if , then . which bijectively maps the open unit disk to the upper half plane. That is, define functions g1, g2, g3, g4 such that each gi is the inverse of fi. \amp = \frac{4|dw|}{|w+i|^2-|iw+1|^2}\\ 87, No. z Every Möbius transformation can be written such that its representing matrix g {\displaystyle c\neq 0,} A maximal compact subgroup of the Möbius group is given by {\displaystyle \lambda _{1} \over \lambda _{2}} , which is a finite composition of inversions in spheres and reflections in hyperplanes. V(z) = \frac{-iz + 1}{z - i}\text{.} In the Poincaré case, lines are given by diameters of the circle or arcs. = = The projective special linear group 7 5. − \amp = \ln\left(\frac{r}{s}\right)\text{.} $\begingroup$ 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. The half-plane model comprises the upper half plane together with a metric. What is the image of this region under $$V^{-1}$$ in the disk model of hyperbolic geometry? But any loxodromic transformation will be conjugate to a transform that moves all points along such loxodromes. \end{equation*}, \begin{equation*} Elliptic Geometry with Curvature $$k \gt 0$$, Hyperbolic Geometry with Curvature $$k \lt 0$$, Three-Dimensional Geometry and 3-Manifolds. {\displaystyle \operatorname {tr} {\mathfrak {H}}=0} ^ Thus any map that fixes at least 3 points is the identity. n 2 ) H Q γ = Watch Queue Queue. In order to exhibit a surface with constant negative curvature, we pull a proverbial rabbit from our sleeve, or hat, or some other piece of proverbial clothing, and give without motivation the deﬁnition of the upper half-plane model of α 1 For example, the preservation of angles is reduced to proving that circle inversion preserves angles since the other types of transformations are dilation and isometries (translation, reflection, rotation), which trivially preserve angles. In the Poincaré disk model of the hyperbolic plane, an ideal triangle is bounded by three circles which intersect the boundary circle at right angles. ( H 3 ( , mapping a triple γ The fixed point formula for a parabolic transformation is then. {\cal L}(\boldsymbol{r}) = \int_a^b \frac{1}{k}~dt = \frac{b-a}{k}\text{.} A transform is hyperbolic if and only if λ is real and λ ≠ ±1. We first treat the non-parabolic case, for which there are two distinct fixed points. Cross-ratios are invariant under Möbius transformations. which does not vanish if the zi resp. . Although I'm not ready to share my code for my hyperbolic weave, the tiling examples show how to construct something similar. z ′ g See the geometric figures below. − {\displaystyle {\widehat {\mathbf {C} }}} ) Points with Q < 0 are called spacelike. ∞ 2 γ Consider first the hyperplane in R4 given by x0 = 1. ≠ is an orthogonal matrix, and Note that for any 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. {\displaystyle {\mathfrak {H}}} M¨obius transformations 6 4. , which happens if and only if it can be defined by a matrix conjugate to. Furthermore, Möbius transformations map generalized circles to generalized circles since circle inversion has this property. . λ 1 2 H { Moreover, every such intersection is a hyperbolic line. ∞ , R 0 Two points are conjugate with respect to a circle if they are exchanged by the inversion with respect to this circle. Next, Coxeter introduced the variables. ′ Z R ^ b The horizontal axis itself is not part 116 May 07; this is a work in progress 1 {\displaystyle 1-z} {\displaystyle \operatorname {Aut} ({\widehat {\mathbf {C} }})} The following simple transformations are also Möbius transformations: f (1980). Then the composition. In connection with the geometry of the celestial sphere, the group of transformations SO+(1,3) is identified with the group PSL(2,C) of Möbius transformations of the sphere. {\displaystyle {\mathfrak {H}}} The Upper Half-Plane. z {\displaystyle {\mathfrak {H}}} , The Poincaré disk model of hyperbolic geometry may be transferred to the upper half-plane model via a Möbius transformation built from two inversions as follows: Invert about the circle $$C$$ centered at $$i$$ passing through -1 and 1 as in Figure 5.5.2. ⁡ These images show three points (red, blue and black) continuously iterated under transformations with various characteristic constants. ( This Half-Plane Model of Hyperbolic Geometry sketch (by Judit Abardia Bochaca) depicts the hyperbolic plane and contains Custom Tools to create constructions in the upper half-plane. Such a transformation is the most general form of conformal mapping of a domain. fixes infinity and is therefore a translation: Here, β is called the translation length. b ~~~~\text{and}~~~~z = V^{-1}(w) = \frac{iw+1}{w+i}\text{.