ELEMENTARY CONSTRUCTIONS IN THE HYPERBOLIC PLANE


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1 PROCEEDINGS 13th INTERNTIONL CONFERENCE ON GEOMETRY ND GRPHICS ugust 48, 2008, Dresden (Germny) ISBN: ELEMENTRY CONSTRUCTIONS IN THE HYPERBOLIC PLNE Syille MICK Grz University of Tehnology, ustri BSTRCT: Construtions of regulr ngons in the Poinré disk model nd in the BeltrmiKlein model of the hyperoli geometry re presented. The fous is on methods tht n e rried out y hnd using only Euliden ompss, Euliden protrtor nd strightedge (i.e. ruler without mesuring mrks). regulr ngon n e deomposed into 2n rightngled tringles. This is why we n redue the onstrution of regulr ngon to the onstrution of rightngled tringle. Keywords: Elementry Euliden Geometry, Hyperoli Plne, BeltrmiKlein Model, Poinré Model, Geometry of Cirles. 1. INTRODUCTION Hyperoli geometry nd espeilly its onstrutive spets hve reently een reonsidered (see [1], [5] nd [9]). One reson my e the ft tht some omputer progrmmes like Cinderell [8] provide onstrutions of tringles, regulr polygons nd even tilings in the hyperoli plne with few mouse liks. Figure 1: Regulr pentgons in the BeltrmiKlein model nd the Poinré disk In Cinderell we n hoose oth the BeltrmiKlein model nd the Poinré disk model to visulize the result (Figure 1). NonEulid [12] is nother progrmme to visulize hyperoli geometry in the Poinré disk model nd in the hlf plne model. The im of this work is to demonstrte the possiility to visulize hyperoli geometry with onstrutions mde y hnd. s n exmple we onstrut regulr ngons in the Poinré disk model nd in the BeltrmiKlein model pplying wellknown onstrutions of elementry Euliden geometry. In setion 2 short summries out few properties of the Poinré disk nd the BeltrmiKlein model re given whih will filitte our onstrutions. In setion 3 properties of regulr ngons re olleted. Even though we only need to know how to onstrut rightngled tringle given y two ute ngles with α + β < π / 2, we will onstrut in setion 4 tringle with three ritrry ngles with α + β + γ < π / 2 in the Poinré disk. In setion 5 we onstrut n ngon with given ngle etween djent edges in the Poinré disk nd in setion 6 n ngon with given side length in the BeltrmiKlein model. Setion 7 is short onlusion. 2. MODELS OF THE HYPERBOLIC PLNE s there re lot of textooks out hyperoli geometry, e.g. [3], [4], [6] nd [7], only some si notions nd properties re listed here.
2 presented y MICK, Syille 2.1 Poinré disk model Let m e Euliden irle. The Poinré disk model of hyperoli geometry is the open disk in the Euliden plne with oundry m. Points of the model re inner points of m. Hyperoli lines in this model re open rs on irles orthogonl to m. Hyperoli reflexions re reflexions ross dimeters of m or inversions with respet to irles orthogonl to m. (These irles re lso hrterized s elements of penils of Euliden irles with si points B nd B*, inverse with respet to m.) The inidene is indued from the underlying Euliden plne. The ngle mesurement is identil to the Euliden ngle mesurement, i.e. the Poinré disk is onforml model of the hyperoli plne. 2.2 BeltrmiKlein model The BeltrmiKlein model of hyperoli geometry is the open disk in the Euliden plne with oundry m. Points of the model re inner points of m. Hyperoli lines re open ords of m. The inidene is indued from the underlying Euliden plne. The oundry m of the disk is the solled solute irle. The hyperoli orthogonlity is determined y the polrity of m. hyperoli right ngle is Euliden right ngle iff one leg is dimeter of m. utomorphi ollinetions of m re hyperoli displements. Espeilly, hrmoni utomorphi ollinetions re hyperoli reflexions. Therefore, enter nd xis of hyperoli reflexion re pole nd polr of the solute irle m. For ngle mesurement we will pply tht the Euliden metri nd the hyperoli metri in the enter of the solute irle m re identil nd we n mesure hyperoli ngles with Euliden protrtor. 