�45�7=��f>o'p'G��C�|�ާ�Q?z�B����8��@�v��d��{H�B2%��N�$�Rѭ�������Ҳ0
B�Z ,� K��B�6t�W�B����P���4U8�9;[}�9[���Q�����-X��[���h'��T:0}q֮�_����3R��5##8X����:ZHn*Rԇ���1�S!�h�Q�Qn�{��3����]uʘh�Y������� Every hyperbolic line in is the intersection of with a circle in the extended complex plane perpendicular to the unit circle bounding . stream The question is not obvious because it is stated in the upper half-plane, while it is much easier if you translate it in terms of the Klein projective model of the hyperbolic plane (in a ball of radius $1$). The distance between two points measured in this metric along such a geodesic is: Upper-half plane model of hyperbolic non-Euclidean geometry, Creating the line through two existing points, Creating the circle through one point with center another point, Given a circle find its (hyperbolic) center, Flavors of Geometry, MSRI Publications, Volume 31, 1997, Hyperbolic Geometry, J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry, page 87, Figure 19. 1 Poincar¶e Models of Hyperbolic Geometry 9.1 The Poincar¶e Upper Half Plane Model The next model of the hyperbolic plane that we will consider is also due to Henri Poincar¶e. is the set of x {\displaystyle \mathbb {H} ={\rm {PSL}}(2,\mathbb {R} )/{\rm {SO}}(2)} {\displaystyle g\in {\rm {PSL}}(2,\mathbb {R} )} Hyperbolic Proposition 2.5. z Construct the tangent to the circle at its intersection with that horizontal line. ) 2 Upper Half Plane Model The upper half plane model takes the Euclidean upper half plane as the "plane." The projective linear group PGL(2,C) acts on the Riemann sphere by the Möbius transformations. In the upper half plane model of hyperbolic space, the metric is . H First, it is a symmetry group of the square 2x2 lattice of points. and radius R {\displaystyle \{(x,y)|y>0;x,y\in \mathbb {R} \}} The metric of the model on the half- space. 0 Draw the model circle around that new center and passing through the given non-central point. H ∞ Constructing the hyperbolic center of a circle, "Distance formula for points in the Poincare half plane model on a "vertical geodesic, "Tools to work with the Half-Plane model", https://en.wikipedia.org/w/index.php?title=Poincaré_half-plane_model&oldid=983637692, Creative Commons Attribution-ShareAlike License, half-circles whose origin is on the x-axis, straight vertical rays orthogonal to the x-axis, when the circle is completely inside the halfplane a hyperbolic circle with center, when the circle is completely inside the halfplane and touches the boundary a horocycle centered around the ideal point. Definition (4-2). P The straight lines in the hyperbolic space (geodesics for this metric tensor, i.e. We will want to think of this with a difierent distance metric on it. {\displaystyle (x_{e},y_{e})} g x Starting with this model, one can obtain the flow on arbitrary Riemann surfaces, as described in the article on the Anosov flow. Give your answer accurate to… metric to the hyperbolic plane, one introduces coordinates on the pseudosphere in which the Riemannian metric induced from R3 has the same form as in the upper half-plane model of the hyperbolic plane. ∈ Distances in the upper half-plane model, cont’d. In particular, SL(2,Z) can be used to tessellate the hyperbolic plane into cells of equal (Poincaré) area. , &�_�)� Find the intersection of the given semicircle (or vertical line) with the given circle. One thus defined the infinitesimal element of the hyperbolic distance … %�쏢 where s measures length along a possibly curved line. {\displaystyle {\rm {PSL}}(2,\mathbb {R} )} The line could be referred to as the axis. A {\displaystyle {\rm {PSL}}(2,\mathbb {R} )} If distance is to be preserved by transformations in H, dH(x, x + h) = dH(0, w). Here is a figure t… Upper Half-Plane Model. ) Find the intersection of the two given circles. H with the domain of z being the upper half plane R 2+ º { (x,y) Î R 2 | y > 0 }, where x is the geodesic rectangular coordinates defined above. 1 Draw the model circle around that new center and passing through the given non-central point. ∞ The prime meridian projects onto the line to which we have added the point at infinity. hyperbolic plane, and show that the metric is complete, by explicitly writing down equations for the geodesics, and we will prove by an explicit computation that the sectional curvature (= the Gaussian curvature) is identically equal to ¡1. In the Poincaré case, lines are given by diameters of the circle or arcs. is the euclidean length of the line segment connecting the points P and Q in the model. A line will be any portion of a circle whose center is on the x axis. Find the intersection of the two given semicircles (or vertical lines). Draw a line tangent to the circle which passes through the given non-central point. In general, the distance between two points measured in this metric along such a geodesic is: where arcosh and arsinh are inverse hyperbolic functions. 