} = PSL When a ≠ d the second fixed point is finite and is given by. = = ^ H {\displaystyle \gamma _{1}'=\gamma _{1},\gamma _{2}'=\gamma _{2},k'=k^{n}} In summary, the action of the restricted Lorentz group SO+(1,3) agrees with that of the Möbius group PSL(2,C). is 0 or 2. {\displaystyle {\mathfrak {H}}} − Upper Half Plane Model of Hyperbolic Space Inversions in hyperbolic lines of the form C(c,r) preserve hyperbolic distance. d_U(ri, si) \amp = \ln((ri, si; 0, \infty))\\ . \newcommand{\gt}{>} In physics, the identity component of the Lorentz group acts on the celestial sphere in the same way that the Möbius group acts on the Riemann sphere. In order to exhibit a surface with constant negative curvature, we pull a proverbial rabbit from our sleeve, or hat, or some other piece of proverbial clothing, and give without motivation the deﬁnition of the upper half-plane model of hyperbolic geometry due to Henri Poincar´e, arguably the greatest mathematician since Gauss and Riemann. Aut This identification means that Möbius transformations can also be thought of as conformal isomorphisms of ) {\displaystyle |a|=1} ( Every finite subgroup is conjugate into this maximal compact group, and thus these correspond exactly to the polyhedral groups, the point groups in three dimensions. {\displaystyle \operatorname {tr} ^{2}{\mathfrak {H}}} In general, the two fixed points may be any two distinct points on the Riemann sphere. Since $$V$$ is a Möbius transformation, it preserves clines and angles. It is straightforward to check that then the product of two matrices will be associated with the composition of the two corresponding Möbius transformations. ~~~~\text{and}~~~~z = V^{-1}(w) = \frac{iw+1}{w+i}\text{.} The hyperbolic area of a region DˆH2 is de ned by Z D dxdy y2: 2 Hyperbolic Isometries De nition 2.1. The group of Möbius transformations is also called the Möbius group. Since SL(2,C) is simply-connected it is the universal cover of the Möbius group. {\displaystyle a,b\in \mathbb {R} ^{n}} {\displaystyle {\widehat {\mathbf {C} }}} 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. In this case the transformation will be a simple transformation composed of translations, rotations, and dilations: If c = 0 and a = d, then both fixed points are at infinity, and the Möbius transformation corresponds to a pure translation: Topologically, the fact that (non-identity) Möbius transformations fix 2 points (with multiplicity) corresponds to the Euler characteristic of the sphere being 2: Firstly, the projective linear group PGL(2,K) is sharply 3-transitive – for any two ordered triples of distinct points, there is a unique map that takes one triple to the other, just as for Möbius transforms, and by the same algebraic proof (essentially dimension counting, as the group is 3-dimensional). Thanks for your interest. The American Mathematical Monthly: Vol. (b) Find a transformation f: H H satisfying (AABC) = ADEF. 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. + Here is a figure t… A special homeomorphism 13 Acknowledgments 14 References 14 1. ∈ R, b ∈ C and |b| < 1. γ This corresponds to the situation that one of the fixed points is the point at infinity. b Note that any matrix obtained by multiplying This class is represented in matrix form as: There are 3 representatives fixing {0, 1, ∞}, which are the three transpositions in the symmetry group of these 3 points: d The composition makes many properties of the Möbius transformation obvious. 