Using the Intermediate Value Theorem to Circumscribe Hyperbolic Triangles
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Abstract:
SummaryWe prove that every triangle in the hyperbolic plane can be circumscribed by an ellipse. The Intermediate Value Theorem is a key ingredient in the proof.Keywords:
Ellipse
Value (mathematics)
The polar decomposition of Möbius transformation of the complex open unit disc gives rise to the Möbius addition in the disc and, more generally, in the ball. Möbius addition and Einstein addition in the ball of a real inner product space are isomorphic gyrogroup operations that play in the hyperbolic geometry of the ball a role analogous to the role that ordinary vector addition plays in the Euclidean geometry of . Möbius (Einstein) addition governs the Poincaré (Beltrami) ball model of hyperbolic geometry just as vector addition governs the standard model of Euclidean geometry. Accordingly, we show in this article that resulting analogies enable Euclidean and hyperbolic geometry to be unified
Hyperbolic space
Non-Euclidean geometry
Absolute geometry
Ball (mathematics)
Ordered geometry
Ultraparallel theorem
Foundations of geometry
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In this study, we give a hyperbolic version of the Smarandache's theorem in the Poincar� upper half-plane model.
Ultraparallel theorem
Poincaré conjecture
Hyperbolic angle
Absolute geometry
Upper half-plane
Foundations of geometry
Hyperbolic space
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Foundations of geometry
Absolute geometry
Non-Euclidean geometry
Ordered geometry
Ultraparallel theorem
Hyperbolic angle
Synthetic geometry
Basis (linear algebra)
Hyperbolic coordinates
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In Euclidean geometry we have the following: Given a line L and a point P not on it, there is exactly one line through P that is parallel to L. It was discovered that assuming this is false produces the equally valid Hyperbolic Geometry, where there are in fact infinitely many lines through P that are parallel to L. This presentation is an introduction to Hyperbolic Geometry, and on modelling it in the Euclidean plane.
Non-Euclidean geometry
Absolute geometry
Foundations of geometry
Ordered geometry
Ultraparallel theorem
Line (geometry)
Hyperbolic angle
Hyperbolic space
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In this paper we study the isoptic curves on the hyperbolic plane. This topic is widely investigated in the Euclidean geometry, but in the hyperbolic geometry there are only a few result. In [13] we have developed a method to investigate the isoptic curves in the hyperbolic geometry and we have applied it to line segments and ellipses.Our goal in this work is to determine the isoptic curves of parabolas in the hyperbolic plane by the above procedure. We use for the computations the classical Beltrami-Cayley-Klein model which is based on the projective interpretation of the hyperbolic geometry and in this manner the isoptic curves can be visualized on the Euclidean screen of computer.
Ellipse
Hyperbolic angle
Ultraparallel theorem
Non-Euclidean geometry
Hyperbolic coordinates
Hyperbolic space
Line (geometry)
Hyperbolic tree
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The purpose the present paper is to construct the hyperbolic trigonometry on Euclidean plane without refereing to hyperbolic plane. In this paper we show that the concept of hyperbolic angle and its functions forming the hyperbolic trigonometry give arise on Euclidean plane in a natural way. The method is based on a key- formula establishing a relationship between exponential function and the ratio of two segments. This formula opens a straightforward pathway to hyperbolic trigonometry on the Euclidean plane. The hyperbolic law of cosines I and II and the hyperbolic law of sines are derived by using of the key-formula and the methods of Euclidean Geometry, only. It is shown that these laws are consequences of the interrelations between distances and radii of the intersecting semi-circles.
Non-Euclidean geometry
Ultraparallel theorem
Spherical trigonometry
Hyperbolic angle
Hyperbolic space
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Hyperbolic space
Hyperboloid
Ultraparallel theorem
Polygon (computer graphics)
Hyperbolic coordinates
Foundations of geometry
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The purpose the present paper is to construct the hyperbolic trigonometry on Euclidean plane without refereing to hyperbolic plane. In this paper we show that the concept of hyperbolic angle and its functions forming the hyperbolic trigonometry give arise on Euclidean plane in a natural way. The method is based on a key- formula establishing a relationship between exponential function and the ratio of two segments. This formula opens a straightforward pathway to hyperbolic trigonometry on the Euclidean plane. The hyperbolic law of cosines I and II and the hyperbolic law of sines are derived by using of the key-formula and the methods of Euclidean Geometry, only. It is shown that these laws are consequences of the interrelations between distances and radii of the intersecting semi-circles.
Non-Euclidean geometry
Spherical trigonometry
Ultraparallel theorem
Hyperbolic angle
Hyperbolic space
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In this note, we present a proof of Smarandaches cevian triangle hyperbolic theorem in the Einstein relativistic velocity model of hyperbolic geometry. 2000 Mathematical Subject Classi
cation: 51K05, 51M10, 30F45, 20N99, 51B10 Keywords and phrases: hyperbolic geometry, hyperbolic triangle, Smarandaches cevian triangle, gyrovector, Einstein relativistic velocity model 1. Introduction Hyperbolic geometry appeared in the
rst half of the 19 century as an attempt to understand Euclids axiomatic basis for geometry. It is also known as a type of nonEuclidean geometry, being in many respects similar to Euclidean geometry. Hyperbolic geometry includes such concepts as: distance, angle and both of them have many theorems in common.There are known many main models for hyperbolic geometry, such as: Poincare disc model, Poincare half-plane, Klein model, Einstein relativistic velocity model, etc. The hyperbolic geometry is a non-Euclidian geometry. Here, in this study, we present a proof of Smarandaches cevian triangle hyperbolic theorem in the Einstein relativistic velocity model of hyperbolic geometry. Smarandaches cevian triangle theorem states that if A1B1C1 is the cevian triangle of point P with respect to the triangle ABC; then PA PA1 PB PB1 PC PC1 = AB BC CA A1B B1C C1A [1]. Let D denote the complex unit disc in complex z plane, i.e. D = fz 2 C : jzj < 1g: The most general Mobius transformation of D is z ! e z0 + z 1 + z0z = e (z0 z); which induces the Mobius addition in D, allowing the Mobius transformation of the disc to be viewed as a Mobius left gyrotranslation z ! z0 z = z0 + z 1 + z0z followed by a rotation. Here 2 R is a real number, z; z0 2 D; and z0 is the complex conjugate of z0: Let Aut(D; ) be the automorphism group of the grupoid (D; ). If we de
ne gyr : D D ! Aut(D; ); gyr[a; b] = a b b a = 1 + ab 1 + ab ; then is true gyrocommutative law a b = gyr[a; b](b a):
Absolute geometry
Foundations of geometry
Ordered geometry
Non-Euclidean geometry
Ultraparallel theorem
Hyperbolic space
Synthetic geometry
Hyperbolic angle
Jordan curve theorem
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In this work it is made an study of two dimensional Non-Euclidean Geometry,
in particular of the Hyperbolic Plane H2 and its representations. First, it is de -
ned H2 through the model of Poincare's Plane (also called Superior Semi-plane)...
Poincaré conjecture
Non-Euclidean geometry
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