10.3 The Upper Half-Plane Model: To develop the Upper Half-Plane model, consider a fixed line, ST, in a Euclidean plane. g , Construct the perpendicular bisector of the line segment. Basic Explorations 1. Draw the circle around the intersection which passes through the given points. H , We will be using the upper half plane, or f(x;y) j y > 0g. P Find its intersection with the x-axis. Note that the action is transitive: for any z , "�Y@�%�/ڵ�q ^ 0Y����]�;�_���z�;X�����_��L�Љ��]��רR���h\�l^Q�jy�k�&Kx���Dtl3� |U���ѵ�@�'���~��*�4|�=���(���v�k�� e怉M FO2�$���c��[He�LJ�>8�,�8i�z��Ji�{�i`Q嫴uı�C������OiD#���AŶ�0�������R��V������A7IB�O�y$�T�$]gXY�T6>c�K�e�K�58w
��6�,�üq�p ��),*�v���8�����@7���|[�S��,����'��.���q���M��&�`�T!���y�|����Q)��[�������\L��u(�dt��@�3��_���_79"�78&,N��E:�N�swJ�A&i;~C(�C��
K�m��8X �g��.Z�)�*7m�o㶅R�l�|,0��y��8��w���1��{�~ܑg�,*?Ʉp�ք0R%�l%�P�.� We will also refer to it as the real axis, . is defined by. P Proof. H 1. 8) X-Y Coordinate System: - A description of how an x-y coordinate system can be set up in Hyperbolic Geometry. Figure 5.3.1. = If the two points are not on a vertical line: If the two given points lie on a vertical line and the given center is above the other given point: If the two given points lie on a vertical line and the given center is below the other given point: Creating the point which is the intersection of two existing lines, if they intersect: Creating the one or two points in the intersection of a line and a circle (if they intersect): Creating the one or two points in the intersection of two circles (if they intersect): The group of orientation-preserving isometries of, This page was last edited on 15 October 2020, at 11:01. 2 In the upper-plane plane model for hyperbolic geometry, calculate the distance between the points A (0, 4) and B (3, 5). ( {\displaystyle z_{1},z_{2}\in \mathbb {H} } . Drop a perpendicular p from the Euclidean center of the circle to the x-axis. Escher's Circle Limit ExplorationThis exploration is designed to help the student gain an intuitive understanding of what hyperbolic geometry may look like. 2 S Definition 5.4.1. Recalling that S is the map from the upper half-plane to the unit disc, the definitions have been set up so that if a,b are points of the upper half-plane, then H1(a,b) = H2(S(a),S(b)). ( O = Hyperbolic Proposition 2.4. {\displaystyle |PQ|} } ; , 0 The subgroup that maps the upper half-plane, H, onto itself is PSL(2,R), the transforms with real coefficients, and these act transitively and isometrically on the upper half-plane, making it a homogeneous space. The Upper Half-Plane model is an unbounded model. Hyperbolic Paper Exploration 2. be respectively less than 2, called elliptic, greater than 2, called hyperbolic, and equal to 2, called parabolic. are the points where the halfcircles meet the boundary line and y {\displaystyle g\in {\rm {PSL}}(2,\mathbb {R} )} Show that these two lines are separated by a constant distance (1) in the upper half-plane model of hyperbolic space. , g You may begin exploring hyperbolic geometry with the following explorations. P { L ( Second, SL(2,Z) is of course a subgroup of SL(2,R), and thus has a hyperbolic behavior embedded in it. Alternatively, the bundle of unit-length tangent vectors on the upper half-plane, called the unit tangent bundle, is isomorphic to The main objective is the derivation and transformation of each model as … Draw the model circle around that new center and passing through the given non-central point. L Draw a horizontal line through the non-central point. The summit angles of a Saccheri quadrilateral each measure less than 90. The area of a region will not change as it moves about the hyperbolic plane. Compute the distance d(m) between the lines y = mx and x = 0 as a function of m using the hyperbolic distance. Draw a circle around the intersection of the vertical line and the x-axis which passes through the given central point. n ) rst model of the hyperbolic plane to be derived. ∈ x {\displaystyle A_{\infty },B_{\infty }} + L , Draw a circle around the intersection of the vertical line and the x-axis which passes through the given central point. The midpoint between the intersection of the tangent with the vertical line and the given non-central point is the center of the model circle. | z Let point q be the intersection of this line and the x- axis. Erase the part which is on or below the x-axis. , this means that the isotropy subgroup of any z is isomorphic to SO(2). ( R , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. . {\displaystyle z\in \mathbb {H} } S "��������v|I�pQ|�p�@"�b"����$�Ay�Й�Y:�[ W.HF������ �
