Hyperbolic Trigonometry and its Application in the Poincaré Ball Model of Hyperbolic Geometry
نویسندگان
چکیده
Hyperbolic trigonometry is developed and illustrated in this article along lines parallel to Euclidean trigonometry by exposing the hyperbolic trigonometric law of cosines and of sines in the Poincaré ball model of n-dimensional hyperbolic geometry, as well as their application. The Poincaré ball model of 3-dimensional hyperbolic geometry is becoming increasingly important in the construction of hyperbolic browsers in computer graphics. These allow in computer graphics the exploitation of hyperbolic geometry in the development of visualization techniques. It is therefore clear that hyperbolic trigonometry in the Poincaré ball model of hyperbolic geometry, as presented here, will prove useful in the development of efficient hyperbolic browsers in computer graphics. Hyperbolic trigonometry is governed by gyrovector spaces in the same way that Euclidean trigonometry is governed by vector spaces. The capability of gyrovector space theory to capture analogies and its powerful elegance is thus demonstrated once more. —
منابع مشابه
Metric and periodic lines in the Poincare ball model of hyperbolic geometry
In this paper, we prove that every metric line in the Poincare ball model of hyperbolic geometry is exactly a classical line of itself. We also proved nonexistence of periodic lines in the Poincare ball model of hyperbolic geometry.
متن کاملThe theorems of Stewart and Steiner in the Poincaré disc model of hyperbolic geometry
In [Comput. Math. Appl. 41 (2001), 135–147], A.A. Ungar employs the Möbius gyrovector spaces for the introduction of the hyperbolic trigonometry. This Ungar’s work plays a major role in translating some theorems from Euclidean geometry to corresponding theorems in hyperbolic geometry. In this paper we explore the theorems of Stewart and Steiner in the Poincaré disc model of hyperbolic geometry.
متن کاملAn Extension of Poincare Model of Hyperbolic Geometry with Gyrovector Space Approach
The aim of this paper is to show the importance of analytic hyperbolic geometry introduced in [9]. In [1], Ungar and Chen showed that the algebra of the group $SL(2,mathbb C)$ naturally leads to the notion of gyrogroups and gyrovector spaces for dealing with the Lorentz group and its underlying hyperbolic geometry. They defined the Chen addition and then Chen model of hyperbolic geomet...
متن کاملTrigonometry on the Unit Ball of a Complex Hilbert Space
1. Introduction. The unit disc furnished with the Poincaré metric provides a model (the Poincaré model) for the hyperbolic geometry. The linear fractional transformations of the unit disc onto itself constitute the group of motions. An analogous phenomenon arises in the unit ball of any complex Hubert space when it is furnished with the Carathéodory-Reiffen metric. The purpose of this note is t...
متن کاملA New Proof of Menelaus’s Theorem of Hyperbolic Quadrilaterals in the Poincaré Model of Hyperbolic Geometry
In this study, we present a proof of the Menelaus theorem for quadrilaterals in hyperbolic geometry, and a proof for the transversal theorem for triangles.
متن کامل