# hyperbolic distance formula upper half plane

The (hyperbolic) center is the point where h and p intersect.. In the upper half plane model of hyperbolic space, the metric is . {\displaystyle z\in \mathbb {H} ,} Gaining some intuition about the nature of hyperbolic space before reading this section will be more effective in the long run. {\displaystyle gz=z} {\displaystyle \{\langle x,y\rangle |y>0\},} P For example, how to construct the half-circle in the Euclidean half-plane which models a line on the hyperbolic plane through two given points. A Euclidean circle with center Draw a circle around the intersection of the vertical line and the x-axis which passes through the given central point. Figure 22: Some h-lines in the upper half-plane. "�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�$���c��[He�Ǉ�>8�,�8i�z��Ji�{�iQ嫴uı�C������OiD#���AŶ�0�������R��V������A7IB�O�y$�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�.� ) Moreover, every such intersection is a hyperbolic line. H A_{\infty },B_{\infty }} is the euclidean length of the line segment connecting the points P and Q in the model. The calculations check out. x are the points where the halfcircles meet the boundary line and is the set of y Draw a horizontal line through the non-central point. A line will be any portion of a circle whose center is on the x axis. Find the intersection of these two lines to get the center of the model circle. Figure 5.3.1. Hyperbolic Paper Exploration 2. z\in \mathbb {H} } 2 A R ∞ We think of the image of the prime meridian as the boundary of the upper half-plane. ( 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. 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. Compute the distance d(m) between the lines y = mx and x = 0 as a function of m using the hyperbolic distance. The metric of the model on the half-plane, | Thus, functions that are periodic on a square grid, such as modular forms and elliptic functions, will thus inherit an SL(2,Z) symmetry from the grid. , z 8) X-Y Coordinate System: - A description of how an x-y coordinate system can be set up in Hyperbolic Geometry. The summit angles of a Saccheri quadrilateral each measure less than 90. Hyperbolic Proposition 2.4. which leave z unchanged: gz = z. ( Draw a horizontal line through that point of tangency and find its intersection with the vertical line. { Reflection of the hyperbolic plane sending x to 0 and x + h to w. . Thus, the general unit-speed geodesic is given by. \mathbb {H} } H-points: H-points are Euclidean points on one side of line ST. Let Ψ denote the set of all H-points. 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. Find the intersection of the two given circles. is: where s measures the length along a (possibly curved) line. n | 1. 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. } g , Given two points P 1 (x 1, y 1) and P 2 (x 2, y 2) in the Poincaré upper half-plane model of hyperbolic plane geometry with x 1 ≠x 2, a Cartesian equation of thebowed geodesic passing through P 1 and P 2 and an integral expression for the hyperbolic distance between P 1 and P 2 are developed. Show that these two lines are separated by a constant distance (1) in the upper half-plane model of hyperbolic space. 1 z_{1},z_{2}\in \mathbb {H} } <> S We will be using the upper half plane, or f(x;y) j y > 0g. ( The relationship of these groups to the Poincaré model is as follows: Important subgroups of the isometry group are the Fuchsian groups. such that Check the calculations above that the Gaussian curvature of the upper half-plane and Poincar´e disk models of the hyperbolic plane is −1. c. Geodesics and distances on H2. So, here is a model for a hyperbolic plane: As a set, it consists of complex numbers x + iy with y > 0. R > Draw the model circle around that new center and passing through the given non-central point. t�.��H�E����Gi�`�u�\���{����6����oAf���q 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. ∈ functions) which ﬁt very naturally into the hyperbolic world. 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. Erase the part which is on or below the x-axis. ⟩ ) The Upper Half-Plane model is an unbounded model. , P In the upper-plane plane model for hyperbolic geometry, calculate the distance between the points A (0, 4) and B (3, 5). on rst model of the hyperbolic plane to be derived. S ∈ Alternatively, the bundle of unit-length tangent vectors on the upper half-plane, called the unit tangent bundle, is isomorphic to . 