The main objective is the derivation and transformation of each model as. You can read about this in thurstons notes which explains in detail the relations between various models of the hyperbolic plane, including a stepbystep way to get between any two of those models such as the upper half plane model and the beltramiklein disc model. Geometers sketchpad by steve szydlik december 21, 2001. This model has the advantage of simplicity, but the disadvantage that angles in the hyperbolic plane are distorted. His model, sometimes known as the beltrami klein model, is much simpler than beltramis original model and has come to be known the klein disk model. The beltramiklein model this model is often referred to as the klein model, because of the extensive work done by the german mathematician felix klein in geometry with this model. On algebraic structures related to beltramiklein model of.
In 1871 klein showed that the last model of beltrami could be constructed by projective geometry. Actually, the radius of the semicircle is irrelevant because scaling is a hyperbolic isometry. These are called conformal because their protractor. Beltrami proved that that hyperbolic geometry was just as consistent as euclidean geometry. Hyperbolic geometry project gutenberg selfpublishing. The goal of this project was to complete a package of tools for automating hyperbolic constructions in each of three models of the hyperbolic plane. The poincar e models of the hyperbolic plane george francisy draft of 8 april 2002 1 conformal models the beltrami klein model of the hyperbolic plane is not as useful for measuring angles as it is for measuring distances. Beltramiklein model and the poincar e circle model are examples of such6fi nite mod els, while the poincares upper half plane model, the weierstrass. Theory of fuchsian groupsmemoir on kleinian groupson the applications of noneuclidean geometry to the theory of quadratic forms. The geodesics are vertical lines and semicircles hitting the boundary at right angles.
In this section, we give an explicit listing of the hyperbolic construction tools that have been developed in geometers sketchpad for the three most common models of hyperbolic geometry. The models measure distance differently, butequal hyperbolic distances are represented byever smaller euclidean distances toward the bounding circle in both. Klein gave an interpretation of beltramis model in terms of projective geometry. Translators introduction theory of fuchsian groupsmemoir on kleinian groupson the applications of noneuclidean geometry to the theory of quadratic. The poincar e models of the hyperbolic plane geometry. Each of our spaces in question, the circle and the hyperbola, are subspaces of. Ellipsoids, complete integrability and hyperbolic geometry 187 nconfocal hypersurfaces, and all the re ected rays tr. As in the klein model above, completing the scripts for the ten standard hyperbolic constructions in the poincare halfplane necessitated the development of other construction tools in this model. Request pdf on algebraic structures related to beltramiklein model of hyperbolic geometry a new approach to the algebraic structures related to hyperbolic geometry comes from einsteins. The beltrami klein model has the advantage that lines in the model resemble euclidean lines. I used this book as my guide to write software creating art with hyperbolic.
Described by eugenio beltrami in 1868 and felix klein in 1871. The klein model or beltramiklein model 1 uses the interior of a unit circle for fully representing the hyperbolic plane. The beltramiklein model of hyperbolic geometry this model is usually called the klein model. Only the beginning and ending angles are important. In 1868 beltrami constructed a riemannian geometry for the poincar e halfplane and disk both named after herni poincar e 18541912 and another model. The interplay between hyperbolic symmetry and history. Once again, lengths are distorted, but angles are correct. Combining the ideas and methods of karzel and ungar give an elegant concept for defect and area in this model. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Isometries of the hyperbolic plane in the beltramiklein disk. Thurston, lecture notes a chapter on orbifolds, 1977.
