Universal Hyperbolic Geometry IV: Sydpoints and Twin Circumcircles
نویسندگان
چکیده
We introduce the new notion of sydpoints into projective triangle geometry with respect to a general bilinear form. These are analogs of midpoints, and allow us to extend hyperbolic triangle geometry to non-classical triangles with points inside and outside of the null conic. Surprising analogs of circumcircles may be defined, involving the appearance of pairs of twin circles, yielding in general eight circles with interesting intersection properties.
منابع مشابه
The Parabola in Universal Hyperbolic Geometry I
We introduce a novel definition of a parabola into the framework of universal hyperbolic geometry, show many analogs with the Euclidean theory, and also some remarkable new features. The main technique is to establish parabolic standard coordinates in which the parabola has the form xz = y2. Highlights include the discovery of the twin parabola and the connection with sydpoints, many unexpected...
متن کاملMetric 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 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 کاملUniversal Approximator Property of the Space of Hyperbolic Tangent Functions
In this paper, first the space of hyperbolic tangent functions is introduced and then the universal approximator property of this space is proved. In fact, by using this space, any nonlinear continuous function can be uniformly approximated with any degree of accuracy. Also, as an application, this space of functions is utilized to design feedback control for a nonlinear dynamical system.
متن کاملUniversal Cellular Automata in the Hyperbolic Spaces
This paper introduces several cellular automata in hyperbolic spaces which are able to simulate any computational device, as they are universal. We sketchily present five universal cellular automata, four of them on two grids of the hyperbolic plane and one in a grid of the hyperbolic 3D space. Key–Words: Theory of Computation, Cellular Automata, universality, hyperbolic geometry.
متن کامل