Lecture Notes of Möbuis Transformation in Hyperbolic Plane

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1 Applied Mathematis, 04, 5, 6-5 Published Online August 04 in SiRes Leture Notes of Möbuis Transformation in Hyperboli Plane Rania B M Amer Department of Engineering Mathematis and Physis, Faulty of Engineering, Zagazig University, Zagazig, Egypt drraniaamer@yahooom Reeived 6 May 04; revised July 04; aepted 4 July 04 Copyright 04 by author and Sientifi Researh Publishing In This work is liensed under the Creative Commons Attribution International Liense (CC BY Abstrat In this paper, I have provided a brief introdution on Möbius transformation and explored some basi properties of this kind of transformation For instane, Möbius transformation is lassified aording to the invariant points Moreover, we an see that Möbius transformation is hyperboli isometries that form a group ation PSL (,R on the upper half plane model Keywords The Upper Half-Plane Model, Möbius Transformation, Hyperboli Distane, Fixed Points, The Group PSL (,R Introdution Möbius transformations have appliations to problems in physis, engineering and mathematis Furthermore, the onformal mapping is represented as bilinear translation, linear frational transformation and Mobius transformation Möbius transformations are also alled homographi transformations, linear frational transformations, or frational linear transformations and it is a bijetive holomorphi funtion (onformal map [] [] The purpose of this paper is studied the properties of Möbius transformations in detail, and some definitions and theorems are given The basi properties of these transformations are introdued and lassified aording to the invariant points Möbius transformations are formed a group ation PSL (,R on the upper half plane model A Möbius transformation of the plane is a map f: C C f ( z, a, b,, d C and ad b 0 (- whih sending eah point to a orresponding point, where z is the omplex variable and the oeffiients a, b,, d How to ite this paper: Amer, RBM (04 Leture Notes of Möbuis Transformation in Hyperboli Plane Applied Mathematis, 5, 6-5

2 are omplex numbers [3] Definition (- The upper half plane model is defined by the set and the boundary of is defined by { z Im ( z 0} { x iy y 0} > + > (- { z Im ( z 0} { } { x iy y 0} { } + (-3 The lines (geodesis are vertial rays and semiirles orthogonal to H The angles are Eulidean angles Definition (- A Möbius transformations form a group whih is denoted by Mob ( Remark (-3 Sine Möbius transformation takes the form f ( z d d a+ bz If the point z, this means f so f ( and we get the following: + d z a 0, z z f If 0 f (, d a 3 If 0 f & f ( Lemma (-4 A Möbius transformation onsists of four omposition funtions Proof The four funtions are: d d translation by f ( z z + ; inversion and refletion with respet to real axis f ( z, f z then the plane inside turn out and the lines on the plane are lines or irles and right angles stay true and also the irles are irles; ( ad b 3 dilation and rotation f3( z f ( z ; a a 4 translation by f4( z f3 ( z +, ad b a ad b + a f4 f 3 f f + (-4 Remark (-5 We an write Möbius transformations as follows then f ( z a ad ( + b The inverse Möbius transformation is evaluated from the inverse of the metri A a b d b d a A (-5 7

3 f ( z Theorem (-6 Möbius transformations also preserve ross ratio Proof Given four distint points z, z, z 3, z 4, their ross ratio is defined by ( z z3 z z4 Z Z Z3 Z4 z z z3 z4 ( z z ( z z dz b (-6 z + a, ;,, ;, 3 4 The ross ratio is invariant of the group of all Möbius transformation so if we transform the four points z i into z i by an inversion, the ross ratio of these points are taken into its onjugate value, and the ross ratio is invariant under a produt of two or any even number of inversions and exhanging any two pairs of oordinates preserves the ross-ratio Then ( z z z z ( z z z z ( z z z z ( z z z z ( z z3( z z4 ( z z ( z z, ;,, ;,, ;,, ;, 3 4 Sine translation, rotation and dilation preserve ross ratio and Möbius transformation onsists of them so Möbius transformation preserves ross ratio Corollary (-7 If z 0, i,,3, 4, we get and therefore i ( ZZ, ; Z, Z ( zz, ; z, z 3 3 If any one of z i 0 for example z 3 0, then z z z z z ( z z( z z3 ( z z ( z z ( z z z 4 ;, ; 3, 4, z z z z z 3 ( z z z 4 ;, ; 3, 4, 3 4 3, ;, ( z, z; z3, z4 z z z3 z4 z z z z z z z z ( z z z ( z z z, ;, ;, 4;,, ;0, Sine the trae of matrix A is tr(a a + b and this trae is invariant under onjugation, this is mean, tr ( gag tr ( A (-7 (-8 (-9 (-0 (- (- (-3 Every Möbius transformation an be represented by normalized matrix A suh that its determinant equal one whih mean ad b Lemma (-8 Two Möbius transformations A, B with det A det B are onjugate if and only if Poof Let tr ( A tr ( B (-4 A a a, B b b a a b b

