Draw the model circle around that new center and passing through the given non-central point. then g = e. The stabilizer or isotropy subgroup of an element 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. Draw the circle around the intersection which passes through the given points. Use dynamic geometry software with the Poincaré Half-plane for the construction investigations (Geometer's Sketchpad, GeoGebra, or NonEuclid). The project focuses on four models; the hyperboloid model, the Beltrami-Klein model, the Poincar e disc model and the upper half plane model. Compute the hyperbolic (Riemannian) metric for the upper half-plane and for the strip |Im z| less than pi/2. , In the upper half-plane. Lemma 9.1 The hyperbolic distance from any point in the interior of Γ to the circle itself is inﬁnite. The arc-length differential determines an area differential and the area of a region will also be an invariant of hyperbolic geometry. 1 Show that these two lines are separated by a constant distance (1) in the upper half-plane model of hyperbolic space. Thus, on Draw a line tangent to the circle going through q. The algebraic formula for a line is . 10.3 The Upper Half-Plane Model: To develop the Upper Half-Plane model, consider a fixed line, ST, in a Euclidean plane. Give your answer accurate to three decimals. L Also, 0, x, and x + h are all on the same hyperbolic line (the real axis), so assuming h > 0. In the upper-plane plane model for hyperbolic geometry, calculate the distance between the points A (0, 4) and B (3, 5). L 2. 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. H 14. {\displaystyle {\rm {PSL}}(2,\mathbb {R} )} 0 {\displaystyle \mathbb {H} } R This model can be generalized to model an Reflection of the hyperbolic plane sending x to 0 and x + h to w. . 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. 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. {\displaystyle \mathbb {H} ={\rm {PSL}}(2,\mathbb {R} )/{\rm {SO}}(2)} Find the intersection of the two given semicircles (or vertical lines). ) 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. g Moreover, every such intersection is a hyperbolic line. ∈ Second, SL(2,Z) is of course a subgroup of SL(2,R), and thus has a hyperbolic behavior embedded in it. L 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. ( ( ⟨ S Definition 5.4.1. z 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. Find the intersection of the given semicircle (or vertical line) with the given circle. / One also frequently sees the modular group SL(2,Z). 9.2.3 Parallel Lines } Gaining some intuition about the nature of hyperbolic space before reading this section will be more effective in the long run. 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. ∈ R Draw a horizontal line through that point of tangency and find its intersection with the vertical line. [2] 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. Hint: Recall the definition of distance in the upper-half plane model. 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. S {\displaystyle |PQ|} %PDF-1.4 P Draw a circle around the intersection of the vertical line and the x-axis which passes through the given central point. Thus, the general unit-speed geodesic is given by. 2 ( Draw a line tangent to the circle which passes through the given non-central point. R 2 . Basic Explorations 1. A {\displaystyle gz_{1}=z_{2}} ∈ 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)). y 2 There are four closely related Lie groups that act on the upper half-plane by fractional linear transformations and preserve the hyperbolic distance. , , S ∈ c. Geodesics and distances on H2. The calculations check out. ( Drop a perpendicular from the given center point 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. Starting with this model, one can obtain the flow on arbitrary Riemann surfaces, as described in the article on the Anosov flow. 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. We will be using the upper half plane, or f(x;y) j y > 0g. , , {\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) 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. Give your answer accurate to… 1. 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. R This group is important in two ways. x ���I��W�NVƘ�0�)x�����A�);i��GK?��ҕJ�D�r�k�������tu�(}6=�J Xs�2b H-points: H-points are Euclidean points on one side of line ST. Let Ψ denote the set of all H-points. ) Find the intersection of the two given circles. 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. t�.��H�E����Gi�`�u�\���{����6����oAf���q S So, here is a model for a hyperbolic plane: As a set, it consists of complex numbers x + iy with y > 0. 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. e Construct the tangent to the circle at its intersection with that horizontal line. x {\displaystyle g\in {\rm {PSL}}(2,\mathbb {R} )} Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part (the y coordinate mentioned above) is positive. S 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 $1$). e The metric of the model on the half- space. Proof. 2 which leave z unchanged: gz = z. P Every hyperbolic line in is the intersection of with a circle in the extended complex plane perpendicular to the unit circle bounding . P Drop a perpendicular p from the Euclidean center of the circle to the x-axis. z r Compute the distance d(m) between the lines y = mx and x = 0 as a function of m using the hyperbolic distance. We will also refer to it as the real axis, . The Upper Half-Plane model is an unbounded model. 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