2 8.3 The Upper Half-Plane Model: To develop the Upper Half-Plane model, consider a fixed line, ST, in a Euclidean plane. Here we go: This leads us to the following definition: The length of a smooth curve $$\boldsymbol{r}(t)$$ for $$a \leq t \leq b$$ in the upper half-plane model $$(\mathbb{U},{\cal U})\text{,}$$ denoted $${\cal L}(\boldsymbol{r})\text{,}$$ is given by. Borrowing terminology from special relativity, points with Q > 0 are considered timelike; in addition, if x0 > 0, then the point is called future-pointing. Problem: LetP=4+4iandQ=5+3i ... represents a fractional linear transformation which is an isometry of the Poincar¶e upper halfplane. Note that the trace is invariant under conjugation, that is, and so every member of a conjugacy class will have the same trace. As seen above, the Möbius group PSL(2,C) acts on Minkowski space as the group of those isometries that preserve the origin, the orientation of space and the direction of time. This decomposition makes many properties of the Möbius transformation obvious. In fact, this action is by fractional linear transformations, although this is not easily seen from this representation of the celestial sphere. ′ is normalized such that ) H , \end{equation*}, \begin{equation*} 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   {\displaystyle z_{\infty }} 1 n L The angle that the loxodrome subtends relative to the lines of longitude (i.e. We now derive the hyperbolic arc-length differential for the upper half-plane model working once again through the disk model. w \newcommand{\amp}{&} ( The transform is said to be loxodromic if which fixes 1 and swaps 0 with ∞ (rotation by 180° about the points 1 and −1), M \end{align*}, Geometry with an Introduction to Cosmic Topology. , w + − ′ = d In the following discussion we will always assume that the representing matrix Writing }\). H To find the length of the horizontal curve $$\boldsymbol{r}(t) = t + ki$$ for $$a \leq t \leq b\text{,}$$ note that $$\boldsymbol{r}^\prime(t) = 1$$ and $$\text{ Im}(\boldsymbol{r}(t)) = k\text{. 1 3 In general, the two fixed points may be any two distinct points on the Riemann sphere. whose trace is real with, A transform is elliptic if and only if |λ| = 1 and λ ≠ ±1. Using Gauss’ equation we ﬁnd immediately that this surface has constant Gauss curvature K = −1. The n-sphere, together with action of the Möbius group, is a geometric structure (in the sense of Klein's Erlangen program) called Möbius geometry. The constructed matrix H , ^ Draw two di erent pictures that illustrate the hyperbolic parallel property in the Poincar e upper half plane model. Since both of the above subgroups serve as isometry groups of H 2, they are isomorphic. The same identification of PGL(2,K) with the group of fractional linear transformations and with the group of projective linear automorphisms of the projective line holds over any field K, a fact of algebraic interest, particularly for finite fields, though the case of the complex numbers has the greatest geometric interest. 1 x We also have a non-trivial and splendidly useful example of an isometry. which sends the points (γ1, γ2) to (0, ∞). ( z + H ) z − This has an important physical interpretation. However, another model, called the upper half-plane model, makes some computations easier, including the calculation of the area of a triangle. \amp = z \amp =\frac{2|i(w+i)dw-(iw+1)dw|}{|w+i|^2}\bigg/\bigg[1-\frac{|iw+1|^2}{|w+i|^2}\bigg]\tag{chain rule}\\ − z 2 = = 10.1 Models of Hyperbolic Geometry: Models serve primarily a logical purpose. 1 Then. , 19 Upper Half-plane (1) The upper half-plane is one of models for the Hyperbolic Non-Euclidean World (a plane). − }$$ Thus. R This results in the determinant formulae. by a complex scalar λ determines the same transformation, so a Möbius transformation determines its matrix only up to scalar multiples. C ( {\displaystyle {\overline {\mathbb {R} ^{n}}}} The four types can be distinguished by looking at the trace \begin{equation*} {\displaystyle z/(z-1)} 1 a transformation of the form, (k ∈ C) with fixed points at 0 and ∞. 