Disk models of noneuclidean geometry beltrami and klein made a model of noneuclidean geometry in a disk, with chords being the lines. A recognition of beltrami comparable to that given the pioneering works of bolyai and lobachevsky seems long overduenot only because beltrami rescued hyperbolic geometry from oblivion by proving it to be logically consistent, but because he gave it a concrete meaning a model that made hyperbolic geometry part of ordinary mathematics. The klein disc model we start with a vector x 2bn r fjjxjj2 solution of the laplacian laplacebeltrami operator on a riemannian manifold of constant negative sectional curvature, namely the hyperboloid model of hyperbolic. In order to do that, some time is spent on neutral geometry as well as euclidean geometry. Geometry of hyperbolic space beltramiklein model conformal ball model the upperhalf space model discrete groups. Beltrami b1868 constructed a euclidean model of the hyperbolic plane, and using di. Apr 20, 2011 this video outlines the basic framework of universal hyperbolic geometry as the projective study of a circle, or later on the projective study of relativistic geometry.
In geometry, the beltramiklein model, also called the projective model, klein disk model, and the cayleyklein model, is a model of hyperbolic geometry in. Aug 12, 2011 combining the ideas and methods of karzel and ungar give an elegant concept for defect and area in this model. If your class didnt follow the book, its not the books fault. We shall recall and explain this idea of cayley and its developments by klein. A consistent model of this axiomatic system implies that the parallel pos. A fourth model of hyperbolic geometry is the klein model. The beltramiklein model, also known as the projective disk model, klein disk model and klein model, is named after eugenio beltrami and felix klein. The beltramiklein model of hyperbolic geometry hyperbolic points. The beltramiklein model k of the hyperbolic plane utilizes a particularly simple such convex set. In particular we give a rigorous and elementary proof for the defect formula stated ungar in comput math appl 53, 2007. The geodesic segments are called the sides of the polygon. Beltramiklein models a chord in the beltramiklein model represents the same hyperbolic line as the orthogonal circular arc with the same endpoints in the poincare circle. Complete integrability and hyperbolic geometry 189 hn, called the projective or beltramiklein model.
In addition three other models poincare halfplane, beltramikleinhilbert. On trigonometry in beltramiklein model of hyperbolic. Sources of hyperbolic geometry history of mathematics, v. For the two dimensions this model uses the interior of the unit circle for the complete hyperbolic plane, and the chords of this circle are the hyperbolic lines. The beltramiklein model of the hyperbolic plane george francisy draft of 19 february 2002 1 introduction in this chapter we introduce the beltramiklein model of the hyperbolic plane. Beltramis models of noneuclidean geometry dipartimento di. Hyperbolic geometry student theses faculty of science and. Defect and area in beltramiklein model of hyperbolic geometry.
The beltramiklein model, also known as the projective disk model, klain disk model and klein model, is named after eugenio beltrami and felix klein. The beltrami klein model of the hyperbolic plane george francisy draft of 19 february 2002 1 introduction in this chapter we introduce the beltrami klein model of the hyperbolic plane. More informed sources refer to the projective model as. As you have discovered, the answers to your questions are yes. Here we shall concentrate on discovering the features of hyperbolic geometry by working with one of its models.
Pdf beltramiklein model of hyperbolic geometry with. A triangle in hyperbolic geometry is a polygon with three sides, a quadrilateral is a polygon with four sides, and so on, as in euclidean geometry. At first i read this as supporting your claim that beltrami did use the tractricoid directly to prove that consistency. The main objective is the derivation and transformation of each model as well as their respective characteristics. The next model is the beltramiklein, or sometimes just called the klein model. The points in the klein model, called kpoints, are the epoints inside the lines in. Poincar e, as i said above, was the rst to use the other two models in order to understand phenomena apparently far from the noneuclidean topic. The beltramiklein model of the hyperbolic plane is not as useful for measuring angles as it is for measuring distances. Defect and area in beltramiklein model of hyperbolic. In the klein beltrami model shown in the figure, top left, the hyperbolic surface is mapped to the interior of a circle, with geodesics in the hyperbolic surface corresponding to chords in the circle. Hyperbolic geometry, in this new series, is made simpler, more logical, more general and. Beltrami proved the relative consistency of hyperbolic geometry in 1868 using differential geometry see the pseudosphere, chapter 10. At this point the hyperboloid model is introduced, related to the other models visited, and developed using some concepts from physics as aids.