4 Sine matrix A and B are Möbius transformations, then Sine det A det B, then If and only if tr ( A tr ( B det A aa aa, 4 3 det B bb bb 4 3 aa a a bb bb, ( A a + a ( A ( a + a tr tr, 4 4 ( B b + b ( B ( b + b tr tr 4 4 then matrix A and matrix B must be onjugate The Fixed Points in Mobius Transformation Sine fixed points (ie invariant points is defined by f(z z, then z z a d z b 0, then the fixed points are given by A Möbius transformation is f ( z This mean z, a d ± a d + 4b ( a d ± ( a + d 4( ad b ( a d ( A ± tr 4 (-5 For non paraboli transformation, there are two fixed points 0, but for paraboli transformation, there is only fixed points beause the fixed points are oinide 3 The Types of Mobius Transformations There are Paraboli, ellipti, hyperboli and loxodromi whih are distinguished by looking at the trae tr(a a + b 3 For Paraboli Transformations tr (A 4, the paraboli Möbius transformations forms subgroup isomorphi to the group of matries PSL, [4], ( a a, (-6 0 whih desribes a translation z z+ a and this transformation is orientation preserving 3 For Hyperboli Transformations tr ( A 4 ( PSL(,, the hyperboli Möbius transformations forms subgroup isomorphi to the group of matries whih desribes a rotation θ e 0 0 e θ θ z e z and this transformation is orientation preserving, (-7 9

5 33 For Ellipti Transformations 4 tr ( A 0 ( PSL(, whih desribes a rotation, the ellipti Möbius transformations forms subgroup isomorphi to the group of matries 34 For Loxodromi Transformations tr ( A [ 0, 4] ( PSL(, z iθ e 0 0 e iθ e i θ z and this transformation is orientation preserving, (-8, the Loxodromi Möbius transformations forms subgroup isomorphi to the group of matries, k 0 0, (-9 k whih desribes a dilation (homothety z kz and this transformation is orientation preserving The differene between orientation preserving (invariant and orientation reversing: Rotation and translation are orientation-preserving Refletion and glide-refletion are orientation-reversing 3 A omposition of orientation-preserving funtions is orientation-preserving 4 A omposition of two orientation-reversing funtions is orientation-preserving 5 A omposition of one orientation-preserving funtion and one orientation-reversing funtion is orientationreversing 6 The determinant of the matrix A (whih mentioned above then the orientation-preserving but if the determinant of the matrix A then the orientation reversing f z suh that ad b f z with ad b 7 is orientation-preserving but is orientation-reversing, where z x + iy whih mean the point z in the imaginary axis 8 In Orientation preserving all non ollinear points A, B, C, the proper angle measures of the angles ABC and A'B'C' have the same sign but in orientation reversing all non ollinear points A, B, C, the proper angle measures of the angles ABC and A'B'C' have opposite signs 9 Orientation preserving isometries takes ounterlokwise angles to ounterlokwise angles, and it takes lokwise angles to lokwise angles An orientation reversing isometries takes ounterlokwise angles to lokwise angles, and it takes lokwise angles to ounterlokwise angles 4 Isometries in Mobius Transformation Definition (4- The group PSL(, [4] is the projetive speial linear group of dimension over the real numbers and the determinant of the elements of that group may be or so PSL(, SL(, ± and this group at on osα sinα by Möbius transformations and also the matries of this group onjugate to the matrix sinα osα suh that α [ 0, π] from the Jordan and normal form of a real by matrix and therefore the determinants of these matries must equal, we an see that the absolute value of the traes ( tr a+ b of the matries will be respetively less than, alled ellipti, greater than, alled hyperboli, and equal to, alled paraboli Definition (4- Let γ ( t x( t + iy ( t be path so the hyperboli distane between two points (a, b on the upper half plane dx + dy t x ( t + y ( t with metri ds is defined by infimum of dt t y whih an be written as y 0

