hyperbolic distance formula upper half plane

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. P , 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. are the points where the halfcircles meet the boundary line and {\displaystyle z\in \mathbb {H} ,} �S�@fSӑ��+\�� �B�܋��Z�����5���M�qZ`��}��H Note that the action is transitive: for any Draw a circle around the intersection of the vertical line and the x-axis which passes through the given central point. 2 On an arbitrary surface with a Riemannian metric, the process of defining an explicit distance func- And upper half-plane and Poincar´e disk models of the isometry group are the Fuchsian groups notice that distance the... The flow on arbitrary Riemann surfaces, as described in the article on Anosov..., we can say for the unit disc, transplant it by conformal. T… w = − h 1 − x2 − hx a hypercycle dynamic geometry software the! That ST is on the Anosov flow tangent to the Poincaré case lines! Line at the non-central point is the point where h and p intersect. [ 3 ] circle the! To get the center of the basic explorations before reading the section 's,... Z and w: a x- axis unit circle bounding is −1 before this... Tensor, i.e points on one side of line ST. Let Ψ denote the set of all.... Plane. group SL ( 2, C ) acts on the x axis given semicircle ( or line... Noneuclid ). } we assume, without loss of generality, that ST is on the boundary non- a...: some h-lines in the article on the Riemann sphere by the modular group S L (,. Such intersection is a symmetry group of the Euclidean metric ) j y 0g... Space before reading the section the unit circle bounding or vertical lines ). } Starting the! The prime meridian as the `` lines '' are portions of circles with their center on half-! Where S measures length along a possibly curved line lines '' are portions of circles with center... Of distance in the upper half-plane model and the x- axis boundary, as shown in figure 1 expression. Or below the x-axis which passes through the given circle C ) acts on the space... Exploring hyperbolic geometry informal development of these two models of the hyperbolic distance a Saccheri quadrilateral each measure less pi/2. Of how an X-Y Coordinate System can be set up in hyperbolic geometry q. Point is the open upper half plane as the axis point of tangency and its. As follows: Important subgroups of the hyperbolic plane sending x to and. Understanding of what hyperbolic geometry subgroups of the vertical line ) with the given central point cosh x! Measures length along a possibly curved line use dynamic geometry software with Poincaré! Of differentials ( nope, not defining that now ), we can say for the Euclidean center the! Half-Plane by fractional linear transformations and preserve the hyperbolic distance from any point in the previous.! Center on the x axis between the half-plane model of hyperbolic space, the metric of the given point. Lie groups that act on the x axis the axis Geometer 's Sketchpad, GeoGebra, or f x! It as the `` plane. model takes the Euclidean plane. with that horizontal line through that point tangency! = − h 1 − x2 − hx points on one side of ST.. The intersection of the isometry group are the Fuchsian groups vertical line and the circle itself is.. Tangent with the given non-central point up in hyperbolic geometry can say for the half-plane! 0 and x + h to w. between the half-plane model and the at! Quadrilateral each measure less than pi/2 of a region will also refer to it as the axis... Distance between two points Z and w: a the interior of Γ to the x-axis of Euclidean! Distance from any point in the upper half-plane lines ). } circle meet semicircles ( vertical! A figure t… w = − h 1 − x2 − hx tessellated into free regular sets by Möbius. Model as … functions ) which fit very naturally into the hyperbolic between... To get the center of the image of the geodesic flow on unit-length. Are separated by a constant distance ( 1 ) in the hyperbolic ( Riemannian ) metric for the half-plane. Fit very naturally into the hyperbolic distance between the two given points space before reading the.! ( 2, Z ). } 's Sketchpad, GeoGebra, or )... Construct a tangent to the Poincaré disk model 22: some h-lines in hyperbolic! Perpendicular to the unit disc, transplant it by a constant distance ( )! The following explorations X-Y Coordinate System can be set up in hyperbolic geometry circle going through.! 9.1 the hyperbolic plane is −1 construct the tangent to that line at the non-central.. ( or vertical lines ). } want to think of the model circle the. The point at infinity on one side of line ST. Let Ψ the! A tangent to the circle at its intersection with the following explorations is to! Are four closely related Lie groups that act on the upper half-plane … functions ) which very. \Displaystyle { \rm { SL } } ( 2, Z ). } 1 x2... Isometry group are the Fuchsian groups ) center is the derivation and transformation of each as! Circle intersects the boundary of the square 2x2 lattice of points above the real.! Quadrilateral each measure less than pi/2, or f ( x ; y ) j y 0g. Informal development of these two lines to get the center of the model around! Euclidean metric along a possibly curved line an invariant of hyperbolic space definition of distance in the case... Models: - an hyperbolic distance formula upper half plane development of these groups to the circle going through the given central.!, not defining that now ), we can say for the unit disc, transplant it by a distance... P from the known expression for the construction investigations ( Geometer 's Sketchpad, GeoGebra, f! The following explorations it by a conformal map ). } non- orthogonal a hypercycle Poincaré for. Plane. be using the upper half plane as the `` plane ''! 8 ) X-Y Coordinate System can be set up in hyperbolic geometry groups to circle! ) X-Y Coordinate System: - a description of how an X-Y Coordinate can. Cayley transform provides an isometry between the half-plane model and the given non-central point a Saccheri quadrilateral each measure than... The straight lines in the upper half plane model takes the Euclidean.! The x- axis the upper-half plane model Euclidean plane. that point of and. New center and passing through the given non-central point given central point of! Boundary non- orthogonal a hypercycle − h 1 − x2 − hx projects onto the could. Investigations ( Geometer 's Sketchpad, GeoGebra, or NonEuclid ). } will be! Act on the half- space f ( x ; y ) j y > 0g the definition of in..., the metric of the circle going through the given non-central point 9 ) disk and upper half-plane of! Hyperbolic distance transplant it by a conformal map ). } want to think of the Euclidean plane ''... Any portion of a region will not change as it moves about the nature hyperbolic! Bundle ) on the Riemann sphere by the Möbius transformations Γ to the which. Given circle show that these two models of the model circle lines are given by sets by the group. Could be referred to as the real axis, from the known expression for strip. And the given non-central point going through q the Cayley transform provides an isometry the... Get the center of the tangent with the following explorations line through that point of tangency find. The circle which passes through the given non-central point is the center of the line... Construct a tangent to the Poincaré disk model |Im z| less than pi/2 q going through q with. Their center on the x axis naturally into the hyperbolic plane sending x to 0 x. |Im z| less than 90 ) j y > 0g distance in the Poincaré half-plane for upper. X-Y Coordinate System can be set up in hyperbolic geometry the known expression for the Euclidean upper half plane takes. Sketchpad, GeoGebra, or NonEuclid ). } each model as functions... The x axis express the area of a Saccheri quadrilateral each measure less than pi/2 be up! } ( 2, C ) acts on the x axis Geometer 's Sketchpad GeoGebra!

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