{\displaystyle A_{\infty },B_{\infty }} The Cayley transform provides an isometry between the half-plane model and the Poincaré disk model. The area of a region will not change as it moves about the hyperbolic plane. ∈ 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�$���h�)k>Z�\s�js�0AUלF�kv��ٹ���2�5ׁ��ѿ_H-�9������� p !F��-�4�7��gd The metric of the model on the half- space. The arc-length differential determines an area differential and the area of a region will also be an invariant of hyperbolic geometry. y Let point q be the intersection of this line and the x- axis. + �я.C����r�Q)�k��N�r�$��v~���#��<0�x�BY���#~�cT����?�S�����0�B�2���!��0����W���wK˗&:�fF���h���v��C:5� j+��!\�t��dFPM���Z! This is the upper half-plane. z ( . R 2 where s measures length along a possibly curved line. 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. 9) Disk and Upper Half-Plane Models: - An informal development of these two models of Hyperbolic Geometry. Erase the part which is on or below the x-axis. 1 ∈ Draw a line tangent to the circle going through q. Draw a horizontal line through the non-central point. , H-points: H-points are Euclidean points on one side of line ST. Let Ψ denote the set of all H-points. Hyperbolic Proposition 2.4. g 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. ) { ( {\displaystyle |PQ|} . This provides a basic description of the geodesic flow on the unit-length tangent bundle (complex line bundle) on the upper half-plane. The algebraic formula for a line is . Find the intersection of these two lines to get the center of the model circle. and radius {\displaystyle z_{1},z_{2}\in \mathbb {H} } ( | "�[email protected]�%�׏/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�.� We will want to think of this with a diﬁerent distance metric on it. R Thus, The Poincar e upper half plane model for hyperbolic geometry 1 The Poincar e upper half plane is an interpretation of the primitive terms of Neutral Ge-ometry, with which all the axioms of Neutral geometry are true, and in which the hyperbolic parallel postulate is true. This model can be generalized to model an The relationship of these groups to the Poincaré model is as follows: Important subgroups of the isometry group are the Fuchsian groups. Let γ( ) ( ) ( )t x t iy t= + be path so the hyperbolic distance between two points (a, b) on the upper half plane with metric 22 2 2 dd d x y s y … t�.��H�E����Gi��u�\���{����6����oAf���q | is the euclidean length of the line segment connecting the points P and Q in the model. S {\displaystyle {\rm {SL}}(2,\mathbb {Z} ).}. S z A line will be any portion of a circle whose center is on the x axis. This group is important in two ways. 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