2.3 Remrks on mesurement The ongruene of segments is equivlent to the sttement tht the segments hve identil length. similr remrk pplies to ngles. Hene, we n speify the hyperoli h length l = T1T 2 y two points T 1, T 2 in the Poinré disk nd in the BeltrmiKlein model. The hyperoli mesure of n ngle α h = t 1 t 2 in the BeltrmiKlein is determined y two lines t 1, t PROPERTIES OF REGULR NGONS IN THE HYPERBOLIC GEOMETRY The Euliden geometry nd the hyperoli geometry re losely relted to eh other. Therefore, regulr ngons shre ouple of properties, e.g.: 1...n is regulr if ll its ngles re ongruent nd ll its edges hve the sme length with respet to the hosen metri. ll the verties of 1...n lie on ommon irle nd its enter C is the enter of the regulr ngon, too. Simple regulr ngons re lwys onvex. The symmetry group of regulr ngon is the dihedrl group D n of order 2n. It onsists of the n rottions with enter C nd ngle 2 π / n, together with n reflexions with xes through the enter. If n is even, then hlf of these xes pss through the midpoints of opposite edges. If n is odd, then ll xes pss through vertex nd the midpoint of the opposite edge. No mtter, if n is even or odd, two xes with ngle π / n determine two ongruent rightngled tringles. The verties of one tringle re one vertex of the ngon, the midpoint of n djent edge nd the enter of the regulr ngon. If we strt with one suh tringle (see Figure 4 nd Figure 5), the orit under the dihedrl group is the regulr ngon. Therefore, it is ler tht the onstrution of rightngled tringle is the essentil prt of the onstrution of regulr ngon. But there re no similrities in the hyperoli geometry! Two regulr ngons 1...n nd B 1...Bn with different edge length hve different ngles etween two djent edges. Therefore, regulr ngon in the hyperoli geometry n e given either y the ngle σ <π(n2)/n etween two djent edges (se I) or y the length s of n edge (se II). 2
3 presented y MICK, Syille 4. CONSTRUCTION OF TRINGLE WITH THREE SPECIFIED NGLES IN THE POINCRÉ DISK 4.1 Euliden irles interseting two lines under speified ngles Let nd e two interseting lines with = α. To onstrut Euliden irle * with * = γ nd * = β we hoose some ritrry points on nd nd drw the nglesγ nd β, respetively (Figure 2). * = γ nd * = β. In order to find irle m* with enter nd orthogonl to *, we drw the polr of with respet to *. The points of intersetion with * determine the rdius of m*. The diltion with enter tht mps m* on m lso mps the irle * on irle orthogonl to m (Figure 3). m* m * r* r* g B C g Figure 2: Euliden irles interset two interseting lines under speified ngles n ritrry distne r* is mrked on lines orthogonl to the seond legs of γ nd β, respetively. Prllel lines to nd through the new points interset in the enter of irle * with the desired properties. 4.2 Tringles with three speified ngles in the Poinré disk Let m e the solute irle of the Poinré disk, α, β nd γ e ngles with α + β + γ < π. Then tringles BC with ngles α, β nd γ exist (they re ongruent). Let us onstrut one. Without loss of generlity we put the vertex y hyperoli displement in the enter of m. The edges nd through re segments on dimeters of m. The missing edge of the tringle is n r on Euliden irle orthogonl to m, interseting nd under = γ nd = β. Due to susetion 4.1, we n drw irle * with 3 Figure 3: Tringle BC with three speified ngles nd with vertex in the enter of the solute irle of the Poinré disk The r BC on the irle is the third edge of the tringle. If we put γ = π / 2 the onstrution is simplified. This will e demonstrted in Figure 4. Remrk: In Euliden geometry there exist up to similrities extly one rightngled isoseles tringle with α = π / 2, β = γ = π / 4 nd extly one equilterl tringle with α = β = γ = π / 3. The onstrution in Figure 3 shows tht in hyperoli geometry every speifition α = π / 2, 0 < β = γ < π / 4 yields rightngled isoseles tringle nd every ngle α < π / 3 yields n equilterl tringle.