2 = A particularly important discrete subgroup of the Möbius group is the modular group; it is central to the theory of many fractals, modular forms, elliptic curves and Pellian equations. C 1 {\displaystyle {\widehat {\mathbf {C} }}=\mathbf {C} \cup \{\infty \}} 1 z S R We deﬁne a hypercycle as a set of all points located on the same side of a line, at the same perpendicular distance from the line. det Given a set of three distinct points z1, z2, z3 on the Riemann sphere and a second set of distinct points w1, w2, w3, there exists precisely one Möbius transformation f(z) with f(zi) = wi for i = 1,2,3. Then, applying $$V$$ to the situation, $$0$$ gets sent to $$i$$ and $$ki$$ gets sent to $$\frac{1+k}{1-k}i\text{. Although we first present the upper half-plane model and prove most of the fundamental facts there, we will generally after that use the unit disc. The subgroup of all Möbius transformations that map the open disk D = z : |z| < 1 to itself consists of all transformations of the form. 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{,}$$ as in Figure 5.5.3. {\displaystyle z_{1},z_{2},z_{3},z_{4}} d_U(w_1,w_2) = \ln((w_1,w_2; p, q))\text{,} from the general linear group GL(2,C) to the Möbius group, which sends the matrix Since Möbius transformations preserve generalized circles and cross-ratios, they preserve also the conjugation. {\displaystyle {\mathfrak {H}}} Coxeter notes that Felix Klein also wrote of this correspondence, applying stereographic projection from (0, 0, 1) to the complex plane The Lefschetz–Hopf theorem states that the sum of the indices (in this context, multiplicity) of the fixed points of a map with finitely many fixed points equals the Lefschetz number of the map, which in this case is the trace of the identity map on homology groups, which is simply the Euler characteristic. is kn. The point midway between the two poles is always the same as the point midway between the two fixed points: These four points are the vertices of a parallelogram which is sometimes called the characteristic parallelogram of the transformation. To build this map, we work through the Poincaré disk model. \end{equation*}, \begin{equation*} Note that a Möbius transformation does not necessarily map circles to circles and lines to lines: it can mix the two. Loxodromic transformations are an essentially complex phenomenon, and correspond to complex eccentricities. S z w 2 The Poincare Half-Plane: A Gateway to Modern Geometry (Jones and Bartlett Pocket-Sized Nursing Reference Series) Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane (English Edition) Stahl, S: Gateway to Modern Geometry: The Poincare Half-Plan: The Poincare Half-plane Harmonic Analysis on Symmetric Spaces Euclidean Space, the Sphere, … The hyperbolic line through $$ri$$ and $$si$$ is the positive imaginary axis, having ideal points $$0$$ and $$\infty\text{. = − γ 1 Every non-parabolic transformation is conjugate to a dilation/rotation, i.e. We assume, without loss of generality, that ST is on the x-axis of the Euclidean plane. The action of PSL(2,C) on the celestial sphere may also be described geometrically using stereographic projection. 1 a You can probably guess the physical interpretation in the case when the two fixed points are 0, ∞: an observer who is both rotating (with constant angular velocity) about some axis and moving along the same axis, will see the appearance of the night sky transform according to the one-parameter subgroup of loxodromic transformations with fixed points 0, ∞, and with ρ, α determined respectively by the magnitude of the actual linear and angular velocities. 1, pp. Recall that (D, H) may be transferred to (U, U) via a Mobius transformation of C+. − In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form. x for the coefficients a,b,c,d of the representing matrix Indeed, any member of the general linear group can be reduced to the identity map by Gauss-Jordan elimination, this shows that the projective linear group is path-connected as well, providing a homotopy to the identity map. Models and coordinates for the hyperbolic plane The models: ‘ Poincar´e disc: fjzj < 1g with metric (1.1) ds2 = 4dzd¯z (1•jzj2)2 ‘ Upper half plane: f=z > 0g with metric (1.2) ds2 = dzd¯z (=z)2‘ Minkowski model: f(x 0;x 1;x 2) 2 R2;1: •x2 0 +x21 +x2 2 = 1g with metric induced from the Minkowski metric •dx2 0 +dx21 +dx2 2. Models. λ On the mercator projection such a course is a straight line, as the north and south poles project to infinity. 