to another triple Geometrically, a Möbius transformation can be obtained by first performing stereographic projection from the plane to the unit two-sphere, rotating and moving the sphere to a new location and orientation in space, and then performing stereographic projection (from the new position of the sphere) to the plane. Define this transformation and then find the image of A the triangle constructed in the previous problem in U under this transformation. {\displaystyle {\widehat {\mathbf {C} }}} {\displaystyle \phi } C det ^ } 2 {\displaystyle {\mathfrak {H}}(z)} The projective special linear group 7 5. A Möbius transformation can be composed as a sequence of simple transformations. w H = What are the “straight lines” in this model? 2 And the second paragraph should be the first either way, as it is a more general description of the subject. Every non-parabolic transformation is conjugate to a dilation/rotation, i.e. Download Citation | The Poincaré Upper Half-Plane | In Chap. z H H z Therefore, the set of all Möbius transformations forms a group under composition. The horizontal axis itself is not part 116 May 07; this is a work in progress 1 The action of PSL(2,C) on the celestial sphere may also be described geometrically using stereographic projection. C ( Since \(z = V^{-1}(w) = \frac{iw+1}{w+i}\) we may work out the arc-length differential in terms of \(dw\text{. Suppose \(w_1\) and \(w_2\) are two points in \(V\) whose pre-images in the unit disk are \(z_1\) and \(z_2\text{,}\) respectively. According to Liouville's theorem a Möbius transformation can be expressed as a composition of translations, similarities, orthogonal transformations and inversions. . From the arc-length differential \(ds = \frac{dw}{\text{Im}(w)}\) comes the area differential: In the upper half-plane model \((\mathbb{U},{\cal U})\) of hyperbolic geometry, the area of a region \(R\) described in cartesian coordinates, denoted \(A(R)\text{,}\) is given by. The existence of the inverse Möbius transformation and its explicit formula are easily derived by the composition of the inverse functions of the simpler transformations. Borrowing terminology from special relativity, points with Q > 0 are considered timelike; in addition, if x0 > 0, then the point is called future-pointing. Other authors include Emil Artin (1957),[12] H. S. M. Coxeter (1965),[13] and Roger Penrose, Wolfgang Rindler (1984)[14] and W. M. Olivia (2002)[15], Minkowski space consists of the four-dimensional real coordinate space R4 consisting of the space of ordered quadruples (x0,x1,x2,x3) of real numbers, together with a quadratic form. − These transformations tend to move all points in S-shaped paths from one fixed point to the other. ) } The components of (5) are precisely those obtained from the outer product. ( The terminology is due to considering half the absolute value of the trace, |tr|/2, as the eccentricity of the transformation – division by 2 corrects for the dimension, so the identity has eccentricity 1 (tr/n is sometimes used as an alternative for the trace for this reason), and absolute value corrects for the trace only being defined up to a factor of ±1 due to working in PSL. It indicates how repulsive the fixed point γ1 is, and how attractive γ2 is. For |k| < 1, the roles are reversed. If Since the determinant of X is identified with the quadratic form Q, SL(2,C) acts by Lorentz transformations. n {\displaystyle {\widehat {\mathbf {C} }}=\mathbf {C} \cup \{\infty \}} The simplest possibility of a fractional multiple means α = π/2, which is also the unique case of 1 g ∞ }\), The area of this \(\frac{2}{3}\)-ideal triangle is thus, With the trig substituion \(\cos(\theta) = x\text{,}\) so that \(\sqrt{1-x^2} = \sin(\theta)\) and \(-\sin(\theta)d\theta = dx\text{,}\) the integral becomes. {\cal L}(\boldsymbol{r}) = \int_a^b \frac{1}{k}~dt = \frac{b-a}{k}\text{.} w {\displaystyle {\mathcal {M}}} 1 {\displaystyle gfg^{-1}(z)=kz} d which sends the points (γ1, γ2) to (0, ∞). 1. Every Möbius transformation can be written such that its representing matrix z Poincar e upper half plane model. = It also plays an important role in hyperbolic geometry, where the Poincaré half-plane model provides a way of … Determine the area of the “triangular” region pictured below. its slope, the "tightness" of the spiral) is the argument of k. Of course, Möbius transformations may have their two fixed points anywhere, not just at the north and south poles. There are several ways to determine f(z) from the given sets of points. x To each (x0,x1,x2,x3) ∈ R4, associate the hermitian matrix, The determinant of the matrix X is equal to Q(x0,x1,x2,x3). ∈ R, b ∈ C and |b| < 1. This figure is determined by two horocycles \(C_1\) and \(C_2\text{,}\) and two hyperbolic lines \(L_1\) and \(L_2\) all sharing the same ideal point. of determinant one is said to be parabolic if, (so the trace is plus or minus 2; either can occur for a given transformation since z It is the domain of many functions of interest in complex analysis, especially modular forms. 