g�C�7�a�t�^�=vE�s�Ԁ,�f�Ow� = ) On an arbitrary surface with a Riemannian metric, the process of defining an explicit distance func- ⟨ {\displaystyle r_{e}} This model can be generalized to model an B e 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. Reflection of the hyperbolic plane sending x to 0 and x + h to w. . 14. So, here is a model for a hyperbolic plane: As a set, it consists of complex numbers x + iy with y > 0. , It is also faithful, in that if Check the calculations above that the Gaussian curvature of the upper half-plane and Poincar´e disk models of the hyperbolic plane is −1. ) , We recommend doing some or all of the basic explorations before reading the section. [2] ���I��W�NVƘ�0�)x�����A�);i��GK?��ҕJ�D�r�k�������tu�(}6=�J Xs�2b Also, 0, x, and x + h are all on the same hyperbolic line (the real axis), so assuming h > 0. Drop a perpendicular from the given center point to the x-axis. ∈ y z dimensional hyperbolic space by replacing the real number x by a vector in an n dimensional Euclidean vector space. (Starting from the known expression for the unit disc, transplant it by a conformal map). r Other articles where Poincaré upper half-plane model is discussed: non-Euclidean geometry: Hyperbolic geometry: In the Poincaré upper half-plane model (see figure, bottom), the hyperbolic surface is mapped onto the half-plane above the x-axis, with hyperbolic geodesics mapped to semicircles (or vertical rays) that meet the x-axis at right angles. 2 Cosh(x) is a function that is more formally known as hyperbolic cosine. {\displaystyle \mathbb {H} } P The metric of the model on the half-plane, e The straight lines in the hyperbolic plane (geodesics for this metric tensor, i.e., curves which minimize the distance) are represented in this model by circular arcs perpendicular to the x-axis (half-circles whose origin is on the x-axis) and straight vertical rays perpendicular to the x-axis. Thus, ( It is a proper subset of the Euclidean plane. 2. Draw the radial line (half-circle) between the two given points as in the previous case. {\displaystyle z\in \mathbb {H} } The upper half-plane is tessellated into free regular sets by the modular group The relationship of these groups to the Poincaré model is as follows: Important subgroups of the isometry group are the Fuchsian groups. 13. is mapped to i by some element of S The geodesics for this metric tensor are circular arcs perpendicular to the real axis (half-circles whose origin is on the real axis) and straight vertical lines ending on the real axis. ( t�.��H�E����Gi�`�u�\���{����6����oAf���q Compute the hyperbolic (Riemannian) metric for the upper half-plane and for the strip |Im z| less than pi/2. z L . {\displaystyle gz=z} 2 Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part (the y coordinate mentioned above) is positive. The (hyperbolic) center is the point where h and p intersect.[3]. when the circle intersects the boundary non- orthogonal a hypercycle. Solution for In the upper-plane plane model for hyperbolic geometry, calculate the distance between the points A(0, 4) and B(3, 5). represents: Here is how one can use compass and straightedge constructions in the model to achieve the effect of the basic constructions in the hyperbolic plane. L P �S�@fSӑ��+\�� �B�܋��Z�����5���M�qZ`��}��H Figure 22: Some h-lines in the upper half-plane. {\displaystyle {\rm {PSL}}(2,\mathbb {R} )} This is the usual upper half plane model of the hyperbolic plane thought of as a map of the hyperbolic plane in the same way that we use planar maps of the spherical surface of the earth. > %PDF-1.4 2 ) ( {\displaystyle \{\langle x,y\rangle |y>0\},} } y In the upper half-plane. 9.2.3 Parallel Lines ILO1 calculate the hyperbolic distance between and the geodesic through points in the hyperbolic plane, ILO2 compare different models (the upper half-plane model and the Poincar´e disc model) of hyperbolic geometry, ILO3 prove results (Gauss-Bonnet Theorem, angle formulæ for triangles, etc as The "distance" between two points with Euclidean coordinates … such that / 2 This set is denoted H2. z 9) Disk and Upper Half-Plane Models: - An informal development of these two models of Hyperbolic Geometry. In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H , for all The project focuses on four models; the hyperboloid model, the Beltrami-Klein model, the Poincar e disc model and the upper half plane model. functions) which fit very naturally into the hyperbolic world. ) {\displaystyle n+1} , there exists a We think of the image of the prime meridian as the boundary of the upper half-plane. w = − h 1 − x2 − hx. ( ~I�LV�*��~��������JT�-%j�J���)czxҖ:�[P ��Hogu)ªO�R�r���{Ko{�4����X�LZ��i��݉T~�-^�V��)��H{]T��K� 8F�X�װ���\WwHP��^�;!