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Chapter 13 the Geometry of Circles

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Chapter 13 the Geometry of Circles Chapter 13 TheR. Connelly geometry of circles Math 452, Spring 2002 Math 4520, Fall 2017 CLASSICAL GEOMETRIES So far we have been studying lines and conics in the Euclidean plane. What about circles, 14. The geometry of circles one of the basic objects of study in Euclidean geometry? One approach is to use the complex numbersSo far .we Recall have thatbeen the studying projectivities lines and of conics the projective in the Euclidean plane overplane., whichWhat weabout call 2, circles, Cone of the basic objects of study in Euclidean geometry? One approachC is to use CP are given by 3 by 3 matrices, and these projectivities restricted to a complex projective the complex numbers 1C. Recall that the projectivities of the projective plane over C, line,v.rhich which we wecall call Cp2, a CPare, given are the by Moebius3 by 3 matrices, functions, and whichthese projectivities themselves correspondrestricted to to a 2-by-2 matrices.complex Theprojective Moebius line, functions which we preservecall a Cpl the, are cross the Moebius ratio. This functions, is where which circles themselves come in. correspond to a 2 by 2 matrix. The Moebius functions preserve the cross ratio. This is 13.1where circles The come cross in. ratio for the complex field We14.1 look The for anothercross ratio geometric for the interpretation complex field of the cross ratio for the complex field, or better 1 yet forWeCP look= Cfor[f1g another. Recall geometric the polarinterpretation decomposition of the cross of a complexratio for numberthe complexz = refield,iθ, where r =orjz betterj is the yet magnitude for Cpl = ofCz U, and{ 00 } .Recallθ is the the angle polar that decomposition the line through of a complex 0 and znumbermakes with thez real -rei9, axis. v.,here See Figurer = Izi 13.1.is the magnitude of z, and 8 is the angle that the line through O and z makes ,vith the real axis. See Figure 14.1.1. Figure 14.1.1 Figure 13.1 %1 -%2 So if we have 3 complex numbers z1, z2=, and1~\ei81 z4, the polar decomposition of the ratio %1 -%4 \ %1 -%4\ z − z z − z 1 2 1 2 iθ1 the angle 61 can be interpreted as the angle= between thee vectors %2-%1 and Z4 -%1 as in z1 − z4 z1 − z4 Figure 14.1.2. 1 2 CHAPTER 13. THE GEOMETRY OF CIRCLES MATH 4520, FALL 2017 for the angle θ1 can be interpreted as the angle between the vectors z2 − z1 and z4 − z1 as in Figure 13.2. 2 CLASSICAL GEOMETRIES 2 CLASSICAL GEOMETRIES Figure 14.1.2 Figure 13.2 So we can write the cross ratio r of the 4 points, Zl, Z2, Z3, Z4, as follows: Figure 14.1.2 So we can write the cross ratio r of the 4 points, z1, z2, z3, z4, as follows: So we can writeZI the-Z2 cross ratio%3 -%4r of the= 4 ~points,Zl -Z2 Zl,~ eZ2, i81 Z3,~Z3 Z4,-Z4 as follows:~ei83 r= ZI -Z4z1 − z2 %3 z-%23 − z4 Zl -Z4z 1 − z2 Z3 -Z2z3 − z4 r = = eiθ1 eiθ3 ZI -Z2 %3 -%4 = ~Zl -Z2 ~e i81 ~Z3 -Z4 ~ei83 r= z1 − z4 z3 − z2 = I Zl -Z211 z1 − zZ34 -Z41 z3ei(81+83)− z2 ZI -Z4 %3 -%2 Zl -Z4 Z3 -Z2 I Zl -z411 z1 − zZ32 -z21z3 − z4 i(θ +θ ) = =I Zl -Z211 Z3 -Z41 ei(81+83) e 1 3 ; z1 − z4 z3 − z2 I Zl -z411 Z3 -z21 where 83 is the angle at Z3 in the quadrilateral determined by the 4 points Zl, Z2, Z3, Z4. where θ is the angle at z in the quadrilateral determined by the 4 points z , z , z , z . See Figurewhere 14.1.3. 833 is the angle at Z3 3in the quadrilateral determined by the 4 points Zl, Z2, Z3,1 Z4.2 3 4 See FigureSee Figure 13.3. 14.1.3. Figure 14.1.3 FigureFigure 14.1.3 13.3 We conclude with a result that connects our complex geometry to circles. We conclude with a result that connects our complex geometry to circles. Theorem 14.1.1: The 4 distinct points %1, %2, %3, %4 in the complex projective line have TheoremWe a real conclude14.1.1: cross ratio withThe if 4 aand distinct result only thatif pointsthey connects all %1,lie on%2, oura %3,single complex %4 circlein the geometryor complexEuclidean to projective circles.line. line have a real crossProof. ratio From if theand discussiononly if theyabove all Zl, lieZ2, on Z3, aZ4 single have a circlereal crossor Euclideanratio if and onlyline. if 81 + 83 Theorem 13.1.1. The 4 distinct points z1, z2, z3, z4 in the complex projective line have a real crossis an integral ratioif multiple and only of ifn. theyBut alla result lieon from a singleEuclidean circle geometry or Euclidean says that line. 81 + 83 is an Proof. Fromintegral the multiple discussion of n ifabove and only Zl, ifZ2, zl, Z3,z2, Z4Z3, have Z4 all alie real on across single ratiocircle ifor andEuclidean only ifline. 