ON THE FIXED POINTS OF AN AFFINE TRANSFORMATION: AN ELEMENTARY VIEW

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1 ON THE FIXED POINTS OF AN AFFINE TRANSFORMATION: AN ELEMENTARY VIEW Abstrct. This note shows how the fixed points of n ffine trnsformtion in the plne cn be constructed by n elementry geometric method. The pproch presented here lso shows how the ffinities of the plne cn be clssified with the help of their fixed points. 1. Introduction Every mthemtics mjor college student hs to be known the fixed point theorem of the similrities of R n : if similrity of R n is not n isometry, then it hs unique fixed point. Perhps most of them dds lso the rgument: the sttement is simple consequence of the fmous Bnch fixed point theorem for contrctions in complete metric spce. However, the fixed points not only of similrities but lso the broder clss of ffine trnsformtions in R n cn be studied with more elementry tools. This note shows how the fixed points of n ffine trnsformtion in the plne cn be constructed by n elementry geometric method. I lerned the bse ide from [1], where n elementry ruler construction is given for the fixed point of similrity in the plne. However, this construction cn be esily pplied for ffine trnsformtions of the plne, too. Perhps this fct ws well-known in the golden ge of clssicl projective geometry, but its erliest source I found in [6]. This pproch lso shows how the ffinities of the plne cn be clssified with the help of their fixed points. I hve successfully presented this tretment severl times, in geometry course for mthemtics mjors who hve studied liner lgebr. 2. Affine trnsformtions Bsic properties of ffine trnsformtions re well-known (reference [5] is good nlytic introduction), however, for the reders convenience, I recll some bsic notions. ZDM Subject Clssifiction. G05, G45, G75. Key words nd phrses. Fixed point theorem, ffine trnsformtion, equiffine trnsformtion. 1

2 2 A bijective trnsformtion F : R n R n is clled ffine, if (1) t R, x, y R n : F (tx + (1 t)y) = tf (x) + (1 t)f (y), or equivlently F = τ v φ for unique liner utomorphism φ Gl(R n ), nd for unique trnsltion τ v : R n R n, x x + v, (v R n ). φ is clled the liner prt of F. A vector x R n is fixed point of F (i.e., F (x) = x) if nd only if φ(x) + v = x, or in more illuminting form, if (2) (φ id)(x) = v. (2) cn be recognized s system of liner equtions, so in the solvble cse rnk r of the liner opertor φ id sys everything on the structure of fixed points, nmely the set of ll fixed points is n ffine subspce of dimension n r in R n. Of course in cse of r = n one does not need to ssume the solvbility of (2), nd one gets immeditely the fixed point theorem of ffine trnsformtions: Theorem 1. An ffine trnsformtion F = τ v φ hs unique fixed point if nd only if φ id is n isomorphism of R n. Moreover, the unique fixed point is (φ id) 1 ( v). Corollry 1 (Fixed point theorem of similrities). If similrity of R n is not n isometry, then it hs unique fixed point. Proof. Let F be similrity with mgnifiction fctor 1 λ > 0. The liner prt of F is λψ, where ψ is n orthogonl trnsformtion of R n. We need only to show tht the kernel of λψ id contins the zero vector lone. If x R n is vector in the kernel of λψ id, then λψ(x) = x, therefore λ ψ(x) = x. Since ψ preserves the norm, we hve (λ 1) x = 0, hence x = 0. The fundmentl theorem of the plne-ffine geometry sttes tht ny two tringles (or equivlently, ny two prllelogrms) of the plne cn be relted by unique ffine trnsformtion. This theorem enbles us to define plne ffine mp through the ssignment of the relted tringles. For exmple, given line t nd two points p, q off t, there is unique ffine trnsformtion tht leves every point on t fixed, nd sends p to q. This ffine trnsformtion is denoted by [t, p q], nd is clled n xil ffine trnsformtion. 3. The construction Herefter for x y R 2 the line tx+(1 t)y (t R) will be denoted by xy, while the segment tx + (1 t)y (t [0, 1]) by [xy].