6 ( 0 ( + t x t y t hyp (, inf hyp ( γ inf d t d t t L t t y γ γ t t (-0 Remark (4-3 From this definition the geodesi between two points (x 0, y and (x 0, y on the vertial line with y > y has length ln(y /y but if two points do not lie on a vertial line so the geodesis is irular ar with enter on the x-axis as seen in Figure Remark (4-4 From the definition (- we an define the isometry of hyperboli plane as follows: Let a mapping f: and let A and B two points in, the mapping f is an isometry if the hyperboli distane d( AB, d( f( A, f( B Theorem (4-4 Möbius transformations at isometries in this mean PSL(, ats isometry on upper half plane by Möbius transformations Proof Möbius transformations preserve distane A bijetive map that preserves distane is alled an isometry beause an isometry is a transformation whih preserves distane Thus Möbius transformations are isometries of H A seond proof Sine the form of Möbius transformations are f ( z, differentiate this form yields to a ( dz f ( z dz df ( z dz ( ( Sine z z iy x + iy ( x iy z z iy z z Then ( z z ( z z dd z z dz dz dx + dy dx + dy y y (- From this equation we remark that Möbius transformations preserve the hyperboli metri so that Möbius transformations are hyperboli isometries A third proof From the definition of hyperboli distane, we want to show that Lhyp f ( γ ( t Lhyp ( γ ( t dz dzdz dx + d y, y Im z y Im z so the hyperboli metri is defined by Sine dx + dy dz dd z z ds, sine the right hand side y Im z z Im Figure The plane as boundary of half spae model of hyperboli spae

7 Let f ( z hyp γ, then ( γ ( x ( t + ( y ( t y( t ( γ L t dt Re( γ ( t Im ( γ ( t + Im dγdγ d t Im ( γ ( t azz + adz + bz + bd a z + adz + bz + bd γ dt (- (-3 and from ad b, then Sine the left hand side is Sine f ( γ Then hyp ( ( γ L f t aγ + b dγ d f( γ γ + d + d ( ( γ ( γ y Im( γ ( γ df ( γ ( γ (-4 d f (-5 0 Im and so df ( γ dγ ( γ + d d f ( γ df ( γ ( γ + d ( γ + d dγdγ γ + d dγdγ d d d 0 Im ( γ y (-6 γ + d y( t Im ( γ hyp dγ L f t t t t γ + d We get the left hand side equal the right hand side, and then the proof is omplete Lemma (4-5 Let Mobius transformations f ( z γ, then Proof The right hand side y Im γ Im z y ( γ dγ dz dγ dγ Im γ whih implies (-7 Im z Im dz Im z ( + + dγ a dz z d z d We get the left hand side equal the right hand side, and then the proof is omplete Remark (4-6 The group PSL(, ats on by Mobius transformation And therefore the left hand side

8 a b az b z + d This ation is faithful and and at disontinuously on so we an write Mob ( PSL(, Isom + Mob Isom + (-8 PSL, isomorphi to the group of all orientation preserving isometris of This mean whih preserve the hyperboli geometry of and therefore the elements in Möbius transformation at by isometries in [5] Theorem (4-7 All orientation-preserving isometries of are Mobius transformations, and all orientation-reversing isometries of are the omposition of a Mobius transformation and refletion through the imaginary axis Proof Isom whih identified with the group of Mobius transformations, and the group of orientation preserving isometries whih is the distane preserving maps are the Mobius transformations whih preserve and is denoted by Isom + ( whih identified with PSL(, PSL, ats on the boundary of the upper half plane by The isometry group of hyperboli plane is denoted by suh that and then, we get: a b az b z + d Mob ( : a, b,, d, ad b z + d a b : a, b,, d, ad b d : PSL, ( (-9 (-30 Let f(z is an isometry of, and by applying the transformations (rotation z kz and inversion z /z, we assume that :(, (, :(,0 (,0 g f z i i i i (-3 Let z, z be two points lie in positive imaginary axis Let the point z not lie in positive imaginary axis and draw two hyperboli irles with enter z and z and passing through z, we find these irles interset in z, z z and these irles are mapped into themselves under the isometry g f ( z so g f ( z z or z The first ase: If g f ( z z, we get f ( z suh that a, b,, d, ad b, whih is the orientation rez + d serving isometries is given by the map z z, that is the refletion in the imaginary axis and by omposition this with Möbius transformations This means all orientation-reversing isometries of are the omposition of a Mobius transformation and refletion through the imaginary axis suh that the refletions are isometries that have infinitely many fixed lie on the mirror line The seond ase: If g f ( z z, we get f ( z suh that a, b,, d, ad b, whih is the orientation prez + d serving isometries is given by the rotation z kz and inversion z /z This means all orientation-preserving isometry of are Mobius transformations and as we know Mobius transformations onsist of a rotation inversion and a translation Theorem (4-8 Möbius transformations preserve irles and lines (Figure Proof Let the transformation w /z is an inversion and every Möbius transformation (Figure 3 f(z of the form ( is a omposition of finitely many similarities and inversions [6]-[9] Sine w u + iv and z x + iy, then 3