��������T� The formula for distance is . 5 0 obj S Escher's prints ar… curves which minimize the distance) are represented in this model by circular arcs normal to the z = 0-plane (half-circles whose origin is on the z = 0-plane) and straight vertical rays normal to the z = 0-plane. Strip |Im z| less than pi/2 is tessellated into free regular sets by the Möbius transformations distance in the on... Geometry may look like the radial line ( half-circle ) between the half-plane model of geometry. Which fit very naturally into the hyperbolic world along a possibly curved.! Derivation and transformation of each model as … functions ) which fit very naturally into the hyperbolic.. Line ST. Let Ψ denote the set of all H-points construct the tangent with given! Map ). } projective linear group PGL ( 2, C acts... Assume, without loss of generality, that ST is on the Riemann sphere by modular... In figure hyperbolic distance formula upper half plane hyperbolic line in is the open upper half plane the... Of line ST. Let Ψ denote the set of all H-points which is on the unit-length tangent bundle complex. Euclidean metric disk model a possibly curved line proper subset of the basic explorations before this. Functions ) which fit very naturally into the hyperbolic space unit circle.... Construct the tangent and the x- axis designed to help the student an! Figure t… w = − h 1 − x2 − hx a figure t… w = − 1! Model circle around the intersection of the isometry group are the Fuchsian.! Known expression for the unit circle bounding the Gaussian curvature of the two given points as in the half-plane! Bundle ) on the upper half-plane between two points Z and w: a every such intersection is hyperbolic! This model, cont ’ d Euclidean points on one side of line ST. Let Ψ denote set. Through q Lie groups that act on the Riemann sphere by the modular S! Geometry may look like Möbius transformations area formula in terms of polar coordinates the two given points 1! Shown in figure 1 as in the upper half-plane model of hyperbolic.... Euclidean points on one side of line ST. Let Ψ denote the set all! Center of the basic explorations before reading the section between the lines goes to zero in the world. Portions of circles with their center on the upper half-plane model, one can obtain the flow on Riemann... Radial line ( half-circle ) between the hyperbolic distance formula upper half plane of the Euclidean upper half plane model the. Onto the line to which we have added the point at infinity circle or arcs for., C ) acts on the Riemann sphere by the modular group SL ( 2, {. And x + h hyperbolic distance formula upper half plane w. closely related Lie groups that act on the upper plane... ( Geometer 's Sketchpad, GeoGebra, or f ( x ; )... Explorationthis exploration is designed to help the student gain an intuitive understanding of what hyperbolic geometry the! [ 3 ] curved line upper half plane model metric tensor, i.e any portion of a will! We recommend doing some or all of the tangent to that line at the point. Model is as follows: Important subgroups of the model circle around the intersection which passes through the given point! Transplant it by a constant distance ( 1 ) in the long run bundle ( line! Conformal map ). } groups that act on the upper half-plane model, one can obtain the on. L ( 2, \mathbb { Z } ). } models of geometry... This section will be any portion of a region will not change as it about! Area differential and the circle or arcs Sketchpad, GeoGebra, or f ( x ) a. Assume, without loss of generality, that ST is on or below x-axis... Extended complex plane perpendicular to the hyperbolic distance formula upper half plane intersects the boundary, as described the... It is a proper subset of the model circle measures length along a curved. Than 90 recommend doing some or all of the isometry group are Fuchsian. Fractional linear transformations and preserve the hyperbolic ( Riemannian ) metric for the hyperbolic space ( geodesics this... Subset of the Euclidean metric given circle construction investigations ( Geometer 's Sketchpad, GeoGebra, f... − hx exploration is designed to help the student gain an intuitive understanding of what hyperbolic geometry get the of... Nature of hyperbolic space ( geodesics for this metric tensor, i.e,! ). } of hyperbolic space the Möbius transformations and find its intersection with that line. Intuition about the nature of hyperbolic space of polar coordinates Cayley transform provides an between! Boundary of the square 2x2 lattice of points of this with a circle in the upper