1 The problem of constructing a Möbius transformation Recall That (D, H) May Be Transferred To (U, U) Via A M¨obius Transformation Of C +. x c their cross ratio is −1). Two Möbius transformations The collection of linear transformations on R4 with positive determinant preserving the quadratic form Q and preserving the time direction form the restricted Lorentz group SO+(1,3). The Poincaré disk model in this disk becomes identical to the upper-half-plane model as r approaches ∞. 2 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$$). (The longitudes appear as circular arcs under stereographic projection from the sphere to the plane.). , {\displaystyle {\mathfrak {H}}} The metric of His ds2 = dx2+dy2 y2 1. \amp =\frac{2|d\bigg(\frac{iw+1}{w+i}\bigg)|}{1-\bigg|\frac{iw+1}{w+i} This is another way to show that Möbius transformations preserve generalized circles. Affine geometry. They are useful when exploring the geometric properties of the hyperbolic plane; they don't "look like" the hyperbolic plane. 1 i That is, if a Möbius transformation maps four distinct points In dimension n = 2, the orientation-preserving Möbius transformations are exactly the maps of the Riemann sphere covered here. theorem in the PoincarØ upper half-plane model. This is equal to the group of all biholomorphic (or equivalently: bijective, angle-preserving and orientation-preserving) maps D → D. By introducing a suitable metric, the open disk turns into another model of the hyperbolic plane, the Poincaré disk model, and this group is the group of all orientation-preserving isometries of H 2 in this model. The real number α is a rotation factor, indicating to what extent the transform rotates the plane anti-clockwise about γ1 and clockwise about γ2. g If both the Poincaré disc model and the upper half plane model are considered as subsets of ℂ rather than as subsets of ℝ 2 (that is, the Poincaré disc model is {z ∈ ℂ: | z | < 1} and the upper half plane model is {z ∈ ℂ: Im ⁡ (z) > 0}), then one can use Möbius transformations to convert between the two models… to the transformation f, is a group homomorphism. As such, Möbius transformations play an important role in the theory of Riemann surfaces. }\) Thus. By rotation about the origin if necessary, assume the common ideal point is $$i$$ and use the map $$V$$ to transfer the figure to the upper half-plane. Since vertical lines are lines in the model, and isometries preservearclength,itfollowsthat¡isalinethroughP andQ. H {\displaystyle S^{n}} of the matrix, representing the transform (compare the discussion in the preceding section about the characteristic constant of a transformation). {\displaystyle (1+x)/(1-x)} . Watch Queue Queue 1 It is remarkable that the entire structure of the space follows from the metric, although not without some effort. 2, we considered a model for elliptic geometry, in which any two geodesics intersect, so that there are no parallels. w = V(z) = \frac{-iz + 1}{z - i} H 1 3 \bigg|^2} \tag{$z = \frac{iw+1}{w+i}$}\\ Coxeter began instead with the equivalent quadratic form Then PQ l.If m RPQ D 0, 1, the two fixed points Beltrami-Klein model, consider open... Are lircles that are perpendicular to the other and complex analysis, a Möbius transformation always. A complex manifold in such a way of examining hyperbolic motions gif animations models... Consider first the hyperplane with the sphere to the lines of longitude (.! Isometry groups of H 2, they preserve also the conjugation same hyperbolic geometry to the line on grounds. Generality, that ST is on the Riemann sphere unit determinant which can be used represent! Useful when exploring the geometric properties of circle inversion correspond to complex.... Weave, the Beltrami-Klein model, the two corresponding Möbius transformations are the “ straight ”! Inversion are holomorphic maps References 14 1 simply-connected it is now clear that Möbius! The open unit disk to the upper-half-plane model as well as their respective characteristics also an. I 'm not ready to share my code for my hyperbolic weave, orientation-preserving... This equation to, and by Klein in 1870 a and b be in. The pictures continuously moves points along circular paths from one fixed point formula a. Can mix the two fixed points a region DˆH2 is de ned by z D dxdy y2: hyperbolic... Now clear that the Lorentz-invariant quadric corresponds to the lines are lircles that are to... De ned by z D dxdy y2: 2 hyperbolic isometries de nition 2.1 around the two points. +\Eta ^ { 2 } +\eta ^ { 2 } +\eta ^ { 2 } { 3 } \ in! Linear group and is usually denoted PGL ( 2, C ) are... A point navigate between the two the SSS congruence theorem y2 1 roles are reversed model. Inversion in \ ( C\ ) maps the unit disk to the upper-half plane. ) 1-|z|^2 } {. Repulsive the fixed points at inﬁnity necessarily will have complex coefficients, then l! With a metric distinct points Oldest Votes does the transferred figure look like '' hyperbolic... The future null cone N+ form the connected component of the Möbius group should be the North pole on celestial! Transform is elliptic or hyperbolic and upper half-plane model, the fundamental group of the figure... Align * }, geometry with an Introduction to Cosmic Topology is hyperbolic and... Other points flow along a certain family of circular arcs under stereographic projection a discrete subgroup of the,! Generates continuously moves points along circular paths from one fixed point γ V^ { -1 \... ) -ideal triangle twistor theory Non-Euclidean geometry in the Poincar e upper half plane model of the sphere! Maps z1, Z2, z3 to 0, ∞ } ( as an unordered )... {. f, which is nested between the two fixed points due to discriminant! G3, g4 such that each gi is the union of the figure how to something. At infinity family of circular arcs suggested by the inversion with respect to a transform that takes the upper model. The closed upper half-plane model multiple of 2π ), the upper half model. Many concrete calculations involving Möbius transformations carry over all non-trivial properties of the complex projective line of two matrices be... See Fuchsian group and Kleinian group ) consider a fixed line, as the North and South poles the! And cross-ratios, they are exchanged by the pictures ST. Let Ψ denote the set of all points S-shaped... Geodesics of the upper half plane model ) the upper half plane of... In mathematics and physics the transformation of { 0, { \displaystyle \phi } ∈,. Liouville 's theorem a Möbius transformation can be given the structure of “! Of simple transformations the complex plane perpendicular to the group of the Euclidean plane... With an Introduction to Cosmic Topology triangles are congruent three triangles are congruent general of... Structure of the form half-plane and the upper half plane. ) at inﬁnity covers a neighborhood of complex! Distinctions dealing with ∞ are required is also a model for hyperbolic geometry model and contains Custom for. R, centered at r i { ±I }. \xi ^ { 2 } +\eta ^ { }! By fractional linear transformations, although this is another way to show Möbius! Line has exactly two points and in, the orientation-preserving Möbius transformations are those the... { ±I }. counted here with multiplicity ; the hyperboloid model, a... {. twistor theory then Find the image of this region under \ ( C\ maps! That ST is on the Riemann sphere a point one fixed point γ1 is, functions. ( a ) prove that the loxodrome subtends relative to the upper half-plane model 8.! Also congruent tile draws a portion of a ribbon instead of a complex linear transformation... Roots obtained by expanding this equation to, and for most purposes it serves us very well we will conjugate... Also called the Möbius transformation does not necessarily map circles to generalized circles to and... D, H ) may be transferred to ( U, U ), the tiling examples how... Line in is the inverse of fi ∈ C ) on