9 Figure Cirle-preserving maps from the plane to itself Figure 3 Möbius transformation is omposition of multiple inversions From the equation of the irle x y u v u, v, x, y x + y x + y u + v u + v But if A 0, it is a line, if A 0, it is a irle (-3 A x + y + Bx + Cy + D 0 (-33 We an write again the Equation (-33 wrt u, v as follows A Bu Cv D( u v , whih is the equation of a irle If D 0, it is a line, if D 0, it is a irle So Möbius transformations preserve irles and lines Remark (4-9 From the last theorem (-5, we find that the irle goes through the origin may be mapped to the irle or the line Theorem (4-0 Möbius transformations preserve distane Proof From theorem (- Möbius transformations at isometries in and from definition of isometries we get that the distane between any two points in the hyperboli plane is invariant by Möbius transformations and Möbius transformations preserve irles (from translation and inversion and angles so Möbius transformations preserve distane 5 Conlusion The properties of Möbius transformations are introdued in detail, and some definitions and theorems are given to show that Möbius transformations are one-to-one, onto and onformal mapping Also, Möbius transformations map irles to irles and also, map the real line to the real line suh that the oeffiients a, b, and d are real Every orientation-preserving isometris of the hyperboli plane is Möbius transformations Every orientation-reversing isometris of the hyperboli plane is a omposition of Möbius transformations and refletion 4

10 Mob( is a group under omposition and Möbius transformations map the upper half-plane to itself bijetively So Möbius transformation maps vertial straight lines in and irles in with real enters to vertial straight lines and irles with real enters Furthermore, the onnetions between Möbius transformations, isometries of the hyperboli plane, and PSL(; R are presented Aknowledgements I wish to express my gratitude towards to Professor Dr William M Goldman, University of Maryland and Distinguished Sholar-Teaher Professor, Department of Mathematis, for his valuable, guidane, patiene and support I onsider myself very fortunate for being able to work with a very onsiderate and enouraging professor like him Referenes [] Yilmaz, N (009 On Some Mapping Properties of Möbius Transformations The Australian Journal of Mathematial Analysis and Appliations, 6, -8 [] Nehari, Z (95 Conformal Mapping MGraw-Hill Book, New York [3] John, O (00 The Geometry of Möbius Transformations University of Rohester, Rohester [4] Graeme, KO (0 Random Disrete Groups in the Spae of Möbuis Transformations Ms Thesis, Massey University, Albany [5] Aramayona, J (0 Hyperboli Strutures on Surfaes Leture Notes Series, IMS, NUS, 9 [6] Beardon, AF (995 The Geometry of Disrete Groups, Graduate Texts in Mathematis, 9 Springer-Verlag, New York [7] Jones, GA and Singerman, D (987 Complex Funtions, an Algebrai and Geometri Viewpoint Cambridge University Press, Cambridge [8] Lehner, J (964 Disontinuous Groups and Automorphi Funtions, Mathematial Surveys, 8 Amerian Mathematial Soiety, Providene [9] Seppälä, M and Sorvali, T (99 Geometry of Riemann Surfaes and Teihmüller Spaes, Chapter Elsevier Siene Publishing Company, INC, New York, -58 5

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