3 ON THE FIXED POINTS OF AN AFFINE TRANSFORMATION 3 s m q d c r d c r d c b b p q b c d b p s d c r b p b c q d s Figure 1. Ruler construction for fixed points of n ffine trnsformtion in the plne Theorem 2. Let F : R 2 R 2, z F (z) = z be n ffine trnsformtion of the plne. Let (, b, c, d) be prllelogrm i.e., let, b, c, d R 2 non-colliner points such tht b = c d. Suppose, moreover, tht rnk(b, b ) = rnk(d, d ) = 2. Let, finlly, b b = p, dc d c = r, bc b c = q, d d = s. With these nottions, we hve: (1) if F hs unique fixed point m, then m = pr qs, (2) if F hs no fixed point, then pr qs but pr qs, (3) if F hs pointwise-fixed line, then this line is pr = qs, (Figure 1). I note tht the proof of Theorem 2 in [6] computes the equtions of lines pr nd qs in specil coordinte-system. From these equtions system of liner equtions is formed, nd the equivlence of this system nd (2) is proven. The following pure nlytic proof differs from the

4 4 one presented in [6], nd tries to grsp the geometric ide of the proof in [1]. Proof. The third sttement nd its converse is obvious. If F is n xil ffine trnsformtion, then the corresponding lines meet ech other on the xis. Conversely, if pr = qs then for some t R we hve (3) p = t + (1 t)b. Since cb d nd c b d, (3) implies: p = ts + (1 t)q = p = t + (1 t)b = p = p, nd nlogously r = r. Thus the line pr = qs is pointwise fixed. Now we prove the first sttement. From the definition of points p nd r, it follows tht there exist α, β, γ, δ R, such tht: (4) (5) tht is (6) (7) From (6) nd (4) we obtin tht p = α + (1 α)b = β + (1 β)b r = γd + (1 γ)c = δd + (1 δ)c, p = α + (1 α)b r = γd + (1 γ)c. (8) p p = (α β)( b ). Similrly, from (7) nd (5) we get (9) r r = (γ δ)(d c ). Since (, b, c, d ) is prllelogrm, it follows tht (10) b = d c = x. For some t R, multiply (8) by t nd (9) by (1 t). This leds to (11) (12) tp tp = t(α β)x, (1 t)r (1 t)r = (1 t)(γ δ)x. Now, dding (11) nd (12) we obtin (13) tp + (1 t)r = tp + (1 t)r + [t(α β) + (1 t)(γ δ)]x. Observe tht if φ id is liner utomorphism, then we cn determine t such tht (14) t(α β)+(1 t)(γ δ) = 0 t[(α β) (γ δ)] = (γ δ). Indeed, rguing by contrdiction, if (α β) (γ δ) = 0 then from (8) nd (9) we get p p = r r = φ(p) p = φ(r) r = (φ id)(p r) = 0,

5 ON THE FIXED POINTS OF AN AFFINE TRANSFORMATION 5 which is impossible. With formul (13) reduces to γ δ t = (α β) (γ δ), (15) tp + (1 t)r = tp + (1 t)r. The point so obtined is denoted by m. Now pplying F to m we get F (m) = F (tp (1 t)r) = tp + (1 t)r = m. It mens tht the fixed point m is on the line pr, nd nlogously on qs. Thus the first sttement is proved. Further nlysis of (13) leds to the second sttement. If α β = γ δ = 0, then from (8) nd (9) we get p = p nd r = r, mening tht F leves p nd r fixed. If n ffine trnsformtion leves fixed two distinct points, then it leves fixed every point on the joining line of these points. This is cse 3, so if F hs no fixed point, then α β = γ δ 0. From (8) nd (9) we get p p = r r φ(p r) = p r φ(p r) (p r) = 0 p r Ker(φ id). Anlogously, q s Ker(φ id). Since dim Ker(φ id) = 1, it follows tht q s p r. This is the second sttement. 4. Clssifiction of ffine trnsformtions of the plne A simple consequence of the fundmentl theorem of plne-ffine geometry is the following: every ffine trnsformtion in the plne is the product of similrity nd n xil ffine trnsformtion. Nmely, let the ffine trnsformtion F be given by (p, q, r) (p, q, r ). Let χ denote similrity for which χ(p) = p nd χ(p) = q. Then F is the product of χ nd the xil ffine trnsformtion [p q, χ(r) r ]. If F hs unique fixed point, then it cn be chosen s p, thus p = p. Theorem 3. If ffine trnsformtion of the plne hs unique fixed point p, then it is the product of rottion round p, centrl dilttion with center p nd n xil ffinity, where p is on the xis. If there re no fixed points, the similrity reduces to trnsltion, nd we hve Theorem 4. Every ffine trnsformtion of the plne without fixed points is the product of trnsltion nd n xil ffine trnsformtion.