We therefore consider the conformal models of henri poincar e. Read on trigonometry in beltramiklein model of hyperbolic geometry, results in mathematics on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. This is the main theme of kleins two papers on the socalled noneuclidean geometry, i and ii 32 34 of klein. Beltrami and the beltramicayleyklein model of the hyperbolic plane 23. In other words, beltrami proved that that hyperbolic geometry was just as consistent as euclidean geometry. A polygon in hyperbolic geometry is a sequence of points and geodesic segments joining those points. We are now ready for the proof that triangles are thin. A universal model for conformal geometries of euclidean. Beltramiklein model of hyperbolic geometry with applications to einsteins theory of special relativity. Points in this model are interpreted to be points interior to, and lines are interpreted to be. The klein model, also known as the projective disc model and beltramiklein model, uses the interior of a circle for the hyperbolic plane, and chords of the circle as lines.
This book presents, for the first time in english, the papers of beltrami, klein, and poincare that brought hyperbolic geometry into the mainstream of mathematics. Felix klein 1849 1925 presented beltramis disk model using projective geometry. This consists of a reinterpretation of the geometrical primitives, points, lines, angles etc. The next model is the beltrami klein, or sometimes just called the klein model. Its set of points consists of all the points of the open disk just as in the. Since it uses the whole infinite halfplane, it is not well suited for playing hyperrogue. The hyperbolic plane department of mathematical durham. But reading that chapter 10, im not so sure any more. Furthermore, we give a formulary for area of circle in the beltramiklein model of hyperbolic geometry.
That is, the euclidean of an angle within the model is not necessarily the angle measure in hyperbolic geometry. In geometry, the beltramiklein model, also called the projective model, klein disk model, and the cayley klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk or ndimensional unit ball and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere. Each of our spaces in question, the circle and the hyperbola, are subspaces of the plane. Isometries and distances in the hyperboloid model 80 11.
In geometry, the beltramiklein model, also called the projective model, klein disk model, and the cayleyklein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk or ndimensional unit ball and lines are represented by the chords, straight line segments with ideal endpoints on the. It sets out to provide recognition of beltrami comparable to that given the pioneering works of bolyai and labachevsky, not only because beltrami rescued hyperbolic geometry from oblivion by proving to be logically consistent, but. You can read about this in thurstons notes which explains in detail the relations between various models of the hyperbolic plane, including a stepbystep way to get between any two of those models such as the upper half plane model and the beltrami klein disc model. The two mathematicians were euginio beltrami and felix klein and together they developed the first complete model of hyperbolic geometry. The use of repeating patterns to teach hyperbolic geometry. Today we discuss the cayleyklein model of the hyperbolic geometry. Along with the klein model 19 and the poincare halfspace model 20, it was proposed by eugenio beltrami, who used these models to show that hyperbolic geometry was equiconsistent with. We develop a formulary for point re flections, addition, defect and area in the beltramiklein model of hyperbolic geometry with geometric approach of karzel cf. Klein made more explicit the connections between the model in the saggio and projective geometry, which beltrami had just mentioned in his article. Poincare discovered a model made from points in a disk and arcs of circles orthogonal to the boundary of the disk.
How did beltrami show the consistency of hyperbolic. It also covers other models including poincares disc model and the beltramiklein model. Kleins construction was motivated by an idea of cayley. We develop a formulary for point re flections, addition, defect and area in the beltrami klein model of hyperbolic geometry with geometric approach of karzel cf. Isometries of the hyperbolic plane in the beltramiklein. This paper concerns hyperbolic geometry as described by the poincare disk model. On trigonometry in beltramiklein model of hyperbolic geometry. Interactive visualization of models of hyperbolic geometry, alark. More informed sources refer to the projective model as the beltrami klein projective disc model. Poincare halfplane model is one of the basic conformal models that are taught in hyperbolic geometry courses.