6.2 Thm: If m RPQ d 0, then PQ l .If m RPQ d 0, then PQ l . a {\displaystyle \gamma _{1},\gamma _{2}} = 1 87, No. ¯ In fact, \(z_2\) gets sent to the point \(ki\) where \(k = |S(z_2)| = |S(V^{-1}(w_2))|\) (and \(0 \lt k \lt 1\)). 2 ) Together with its subgroups, it has numerous applications in mathematics and physics. This is the upper half-plane. In the following discussion we will always assume that the representing matrix ( ζ λ d tr }\) Thus. z H The Upper Half Plane Model for Hyperbolic Geometry. that the matrix be invertible. λ H a k ) In general, the two fixed points may be any two distinct points on the Riemann sphere. ) R det Define This Transformation And Then Find The Image Of A The Triangle Constructed In The Previous Problem In U Under This Transformation. z PSL The same terminology is used for the classification of elements of SL(2, R) (the 2-fold cover), and analogous classifications are used elsewhere. The Poincare Half-Plane: A Gateway to Modern Geometry (Jones and Bartlett Pocket-Sized Nursing Reference Series) Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane (English Edition) Stahl, S: Gateway to Modern Geometry: The Poincare Half-Plan: The Poincare Half-plane Harmonic Analysis on Symmetric Spaces Euclidean Space, the Sphere, … has the same characteristic polynomial X2−2X+1 as the identity matrix, and is therefore unipotent. The Poincaré disk model is one way to represent hyperbolic geometry, and for most purposes it serves us very well. If we take the one-parameter subgroup generated by any hyperbolic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes the same two points. H ( n ( $\begingroup$ Given that he talks about the triangle living in the upper half plane, and specifically calls it the upper half plane model in the title, it's safe to assume he's talking about hyperbolic geometry here. {\displaystyle \operatorname {Aut} ({\widehat {\mathbf {C} }})} An explicit equation can be found by evaluating the determinant, by means of a Laplace expansion along the first row. Copy link Quote reply neozhaoliang commented May 22, 2020. V(z) = \frac{-iz + 1}{z - i}\text{.} H P 2. , z , {\displaystyle {\mathcal {M}}\cong \operatorname {PSL} (2,\mathbf {C} )} We will be using the upper half plane, or f(x;y)jy>0g. x 10.1 Models of Hyperbolic Geometry: Models serve primarily a logical purpose. f If ρ = 0, then the fixed points are neither attractive nor repulsive but indifferent, and the transformation is said to be elliptic. They are useful when exploring the geometric properties of the hyperbolic plane; they don't "look like" the hyperbolic plane. {\displaystyle {\mathfrak {H}}} , Using Gauss’ equation we find immediately that this surface has constant Gauss curvature K = −1. ∞ If one of the zi is ∞, then the proper formula for = z is a combination of a (homothety and a rotation) Thus, the two acute angles of a Saccheri quadrilateral are also congruent. If ad = bc, the rational function defined above is a constant since. to the projective special unitary group PSU(2,C) which is isomorphic to the special orthogonal group SO(3) of rotations in three dimensions, and can be interpreted as rotations of the Riemann sphere. The stars seem to move along longitudes, away from the South pole toward the North pole. The upper half-plane model. It is straightforward to check that then the product of two matrices will be associated with the composition of the two corresponding Möbius transformations. When sailing on a constant bearing – if you maintain a heading of (say) north-east, you will eventually wind up sailing around the north pole in a logarithmic spiral. This can be used to iterate a transformation, or to animate one by breaking it up into steps. \end{equation*}, \begin{align*} {\displaystyle {\mathfrak {H}}} α z z {\displaystyle \lambda =e^{i\alpha }} = NonEuclid supports two different models of the hyperbolic plane: the Disk model and the Upper Half-Plane model. with the complex number λ not equal to 0, 1 or −1, corresponding to a dilation/rotation through multiplication by the complex number k = λ2, called the characteristic constant or multiplier of the transformation. d_U(w_1, w_2) = d_H(z_1,z_2) = \ln((z_1, z_2 ; u, v))\text{,} z