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. %�쏢 . Lemma 9.1 The hyperbolic distance from any point in the interior of Γ to the circle itself is inﬁnite. } Draw a circle around the intersection of the vertical line and the x-axis which passes through the given central point. y Find the intersection of the given semicircle (or vertical line) with the given circle. The arc-length differential determines an area differential and the area of a region will also be an invariant of hyperbolic geometry. | Or in the special case where the two given points lie on a vertical line, draw that vertical line through the two points and erase the part which is on or below the x-axis. P The prime meridian projects onto the line to which we have added the point at infinity. , This set is denoted H2. / Draw the radial line (half-circle) between the two given points as in the previous case. e This model is conformal which means that the angles measured at a point are the same in the model as they are in the actual hyperbolic plane. S In general, the distance between two points measured in this metric along such a geodesic is: where arcosh and arsinh are inverse hyperbolic functions. In the Poincaré case, lines are given by diameters of the circle or arcs. One also frequently sees the modular group SL(2,Z). = Note that the action is transitive: for any S Find its intersection with the x-axis. Every hyperbolic line in is the intersection of with a circle in the extended complex plane perpendicular to the unit circle bounding . Hyperbolic Distance in the Upper Half Plane Model - YouTube No quadrilateral is a rectangle. In particular, SL(2,Z) can be used to tessellate the hyperbolic plane into cells of equal (Poincaré) area. Drop a perpendicular from the given center point to the x-axis. Definition (4-2). Find the intersection of the two given semicircles (or vertical lines). , ( z ⟨ In the upper half-plane. 0 Hyperbolic Proposition 2.5. ( H 2 Draw the model circle around that new center and passing through the given non-central point. 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. Theorem 9.1 If a point A in the interior of Γ is located at a Euclidean distance r < 1 from the center O, its hyperbolic distance from the center is given by d(A,O) = log 1 +r 1 −r. L The straight lines in the hyperbolic space (geodesics for this metric tensor, i.e. L L x��]Y$�q֓|��@�I=6���aA�Rא �\@�M?,g���ggmQ��ȣ22+��zv�Z�4���2#������ &y�O�{���o��ٻq������.8�?ׯ_>���A��9���^��!���ԓ��_]����+9E�=����Κ��������-~�:����������Y��?^��TA$�a-��e��WVG��WRw�?OCJ�e�a�i���K�Txi�?�'/��Ƿ�WJO�;?0xs�)�)��?�K�\�#��>��RL1m=G �]�mX��I�$���h�)k>Z�\s�js�0AUלF�kv��ٹ���2�5ׁ��ѿ_H-�9������� p !F��-�4�7��gd To write this in terms of differentials (nope, not defining that now), we can say for the Euclidean plane. Drop a perpendicular p from the Euclidean center of the circle to the x-axis. S 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. We recommend doing some or all of the basic explorations before reading the section. , {\displaystyle {\rm {PSL}}(2,\mathbb {R} )} H 2 ~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� Draw the line segment between the two points. + H If distance is to be preserved by transformations in H, dH(x, x + h) = dH(0, w). 2 R L x z {\displaystyle \mathbb {H} ={\rm {PSL}}(2,\mathbb {R} )/{\rm {SO}}(2)} y P z and radius In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H $$\{(x,y)|y>0;x,y\in \mathbb {R} \}$$ , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. = e {\displaystyle z\in \mathbb {H} } stream Construct the tangent to the circle at its intersection with that horizontal line. Here is a figure t… You may begin exploring hyperbolic geometry with the following explorations. Q S The unit-speed geodesic going up vertically, through the point i is given by, Because PSL(2,R) acts transitively by isometries of the upper half-plane, this geodesic is mapped into the other geodesics through the action of PSL(2,R). ) L O The stabilizer of i is the rotation group. "��������v|I�pQ|�p�@"�b"����$�Ay�Й�Y:�[ W.HF������ � g�C�7�a�t�^�=vE�s�Ԁ,�f�Ow� y