81 + 83 is an integral multiple of n. But a result from Euclidean geometry says that 81 + 83 is an Proof.CorollaryFrom the14.1.2: discussion The image above ofaz 1,circlez2, zor3, Euclideanz4 have aline real under cross a Moebius ratio if andfunction only f ifofθ1 + θ3 integral multiple of n if and only if zl, z2, Z3, Z4 all lie on a single circle or Euclidean line. is anthe integral complex multiple projective of lineπ. Butis a circle a result or Euclidean from Euclidean line. geometry says that θ1 + θ3 is an integral multiple of π if and only if z , z , z , z all lie on a single circle or Euclidean line. CorollaryProof. 14.1.2: Choose The your imagefavorite ofa circle circle 1or 2Euclideanor 3Euclidean4 line andline fix under three adistinct Moebius points function Zl, z2, f of the CorollarycomplexZ3, on projective it. 13.1.2. A fourth Theline point imageis aZ4 circle lies of aon or circle that Euclidean circle or Euclidean or line.Euclidean line line under if and a Moebiusonly if the function cross f of ratio of Zl , Z2, Z3, z4 is real. Similarly, the cross ratio of f(Zl)' f(Z2), f(Z3), f(Z4) is real if Proof.the complexChoose projectiveyour favorite line iscircle a circle or Euclidean or Euclidean line line. and fix three distinct points Zl, z2, Z3, on it. A fourth point Z4 lies on that circle or Euclidean line if and only if the cross ratio of Zl , Z2, Z3, z4 is real. Similarly, the cross ratio of f(Zl)' f(Z2), f(Z3), f(Z4) is real if " " 13.2. INVERSION 3 Proof. Choose your favorite circle or Euclidean line and fix three distinct points z1, z2, z3, on it. A fourth point z4 lies on that circle or Euclidean line if and only if the cross ratio of z1, z2, z3, z4 is real. Similarly, the cross ratio of f(z1), f(z2), f(z3), f(z4) is real if and only if f(z4) lies on the circle or Euclidean line determined by f(z1), f(z2), f(z3). By Lemma 13.6.1 (the invariance of the cross ratio), the cross ratio of z1, z2, z3, z4 and f(z1), f(z2), f(z3), f(z4) are the same. So z4 lies on the circle or Euclidean line through z1, z2, z3 if and only if f(z4) lies on the circle or Euclidean line through f(z1), f(z2), f(z3). THE GEOMETRY OF CIRCLES 3 and only if f(%4) lies on the circle or Euclidean line determined by f(%l), f(%2), f(%3). By 13.2Lemma Inversion 13.5.1 (the invariance of the cross ratio), the cross ratio of %1,%2, %3, %4 and f(%l), f(%2), f(%3), f(%4) are the same. So %4lies on the circle or Euclidean line through %1,%2, Recall from%3 Chapterif and only 13if f(%4) that lies any on the Moebius circle or Euclidean function line can through be regardedf(%l), f(%2), asf(%3). the composition of translations, multiplications by a constant, and taking the multiplicative inverse. Consider the Moebius14.2 function Inversionf(z) = 1=z. For the sake of tradition and for the sake of understanding the function more simply, we define a slightly different function. We call inversion the Recall from Chapter 13 that any Moebius function can be regarded as the composi- function definedtion of bytranslations, multiplications by a constant, and taking the multiplicative inverse. Consider the Moebius function j(z) =β l/z.(z )For = 1the=z:¯ sake of tradition and for the sake of understanding the function more simply, we define a slightly different function. We call Since complexinver.sion conjugation the function isdefined just aby rigid reflection about the real axis, β takes circles and .8(z) = l/z. Euclidean lines to circles and Euclidean lines as well. Note thatSincejzj complex2β(x) =conjugationzz=¯ z¯ = isz .just So a zrigidand reflectionβ(z) areabout on the a real ray axis, from .8 takes the circles origin. and When jzj = 1, Euclidean lines to circles and Euclidean lines as well. then β(z) = zNote. Inversion that IzI2.8(x) is like = zz/z a “reflection”= z. So z and about .8(z) are a on circle. a ray Figurefrom the 13.4origin. showsWhen the inversion of some linesrzl and= 1, then circles. .8(z) = z. Inversion is like a "reflection" about a circle.
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