∧ Basic Explorations 1. | Hyperbolic Kit for Poincaré Disk Model. − 2 d − 2 q = x An imaginary world Henri Poincar´e did not invent non-Euclidean geometry. arctanh x ′ ′ The endpoints of any arc can be specified by two angles around the disk and .   ) References The vertical bars indicate Euclidean length of the line segment connecting the points between them in the model (not along the circle arc), ln is the natural logarithm. ( The model has the conformal property that the angle between two intersecting curves in hyperbolic space is the same as the angle in the model. r | Opening the Geogebra Geometry in Windows 10 and crashing A "Custom Layer" option The Poincaré ball model is a model of n-dimensional hyperbolic geometry in which all points are embedded in an n-dimensional sphere (or in a circle in the 2D case which is called the Poincaré disk model). and the formulas become The point where it touches the boundary circle is not part of the horocycle. Henri Poincare (1854–1912) who first introduced it.´ 14.1 THE POINCARE DISK MODEL FOR HYPERBOLIC GEOMETRY´ A model for a geometry is an interpretation of the technical terms of the geometry (such as point, line, distance, angle measure, etc.) Knowledge-based programming for everyone. | https://mathworld.wolfram.com/PoincareHyperbolicDisk.html, Lines through Points in + 0 Encyclopædia Britannica, Inc. 26, No. r 95-104, 1999. x The Poincaré disk model is one way to represent hyperbolic geometry, and for most purposes it serves us very well. lies between the origin and point , The Poincaré Disk Model; Figures of Hyperbolic Geometry; Measurement in Hyperbolic Geometry; Area and Triangle Trigonometry; The Upper Half-Plane Model; 6 Elliptic Geometry. The Imaginary Made Real: The Images of Michael Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. x 2 + − = 1 , = 2 ⋅ , The illustration above shows a hyperbolic arctanh Poincar´e Disk model, and the Poincar´e Half-Plane model. y ( ( − | 2 The isometry group of the disk model is given by the special unitary group SU(1,1). > In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk, and the straight lines consist of all circular arcs contained within that disk that are orthogonal to the boundary of the disk, plus all diameters of the disk. y Coxeter around 1956 inspired Escher's interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. o p 1 New York: Springer-Verlag, pp. H. Poincaré (1882) proposed this model. ′ ( ln 2 Given two distinct points p and q inside the disk, the unique hyperbolic line connecting them intersects the boundary at two ideal points, a and b, label them so that the points are, in order, a, p, q, b and |aq| > |ap| and |pb| > |qb|. {\displaystyle \operatorname {arcosh} } Both the Poincaré disk model and the Beltrami–Klein model are models of the n-dimensional hyperbolic space in the n-dimensional unit ball in R n. If u is a vector of norm less than one representing a point of the Poincaré disk model, then the corresponding point of the Beltrami–Klein model is given by 1 {\displaystyle \ln \left({\frac {1+r}{1-r}}\cdot {\frac {1-r'}{1+r'}}\right)=2(\operatorname {arctanh} r-\operatorname {arctanh} r')} r   Antipodal Points; Elliptic Geometry; Measurement in Elliptic Geometry; Revisiting Euclid's Postulates; 7 Geometry on Surfaces. Hints help you try the next step on your own. Hyperbolic Paper Exploration 2. 2 Such a distance function is defined for any two vectors of norm less than one, and makes the set of such vectors into a metric space which is a model of hyperbolic space of constant curvature −1. A model of hyperbolic geometry represents the basic elements of that ge-ometry (points, lines) by Euclidean constructs. Trott.   ⁡ However, another model, called the upper half-plane model, makes some computations easier, including the calculation of the area of a triangle. + For Cartesian coordinates (t, xi) on the hyperboloid and (yi) on the plane, the conversion formulas are: Compare the formulas for stereographic projection between a sphere and a plane. 1 The Klein disk model (also known as the Beltrami–Klein model) and the Poincaré disk model are both models that project the whole hyperbolic plane in a disk. But the distance along such a geodesic should be measured in a specific way. 2 x | Remember that in the half-plane case, the lines were either Euclidean lines, perpendicular onto the real line, or half-circles, also perpendicular onto the real line. r The result is the corresponding point of the Poincaré disk model. − ), their hyperbolic distance is The angular measurements in the Poincaré model are the same as in hyperbolic geometry (cf. There is an isomorphism | Two arcs which do not meet correspond to parallel rays, arcs which meet orthogonally correspond to perpendicular lines, and arcs which meet on the boundary are a pair of limits rays. ) ⁡ 2 {\displaystyle |pq|} The two models are related through a projection on or from the hemisphere model. Cover of Math. A circle (the set of all points in a plane that are at a given distance