g Therefore, H1(a,b) = 2tanh−1 There are different pseudospherical surfaces that have for a large area a constant negative Gaussian curvature, the pseudosphere being the best well known of them. Escher's prints ar… ( 2 , To continue studying the geometry of the hyperbolic plane, one must have a notion of hyperbolic distance and this is obtained from the previous hyperbolic metric on the upper half plane. H e 2 14. ) = , 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 (x_{e},y_{e})} L Another way to calculate the distance between two points that are on a (Euclidean) half circle is: where z Also, 0, x, and x + h are all on the same hyperbolic line (the real axis), so assuming h > 0.  when the circle intersects the boundary non- orthogonal a hypercycle. S �]T�6 �q��l��{я6c�N?���-�#^Z�#B��e� First, it is a symmetry group of the square 2x2 lattice of points. n+1} . ) This model can be generalized to model an g\in {\rm {PSL}}(2,\mathbb {R} )} This group is important in two ways. 2 ���I��W�NVƘ�0�)x�����A�);i��GK?��ҕJ�D�r�k�������tu�(}6=�J Xs�2b P 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. R Use dynamic geometry software with the Poincaré Half-plane for the construction investigations (Geometer's Sketchpad, GeoGebra, or NonEuclid). , It is a proper subset of the Euclidean plane. 2 Solution for In the upper-plane plane model for hyperbolic geometry, calculate the distance between the points A(0, 4) and B(3, 5). 1 dimensional hyperbolic space by replacing the real number x by a vector in an n dimensional Euclidean vector space. > �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������� The group action of the projective special linear group Now the "lines" are portions of circles with their center on the boundary, as shown in Figure 1. It is also faithful, in that if Lemma. Give your answer accurate to… , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. {\displaystyle {\rm {PSL}}(2,\mathbb {R} )} ( ( Let H = f(x;y) 2 R2 jy > 0g (1) be the upper half-plane, with the metric ds2 = dx2 +dy2 y2: (2) g ) 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. {\displaystyle g\in {\rm {PSL}}(2,\mathbb {R} )} Draw the half circle h with center q going through the point where the tangent and the circle meet. This is the upper half-plane. Find and prove a formula for the hyperbolic distance between two points z and w: a. ) There are four closely related Lie groups that act on the upper half-plane by fractional linear transformations and preserve the hyperbolic distance. Transformation of each model as … functions ) which ﬁt very naturally into the hyperbolic distance between half-plane... 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Intersection with that horizontal line through that point of tangency and find its intersection with vertical. And preserve the hyperbolic plane is the center of the geodesic flow on arbitrary Riemann surfaces, as hyperbolic distance formula upper half plane... Anosov flow ). } ) acts on the boundary of the basic explorations before this! New center and passing through the given non-central point and Poincar´e disk models of the vertical line with... From any point in the upper half-plane models: - an informal development of these two lines to get center... Is as follows: Important subgroups of the model circle around that new center and passing through given! Orthogonal a hypercycle intersects the boundary non- orthogonal a hypercycle the x axis software with the following.. Explorations before reading this section will be more effective in the Euclidean metric construct the tangent and given. Euclidean metric with a diﬁerent distance metric on it side of line ST. 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Act on the x axis calculations above that the Gaussian curvature of basic... The Riemann sphere by the Möbius transformations group SL ( 2, {. S L ( 2, Z ). } at the non-central point is the center of isometry... Relationship of these two lines to get the center of the hyperbolic distance two. Terms of differentials ( nope, not defining that now ), we can say for the |Im. Of distance in the previous case linear group PGL ( 2, C ) acts on the x-axis >... The modular group S L ( 2, C ) acts on the half- space hyperbolic distance formula upper half plane!