From this decomposition, we see that Möbius transformations carry over all non-trivial properties of circle inversion. the cross-ratio of Affine geometry. g ≅ 1 g R Upper Half-Plane by Audrey Terras (2013-09-13) Non-Euclidean Geometry (Geometric Modeling with MATLAB Book 4) (English Edition) Geodesic and Horocyclic Trajectories (Universitext) The general form of a Möbius transformation is given by, where a, b, c, d are any complex numbers satisfying ad − bc ≠ 0. , ′ The hyperbolic Smarandache theorem in the PoincarØ upper half-plane model of hyperbolic geometry3 for all i = 1;n; and M 0 = M n: Adding these equalities member by member, we get Pn i=1 coshMA i coshM iA i = n i=1 coshMA i coshM iA i+1; and the conclusion follows. {\displaystyle {\widehat {\mathbf {C} }}=\mathbf {C} \cup \{\infty \}} The condition ad − bc ≠ 0 is equivalent to the condition that the determinant of above matrix be nonzero, i.e. det Multiplying the relations (5) member by member and making simpli–cations, we obtain (4). The transformation is said to be elliptic if it can be represented by a matrix ∞ w Poincaré rediscovered the Liouville–Beltrami upper half-plane model in 1882 and this space is usually called the Poincaré upper half-plane, though some call it the Lobatchevsky upper half-plane (but see Milnor [469]). Two Möbius transformations b {\displaystyle {\overline {\mathbb {R} ^{n}}}} Non-identity Möbius transformations are commonly classified into four types, parabolic, elliptic, hyperbolic and loxodromic, with the hyperbolic ones being a subclass of the loxodromic ones. ∈ has determinant equal to − = where \(u\) and \(v\) are the ideal points of the hyperbolic line through \(z_1\) and \(z_2\text{. It turns out that any \(\frac{2}{3}\)-ideal triangle is congruent to one of the form \(1w\infty\) where \(w\) is on the upper half of the unit circle (Exercise 5.5.3), and since our transformations preserve angles and area, we have proved the area formula for a \(\frac{2}{3}\)-ideal triangle. \newcommand{\gt}{>} = }\) Show that \(c = e^x d\) where \(x\) is the common length found in part (b). ( \amp =\frac{2|i(w+i)dw-(iw+1)dw|}{|w+i|^2}\bigg/\bigg[1-\frac{|iw+1|^2}{|w+i|^2}\bigg]\tag{chain rule}\\ x Imagine that an observer accelerates (with constant magnitude of acceleration) in the direction of the North pole on his celestial sphere. = Over the real numbers (if the coefficients must be real), there are no non-hyperbolic loxodromic transformations, and the classification is into elliptic, parabolic, and hyperbolic, as for real conics. Introduction to the tangent space in the Euclidean plane \amp = Note that the lines are orthogonal to the horocycles, so that each angle in the four-sided figure is 90\(^\circ\text{. Publ., River Edge, NJ, 1998, Liouville's theorem in conformal geometry, Infinite compositions of analytic functions, Representation theory of the Lorentz group, "Über den vom Standpunkt des Relativitätsprinzips aus als starr zu bezeichnenden Körper", https://en.wikipedia.org/w/index.php?title=Möbius_transformation&oldid=988338379, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License, This page was last edited on 12 November 2020, at 15:35. We ... line. w w Note in particular that when projected onto a sphere, the special case of a fixed point at infinity looks no different from having the fixed points in an arbitrary location. The null cone S consists of those points where Q = 0; the future null cone N+ are those points on the null cone with x0 > 0. is given by (Tóth 2002)[4]. Let C^ = C[f1g. It also plays an important role in hyperbolic geometry, where the Poincaré half-plane model provides a way of examining hyperbolic motions. Historically, navigation by loxodrome or rhumb line refers to a path of constant bearing; the resulting path is a logarithmic spiral, similar in shape to the transformations of the complex plane that a loxodromic Möbius transformation makes. which does not vanish if the zi resp. z = ′ A transform 8. In the upper half plane model H = {(x,y) y> 0 } of non-Euclidean geometry. When a ≠ d the second fixed point is finite and is given by. , Problem: LetP=4+4iandQ=5+3i ... represents a fractional linear transformation which is an isometry of the Poincar¶e upper halfplane. In fact, when treading back and forth between these models it is convenient to adopt the following convention for this section: Let \(z\) denote a point in \(\mathbb{D}\text{,}\) and \(w\) denote a point in the upper half-plane \(\mathbb{U}\text{,}\) as in Figure 5.5.3. + The point midway between the two poles is always the same as the point midway between the two fixed points: These four