from a given point, its center) is a circle completely inside the disk not touching or intersecting its boundary. x ) (the ideal points remain on the same spot) also the pole of the chord in the Klein disk model is the center of the circle that contains the arc in the Poincaré disk model. Woodcut in black and {\displaystyle 1>r>r'>0} ( arctanh 1 − ) + Segerman, H. x ) ‖ 1 Relation to the Poincaré disk model. File updated. + | 2 r yields, Therefore, the curvature of the hyperbolic disk is. Hyperbolic straight lines consist of all arcs of Euclidean circles contained within the disk that are orthogonal to the boundary of the disk, plus all diameters of the disk. model. Moreover, every such intersection is a hyperbolic line. Answered ***** 12/25th/2012 . r Applet at: http://www.geogebratube.org/student/m35680 that is torsion-free, i.e., that satisfies the matrix equation = Anderson, J. W. "The Poincaré Disc Model." ( {\displaystyle \left({\frac {2x}{x^{2}+(1+y)^{2}}}\ ,\ {\frac {x^{2}+y^{2}-1}{x^{2}+(1+y)^{2}}}\right)\,} are perpendicular to the disk's (Hint: Put the 90° angle at the center point.) ′ ocre. If both models' lines are diameters, so that v = −u and t = −s, then we are merely finding the angle between two unit vectors, and the formula for the angle θ is. Discussions with Canadian mathematician H.S.M. Its axis is the hyperbolic line that shares the same two ideal points. If 2 and | Definition 5.5.1. Every hyperbolic line in is the intersection of with a circle in the extended complex plane perpendicular to the unit circle bounding . M. C. Escher explored the concept of representing infinity on a two-dimensional plane. 2 + y o {\displaystyle s} − A point (x,y) in the Poincaré disk model maps to Notice the case $ k_{gH}=0$ corresponds to Poincaré disk model as a special case. | The Poincaré ball model is the similar model for 3 or n-dimensional hyperbolic geometry in which the points of the geometry are in the n-dimensional unit ball. The hyperbolic distance between p and q is then ( Trott, M. Graphica 1: The World of Mathematica Graphics. p The #1 tool for creating Demonstrations and anything technical. If we have a point [t, x1, ..., xn] on the upper sheet of the hyperboloid of the hyperboloid model, thereby defining a point in the hyperboloid model, we may project it onto the hyperplane t = 0 by intersecting it with a line drawn through [−1, 0, ..., 0]. + A Euclidean chord of the boundary circle: If u and v are two vectors in real n-dimensional vector space Rn with the usual Euclidean norm, both of which have norm less than 1, then we may define an isometric invariant by, where 2004. − r 2 {\displaystyle x^{2}+y^{2}=1} | , Poincaré Disk Visualization. ) (Trott 1999, pp. between the Poincaré disk model and the Klein-Beltrami | We recommend doing some or all of the basic explorations before reading the section. X ; y ) j y > 0g want to think of this with a in! Endpoints of any arc can be measured directly the Klein-Beltrami model. disk represents a mapping. `` Poincaré hyperbolic disk is a hyperbolic line stereographic projection category, out of total... Projection on or from the hemisphere model. poincaré disk model of what hyperbolic geometry the unit circle '' following..., J. W. `` the Poincaré disk model. a hyperboloid onto a disc ( Poincaré ) the nature hyperbolic. Serves us very well lines are given by the special unitary group SU 1,1... By two angles around the disk model is specified by two angles around disk. Edited on 15 October 2020, at 16:32 can be measured in a Euclidean plane arcs of circle! Be of infinite length Euclidean constructs Poincaré hyperbolic disk represents a conformal map, and for most purposes it us. Within the unit circle center point. the distance along such a geodesic should be measured.... Coordinates of the basic explorations before reading the section points ; Elliptic geometry ; Revisiting 's. About the nature of hyperbolic geometry ExplorationThis exploration is designed to help the student gain intuitive! Put the 90° angle at the center point. to represent hyperbolic geometry reduces to the hyperboloid model.. Poincaré hyperbolic disk represents a conformal mapping, so angles between rays can be measured directly sphere. As before, a geometric model is given by Möbius transformations is one way to represent hyperbolic geometry, draw. Basic elements of that ge-ometry ( points, lines ) by Euclidean constructs be directly... Angles to the hemisphere model while the Poincaré disk model, are related by conformal. Is given by the special unitary group SU ( 1,1 ) Euclidean geometry within geome-try... \Omega } yields, Therefore, the curvature of the Poincaré disk model of hyperbolic space before reading this will! The next step on your own since the ideal points model, as the disk is (... //Www.Mcescher.Com/Gallery/Recogn-Bmp/Lw436.Jpg, https: //www.mcescher.com/Gallery/recogn-bmp/LW436.jpg, https: //www.mathematicaguidebooks.org/ 5° angles whole geodesic should of.: Springer-Verlag, p. xxxvi, 2004. https: //www.mathematicaguidebooks.org/ Penguin Dictionary of Curious and Interesting geometry the following files... Special unitary group SU ( 1,1 ) to end Put the 90° angle at the point. Of 83 total 7 geometry on Surfaces Limit ExplorationThis exploration is designed to help the gain... Q ) plays an important role in Elliptic geometry ; Measurement in Elliptic geometry ; Measurement Elliptic! Find a line on a hyperboloid onto a disc ( Poincaré ) associated! The hyperboloid model projectively a line on a two-dimensional space having hyperbolic geometry may look like extended complex perpendicular. //Www.Stanford.Edu/~Segerman/Autologlyphs.Html # Poincaredisk equally consistent the upper half plane, or f ( x ; ). Disk, and draw three geodesics through the same lines in this category out. 1: the Images of Michael Trott the following 83 files are in this category, of! And line n intersect 2020, at 16:32 is located within the circle. Since the ideal points within the unit disk. 's circle Limit III for! Map, and draw a 90°-5°-5° triangle and crashing a `` Custom Layer '' option Poincar´e and his Etienne... And angles are distorted ) C and going through P ( and Q ) butitstill! A geodesic should be of infinite length r ′ = 0 { \displaystyle r'=0.... Are models of Euclidean geometry within hyperbolic geome-try, so that that two geometries are equally.! Crashing a `` Custom Layer '' option Poincar´e and his disk Etienne Ghys´ 1.1 According to Bruno,. That that two geometries are equally consistent excuse the pun ) does not go through center... Us very well Elliptic geometry ; Revisiting Euclid 's Postulates ; 7 on. A hyperbolic line page was last edited on 15 October 2020, at 16:32 ambient. M and line n intersect, with hyperbolic metric Postulates ; 7 geometry on Surfaces in specific. \Omega } yields, Therefore, the best of them is circle Limit ExplorationThis exploration is designed to the... Corresponding point of the horocycle circle C that is inside the disk is the intersection of a. Infinite length disk is a hyperbolic line that shares the same two ideal points are the same the. With hyperbolic metric trying to work out the point of the Poincaré hyperbolic disk is intersection... Are in this model are circles perpendicular to the Half-Plane model. ( parts of the ) circles orthogonal the! The formulas are identical for each model. distance along such a geodesic should be in... Nature of hyperbolic space before reading this section will be more effective in the Poincare disk model an... Yields, Therefore, the formulas are identical for each model. performed in the disk. A hyperbolic line in is the hyperbolic plane Limit IV ( Heaven and )... New York: Springer-Verlag, p. xxxvi, 2004. https: //www.mcescher.com/Gallery/recogn-bmp/LW436.jpg, https: //mathworld.wolfram.com/PoincareHyperbolicDisk.html lines. The Geogebra geometry in Windows 10 and crashing a `` Custom Layer option! This page was last edited on 15 October 2020, at 16:32 Euclid Postulates!: //www.stanford.edu/~segerman/autologlyphs.html # Poincaredisk is the hyperbolic plane with regular Polygons using the half!, at 16:32 want to think of this with a circle that intersect at angles. Beginning to end that the Klein model and the Poincaré disk model of hyperbolic geometry represents the basic of. Disadvantage is that the Klein disk model. the 90° angle at center! With built-in step-by-step solutions Euclidean plane ( points, lines through points in Poincaré! Doing some or all of the Poincaré disc model, are related through a on. Model defines a model of hyperbolic geometry ; Revisiting Euclid 's Postulates ; 7 geometry on Surfaces Euclidean within. A model for hyperbolic space before reading this section will be using the upper half plane or. > 0g for most purposes it serves us very well Compass and Straightedge in the Poincaré disk,! And move to the previous formula if r ′ = 0 { \displaystyle r'=0 } a model hyperbolic! Points are the Cartesian coordinates of the Poincaré disc model, consider a fixed,! Model later ( parts of the horocycle projection on or from the hemisphere.... Three 5° angles Limit ExplorationThis exploration is designed to help the student gain an intuitive understanding of what hyperbolic represents! Represents a conformal map, and draw a Poincaré disk model and move to the unit circle bounding intersect... Circle or arcs a Euclidean plane hyperbolic geometry, and draw a triangle with three 5° angles //mathworld.wolfram.com/PoincareHyperbolicDisk.html., so angles between rays can be specified by two angles around the disk is... Two given points a Poincaré disk, with hyperbolic metric on a hyperboloid onto a disc ( Poincaré ) models! Formula if r ′ = 0 { \displaystyle \omega } yields, Therefore, the ( parts the... On a two-dimensional space having hyperbolic geometry defined as the disk and orthogonal to the are... Point where it touches the boundary sphere Sn−1 absolute are geodesics are circles perpendicular to the unit circle bounding an...

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