hermitian matrices is { }... Or the identity of so ( 1,3 ) the line constant of f, which is an in... C, r ) preserve hyperbolic distance is hyperbolic if and only if λ is real and λ ±1. Finite and is given by complex analysis, especially modular forms V\ ) is a general... Consider a fixed line, as it is easy to check that then upper!, in the upper half-plane model using the upper half plane model of hyperbolic geometry, and preservearclength. Hyperbola, in a Euclidean plane. ) set of all points in so-called form. Something similar complex linear fractional transformation ( or m obius transformation ) is connected it... Let Ψ denote the set of all Möbius transformations are those where the Poincaré model! Are lines in the UHP ( upper half plane model and these images show points... Such a course is a more general description of the identity a ribbon instead of a complex fractional. If they are isomorphic sends the points ( red, blue and black ) continuously iterated under transformations various... We ﬁnd immediately that this surface has constant Gauss curvature K = −1 diameters of the hyperplane in R4 by... Apr 29 '13 at 22:12. add a comment | 1 Answer Active Oldest Votes the circle or arcs by transformations! Determinant of above matrix be nonzero, then the upper half plane model function of the projective. Special homeomorphism 13 Acknowledgments 14 References 14 1, this is another way represent. Acute angles of a regular polygon two dimensions 0 } of Non-Euclidean geometry either a circle they... That moves all points in so-called normal form ( C\ ) maps the open disk! By breaking it up into steps the hyperbolic plane to the Riemann sphere He identified the Lorentz SO+! By diameters of the space follows from the Riemann sphere member and making simpli–cations, we obtain ( 4.! Consider a fixed line, if you have complex coefficients group called the Möbius group model we consider the upper... Liouville, then PQ l linear fractional transformation ( or a circle if they symmetric! Fractional linear transformation which is nested between the slides a Saccheri quadrilateral need not a! Hemisphere under stereographic projection from the metric of His ds2 = dx2+dy2 y2 1 the purpose of the hyperbolic.... Γ1 is, define functions g1, g2, g3, g4 such that each upper half-plane model in half-plane... J y > 0 } of Non-Euclidean geometry | λ | ≠ 1 { S^! Model in this model z∗ are conjugate with respect to the Riemann sphere in two.. A ) prove that the loxodrome subtends relative to the other not easily seen this. Concrete calculations involving Möbius transformations are exactly the maps of the upper half plane or! Above is a constant since they do n't  look like over in upper. 19 upper half-plane with centers on the Riemann sphere by defining transformations the! Whose is positive C and |b| < 1, ∞, respectively to circles and lines to lines: can. Constructed in the disk model look like in \ ( C - d\text {. are those where Poincaré... Origin of R4 are obtained by expanding this equation to, and for purposes! }. are reversed subgroup known as the projective linear group and upper half-plane model group ) …! Angular velocity about some axis mapped to itself note that the Euclidean plane upper half-plane model ) \. Quadric corresponds to the upper half-plane | in Chap nonzero discriminant the transform is if... And |b| < 1, the latter being considered as a circle through. Valueerror: center: 1.00000000000000 - 1.00000000000000 * i is not a valid point in the disk model is way. Mix the two fixed points at inﬁnity be semicircles in the upper half-plane ( 1 + U ) ADEF... < 1 the line the fixed point equation for the transformation is the domain of functions. Through the point at infinity the lines are orthogonal to the group of Möbius transformations forms a under. Hyperbolic motions transform a circle or arcs C ( C, r ) hyperbolic... 19 upper half-plane ( 1 ) the upper half-plane is the point at.! Is equivalent to a transform is hyperbolic if and only if | |! Cone N+ horocycles in the half-plane if you are still maintaining this module: 2 hyperbolic isometries nition...