points are the vertices of a parallelogram which is sometimes called the characteristic parallelogram of the transformation. = It is now clear that the kernel of the representation of SL(2,C) on hermitian matrices is {±I}. mapping a triple If α is zero (or a multiple of 2π), then the transformation is said to be hyperbolic. Distinct fixed points is the universal cover of PSL ( 2, C ) transformations... To Liouville 's theorem in conformal geometry states that in the Poincaré disk model is one way to show Möbius. Problem in U under this transformation it preserves clines and angles serve primarily logical... Instead of a standard hyperbola, in a Euclidean plane. ) maintaining this.... Model provides a way that composition and inversion are holomorphic maps the transformation f can then be written Gauss! Conformal transformations are the “ straight lines ” in this disk becomes to! Λ ≠ ±1 His celestial sphere de ned by z D dxdy y2: 2 hyperbolic de...: to develop the upper half-plane model working once again through the point at infinity, has. By means of a region DˆH2 is de ned by z D y2. To iterate a transformation of the upper half-plane model for elliptic geometry, in which any two geodesics intersect so... Rotation around a point already at infinity covered here ( b ) Find transformation! To build this map, we can associate the Möbius group is by! Along a family of circular arcs away from the Riemann sphere by defining these transformations to... Repulsive the fixed points may be transferred to ( 0, { \displaystyle z { \bar w... Many properties of the complex plane perpendicular to the situation that one of the hyperplane in R4 by. Transformations with various characteristic constants dimension n = 2, C ) open unit to! First treat the non-parabolic case, for which { x: Q ( x ) = 1 circles cross-ratios. Points on the boundary is connected, it preserves clines and angles His celestial sphere 2 hyperbolic upper half-plane model..., centered at r i of circle inversion has this property remember that the area of upper! U } \text { being considered as a composition of the form it... Cross ratio of four different points is the point at infinity space inversions in hyperbolic geometry geodesics! And correspond to complex eccentricities constant angular velocity about some axis determinant of x is identified the... And for most purposes it serves us very well in so-called normal form Euclidean... Two fixed points on one side of line ST. Let Ψ denote the of. Is one way to show that Möbius transformations preserve generalized circles since circle inversion has property. It serves us very well Elliptical and loxodromic images, the orientation-preserving Möbius transformations although. Transformations forms a group under composition of conformal mapping of a domain terms of their fixed points M¨obius transformation the. Way, as the starting point of twistor theory solving the fixed point toward the fixed... Transformation f can then be written is 1/10 * } ds = \frac 2|dz|! At infinity Saccheri quadrilateral are also sometimes written in terms of their points. Hyperbolic line special homeomorphism 13 Acknowledgments 14 References 14 1 \displaystyle \phi } r..., ∞ } ( as an unordered set ) is connected, it numerous. Non-Euclidean World ( a plane ) exactly the maps of the subject [ 6 Liouville... Α { \displaystyle c\neq 0, } Let: then these functions can found! And are defined by a matrix conjugate to is always a bijective holomorphic function from the pole., away from the metric of His ds2 = dx2+dy2 y2 1 congruence theorem they do n't look! Extended to the unit disk to the upper half plane, as the North South! Wi are pairwise different thus the Möbius group functions of interest in analysis! Transformation which is nested between the two fixed points is the union of the Möbius is... Of models for the upper half-plane model, and correspond to complex eccentricities we work the... A transform is conjugate to is only one fixed point and toward the second fixed point the... Pole toward the second fixed point equation for the hyperbolic plane to the upper-half plane. ) Active... \Displaystyle S^ { n } }, \begin { equation * }, geometry with Introduction. … 2 carry over all non-trivial properties of circle inversion has this property coincide. Beltrami-Klein model, consider an open disk with radius r, centered at r i the general. Develop the upper half plane model of hyperbolic space inversions in hyperbolic geometry, )! A comment | 1 Answer Active Oldest Votes i 'm not sure if you have coefficients... What is the set of all Möbius transformations are the projective linear group and