6 6 Proof. As consequence of the fixed point theorem, in this cse dim Ker(φ id) 1. Let q p Ker(φ id), (p, q R 2, p q). Then F (q) F (p) = q p, therefore (p, q) cn be trnslted to (F (p), F (q)). The trnsltion, determined by p F (p), is denoted by τ. Now the required ffine trnsformtion cn be given s (p, q, r) (F (p), F (q), F (r)). This ffinity is the product of the trnsltion τ nd the xil ffine trnsformtion [F (p)f (q), τ(r) F (r)] (Figure 2.) τ(r) F (r) r F (p) = τ(p) F (q) = τ(q) p q Figure 2. An ffine trnsformtion without fixed point is the product of trnsltion nd n xil ffinity 5. The equiffine cse An ffine trnsformtion of the plne tht preserves the re is clled equiffine trnsformtion. Let p, q, r be non-colliner points in R 2. The unique ffine trnsformtion tht leves p fixed, while interchnges q nd r, is clled n ffine reflection nd denoted by [p, q r]. In ddition, let s be the midpoint of the segment qr. The bove ffine reflection leves fixed every point of ps but no other points. Clerly, every Eucliden reflection is n ffine reflection. Let p, q, r be non-colliner points in R 2. The unique ffine trnsformtion leving every point fixed on the line through p nd prllel to qr nd tht tkes q to r, is denoted by [p; q r] nd is clled sher. The sher [p; q r] hs the line through p prllel to qr s its set of fixed points. From the fixed point-construction one cn derive simple proof of Veblen s well-known theorem ([7, Theorem 49 of Chp. III]): Theorem 5 (Veblen). Every equiffine trnsformtion of the plne is the product of two ffine reflections.

7 ON THE FIXED POINTS OF AN AFFINE TRANSFORMATION 7 Proof. If the equiffine trnsformtion of the plne differs from trnsltion, hlf-turn or sher, then it cn be given by (, b, c, d) (b, c, c, d ), d = d c, where (, b, c, d) is prllelogrm (see [1, Chp. 13.4]). The fixed point construction is illustrted by Figure 3. Let n be the midpoint of the segment [b]. Join n with the unique fixed point m, or in the fixed point free cse drw prllel line through n with pr (qs). This line is denoted by t. In the cse of the unique fixed point, the ffinity cn be given by (, b, m) (b, c, m), moreover in the fixed point free cse by (, b, d) (b, c, d ). In both cses the ffinity is the composition of the ffine reflections [t, b] nd [p, c]. m t d = r d = s c n b = = p c = b = q t d = r d = s c n b = = p c = b = q Figure 3

8 8 The cse of sher cn be esily hndled: the sher [p; q r] is just the product of the ffine reflections [p; q s] nd [p; s r], where s is on the imge of qr by the hlf-turn with center p. A trnsltion or hlf-turn is the product of two Eucliden reflections, so the sttement is obvious. The equiffine geometry nd the equiffine group hs been extensively studied. A recent pproch to the geometric xiomtiztion of equiffine geometry is presented in pper [4]. A higher dimensionl generliztion of Veblen s theorem cn be found in [2]. Acknowledgments I m deeply indebted to P.T. Ngy (University of Debrecen, Hungry) for inspiring converstions on this topic. I lso pprecite the help of J. Horváth (University of Sopron, Hungry) t very erly stge of this pper. His joint work with M. Holli [3] seems to me the closest in spirit to my own view. I m lso grteful to the referees for their comments which improved the finl ppernce of this pper. References [1] H. S. M. Coxeter. Introduction to Geometry. John Wiley & Sons, Inc., Second edition, [2] E. W. Ellers. Decomposition of equiffinities into reflections. Geometrie Dedict, 6: , [3] M. Holli nd J. Horváth. Affine trnsformtions in the plne nd in the spce. In Studies on Mthemticl Eduction, pges Fculty of Science, ELTE, Hungry, Hungrin. [4] V. Pmbuccin. Wht is plne equiffine geometry? Aequtiones Mthemtice, 66:90 99, [5] P. J. Ryn. Eucliden nd non-eucliden Geometry. Cmbridge University Press, [6] Z. A. Skopec. On fixpoints of ffine mppings. Mtemtik v skole, 3:5 10, Russin. [7] O. Veblen nd J. W. Young. Projective Geometry. Vol. I., II. Blisdell Publishing Compny, Zoltán Kovács, Institute of Mthemtics nd Computer Science, College of Nyíregyház, 4401 Nyíregyház, Pf. 166., Hungry E-mil ddress: kovcsz@nyf.hu