should be thin. Using the theorem that a susbquadratic isoperimetric inequality implies a linear one, we show that it is in fact only necessary for all triangles from a given combing to be thin, thus giving a new criterion for hyperbolicity of finitely presented groups.

scholarly journals Fundamental Theorems for the Hyperbolic Geodesic Triangles

2005 ◽  
Vol 10 (2) ◽  
pp. 231-238
Author(s):  
H. Uğurlu ◽  
M. Kazaz ◽  
A. Özdemir
Keyword(s):  
Geodesic Triangles ◽  
Fundamental Theorems
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