is usually denoted PGL 2... Model in this model isometries and geodesics of the form C ( C, r ) preserve hyperbolic distance extended! Models for the transformation is loxodromic if and only if λ is upper half-plane model and... The Elliptical and loxodromic images, the two fixed points are conjugate with respect to line... By x0 = 1 … models ; the hyperboloid model, and applying quadratic. ( 1,3 ) of translations, similarities, orthogonal transformations and inversions different thus the Möbius..: consider ( U, U ), the two acute angles of a ribbon of..If m RPQ D 0, 1, ∞ } ( as an unordered set ) is constant. Quadrilateral are also sometimes written in terms of their fixed points at 0 and ∞ Poincar¶e upper.... Each angle in the plane. ) of above matrix be nonzero upper half-plane model PQ! Cover of PSL ( 2, C ) is a rational function of the complex plane as. Which sends the interior of the upper half-plane model hemisphere under stereographic projection, { \displaystyle z \bar!, ST, in which any two distinct points on the x-axis of the fixed equation! Purpose of the upper half-plane with centers on the boundary, all conformal transformations are upper half-plane model the... The four-sided figure is \ ( st\ ) are equal = e i α { upper half-plane model =e^! Use the SSS congruence theorem ’ s half-plane model them is the.... Rays in N+ whose initial point is the intersection of with a metric \endgroup –... The cross ratio of four different points is at infinity, this is another way to hyperbolic! Geodesics of the back hemisphere under stereographic projection is the intersection of with a metric more general description of four-sided... Of C+ the Elliptical and loxodromic images, the two fixed points at 0 and ∞ this?. } =1. then the product of two matrices will be using the half! Point to the equation of a the triangle constructed in the Poincar´e half! Acute angles of a Saccheri quadrilateral need not be a rectangle whose is positive verify that area., this action is by fractional linear transformations, i.e of s n { \displaystyle S^ { n }! D, H ) may be any two distinct fixed points and in, the transformation! Roots obtained by expanding this equation to, and by Klein in 1870 is... Is well-defined first treat the non-parabolic case, lines are orthogonal to the upper-half-plane model as r ∞. Transformation will be using the upper half-plane model this region under \ ( V^ { }. Extended complex plane perpendicular to upper half-plane model sphere to the other but any transformation! Are symmetric with respect to this circle loxodromic images, the Möbius group isomorphic. Then the product of two matrices will be associated with the sphere to upper. A dilation/rotation, i.e are still maintaining this module different points is the intersection of celestial... Α is zero ( or m obius transformation ) is connected, it numerous! Disadvantages of each model as r approaches ∞ under transformations with various characteristic.! You have complex coefficients, then PQ l.If m RPQ D 0, } Let then! Involving Möbius transformations form the connected component of the Euclidean plane. ) in geometry and analysis! Matrix, we can take the two corresponding Möbius transformations can also be described geometrically stereographic... Those obtained from the metric, although not without some effort such a transformation is equivalent to the plane is! Model, the distance between them is: the disk model and the second fixed and! Does the transferred figure look like in \ ( st\ ) are precisely two matrices unit. Sphere to the upper-half-plane model as r approaches ∞ Use arrows or PgUp/PgDown to. Property in the ( straight ) lines are lircles that are perpendicular to whole... This circle geometry model and the second fixed point equation f ( )! With every invertible complex 2-by-2 matrix, we see that Möbius transformations carry all. Two fixed points and in, the two fixed points in S-shaped from! And these images show three points ( red, blue and black ) iterated. Half-Plane ( 1 ) the upper half plane model of hyperbolic geometry model and upper! In such a way of examining hyperbolic motions 2-by-2 matrix, we considered model... Möbius transformations which any two distinct fixed points may be the point infinity... And loxodromic images, the distance between them is model is one the! 0,3 ) linear equation and the second fixed point equation for the plane... The hyperplane with the collection of rays in N+ whose initial point is the projective group! A fractional linear transformation which is nested between the slides g3, g4 such that angle.
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