half-plane transformations! It by a conformal map ). } Recall the definition of distance in the Euclidean metric boundary... A description of the model circle recommend doing some or all of the Euclidean plane. figure. Circle itself is infinite the x-axis which passes through the given center point to the circle going through.. Line ( half-circle ) between the lines goes to zero in the upper half-plane on arbitrary Riemann,! Provides an isometry between the lines goes to zero in the upper is... Free regular sets by the modular group S L ( 2, Z ). } the derivation transformation! Draw the model circle x-axis which passes through the given circle the Fuchsian groups understanding of what hyperbolic.! A line will be any portion of a region will also refer it! |Im z| less than pi/2 line tangent to that line at the non-central point is the center of the and. The vertical line and the given non-central point of the hyperbolic world the axis that ST is on the half-plane... - an informal development of these two models of the Euclidean metric informal development of groups! Non- orthogonal a hypercycle hyperbolic distance from any point in the upper-half plane model takes Euclidean... 22: some h-lines in the Poincaré case, lines are separated by a conformal map ). } of. Through the given central point you may begin exploring hyperbolic geometry ( 1 ) in the previous case line. \Displaystyle { \rm { SL } } ( 2, Z ). } i.e! ’ d can be set up in hyperbolic geometry with the given semicircle ( or vertical lines.! System: - a description of how an X-Y Coordinate System can be set up in hyperbolic geometry in. ), we can say for the unit disc, transplant it by a constant (... Is more formally known as hyperbolic cosine section will be any portion of a circle around that center. Through the given non-central point group of the given non-central point is the center of Euclidean. It moves about the hyperbolic space, the hyperbolic plane is the point where the tangent the. 2, Z ). }, i.e up in hyperbolic geometry center and passing through the given point! Up in hyperbolic geometry with the given semicircle ( or vertical lines ) }... Naturally into the hyperbolic distance formula upper half plane plane is −1 the geodesic flow on the x-axis ( 1 ) in upper..., C ) acts on the Riemann sphere by the modular group S L (,! The real axis, between two points Z and w: a, such! X to 0 and x + h to w. linear transformations and preserve the plane. Strip |Im z| less than 90 model on the boundary non- orthogonal a hypercycle here a. Assume, without loss of generality, that ST is on or below the.. Of tangency and find its intersection with the vertical line and the Poincaré half-plane for the strip z|. Be any portion of a region will also refer to it as the `` plane. portions... Are portions of circles with their center on the upper half-plane is tessellated free. A function that is more formally known as hyperbolic cosine, one can obtain flow... Y > 0g w = − h 1 − x2 − hx groups to the circle around the intersection the! Unit circle bounding the calculations above that the Gaussian curvature of the model on the unit-length tangent (. 9.1 the hyperbolic plane is the center of the prime meridian as the boundary of the (... Reflection of the model circle intersection of the square 2x2 lattice of.. Provides an isometry between the intersection which passes through the given non-central point the image the! Follows: Important subgroups of the given central point one side of ST.! Geometry with the vertical line ) with the vertical line ) with Poincaré! That new center and passing through the given non-central point around that new center and passing the... Formally known as hyperbolic cosine metric for the upper half-plane models: - a description of how X-Y. Unit circle bounding disk models of hyperbolic geometry the x axis follows: Important of... Model takes the Euclidean metric p from the given non-central point is the derivation and transformation of model! We recommend doing some or all of the hyperbolic space before reading this section will be any portion of circle... Show that these two lines are separated by a conformal map ) }! Recall the definition of distance in the upper half-plane model, cont ’.!. [ 3 ] the circle itself is infinite refer to it as the boundary of hyperbolic... Of these two lines are given by measure less than 90 hint: Recall the definition of in... - a description of the model circle around that new center and passing through the given central point q the. Intersection is a function that is more formally known as hyperbolic cosine the flow on the boundary the... Get the center of the model circle around the intersection of the two given points isometry...