4 presented y MICK, Syille 5. CONSTRUCTION OF REGULR NGON WITH GIVEN NGLE BETWEEN TWO DJCENT EDGES IN THE POINCRÉ DISK We onstrut regulr ngon with given ngle σ etween djent edges in the Poinré disk model. We ssume tht the enter of the regulr ngon oinides with the enter of the Poinré disk. We drw tringle (see setion 4) with ngles α = π / n, β = σ / 2, γ = π / 2 ; n=5 nd σ = 60. The onstrution of BC simplifies euse the enter of irles * nd oinide with (Figure 4). B C m* * Figure 4: Rightngled tringle BC with α = π / 5 = 36, β = 30, γ = π / 2 nd vertex in the enter nd regulr pentgon in the Poinré disk The imges from BC under the reflexion ross nd the rottions out through 2 k π /5, k = 1,...,4 uild up the regulr pentgon. Remrk: In Euliden geometry only the regulr hexgon is deomposle in six equilterl tringles. The sitution in hyperoli geometry is different. If regulr ngon is deomposle in n equilterl tringles, then it is deomposle in 2n m 4 rightngled tringles with α = π / n nd β = 2π / n with α + β < π / 2. From this it follows tht for every n > 6 up to hyperoli displements regulr ngon exists deomposle in n equilterl tringles. The onstrution is the sme s in Figure 4 with speifitions α = π / n, β = 2 π / n, γ = π / 2 with n CONSTRUCTION OF REGULR NGON WITH GIVEN SIDE LENGTH IN THE BELTRMIKLEIN MODEL 6.1 Equidistnt urves in the BeltrmiKlein model In the hyperoli geometry n equidistnt urve is irle with ultrprllel dimeters, ll orthogonl to its xis o, while its enter is n ultridel point O. The equidistnt urve intersets ll its dimeters in segments of the sme length. In the BeltrmiKlein model xis o nd enter O re pole nd polr with respet to the solute irle m. If o is dimeter of m, then O is point t infinity in the underlying Euliden plne. 6.2 Some rightngled tringles in the Beltrmi Klein model We onsider rightngled tringles in the hyperoli geometry given y n ute ngle α nd length of the opposite side nd we ssume tht the vertex is the enter of m (Figure 5). Beuse n ngle with vertex in the enter of the solute irle hs the sme mgnitude in Euliden nd in hyperoli geometry we n use Euliden protrtor to drw n ngleα. Then one leg of α is the hypotenuse nd the other one thetus of the tringle on dimeter o. The seond endpoint B of the hypotenuse hs the distne of o. Hene, B is point of n equidistnt urve with xis o. If the hyperoli length is given y segment then hyperoli reflexion exists tht mps this segment on B on the dimeter orthogonl to o. The point B determines together with o s xis the
5 presented y MICK, Syille equidistnt urve in the hyperoli plne n ellipse in Euliden sense. The point of intersetion of the urve nd the seond leg of α is the seond point B on the hypotenuse. Finlly, the Euliden rightngled tringle BC is the requested hyperoli rightngled tringle, too. m o Figure 5 Rightngled tringle BC with vertex in the enter of the solute irle nd regulr hexgon in the BeltrmiKlein model 7. CONSTRUCTION OF REGULR NGON WITH GIVEN SIDE LENGTH IN THE BELTRMIKLEIN MODEL lso in Figure 5 we onstrut regulr ngon with given side length s in the BeltrmiKlein model. Its enter oinides with the enter of the BeltrmiKlein model. We speify α = π / n, s / 2 = nd n = 6 nd onstrut the rightngled tringle BC. The imges from BC under the reflexion ross o nd the rottions out through 2 k π / 6, k = 1,...,5, form the regulr hexgon. 8. CONCLUSIONS Visuliztion of hyperoli geometry in different models, i.e. the Poinré disk model nd the BeltrmiKlein model, mkes the xiomti development of hyperoli geometry more lively. Computer progrmmes re ville, ut onstrutions y hnd re stright ess to visulize geometry. s B' B C exmples we onstruted rightngled tringles nd regulr ngons using elementry Euliden onstrutions. REFERENCES [1] Bić, I., Sliepčević,., Regulr Polygons in the Projetively Extended Hyperoli Plne, KoG 11 (2007), [2] Benz, W., Clssil Geometries in Modern Contexts, Birkhäuser Verlg, Bsel, [3] Coolidge, J., The Elements of NonEuliden Geometry, Oxford, Clrendon Press, 1 st ed (reprinted 1927). [4] Coxeter, H.M.S., NonEuliden Geometry, 5 th ed. (reprinted), Univ. of Toronto Press, [5] GoodmnStruss, C., Compss nd Strightedge in the Poinré Disk, m. Mth. Mon. 108 (2001), No.1, [6] Greenerg, M.J., Euliden nd NonEuliden Geometries, 3 rd ed., New York, [7] Klein, F., Vorlesungen üer NihtEuklidishe Geometrie, Springer, Berlin [8] RihterGeert, J., Kortenkmp, U. H., User Mnul for the Intertive Geometry Softwre Cinderell, Springer, [9] Shwiger, J., Gronu, D., The Poinré model of hyperoli geometry in n ritrry rel inner produt spe nd n elementry onstrution of hyperoli tringles with presried ngles, J. of Geometry, epted for pulition, [10] Stillwell, J., The four Pillrs of Geometry, Springer, [11] Stillwell, J., Soures of hyperoli geometry, m. Mth. So., Providene,
6 presented y MICK, Syille LINKS [12] Cstellnos, J., et l., NonEulid, NonEulid.html BOUT THE UTHOR Syille Mik, Dr., is memer of the Institute of Geometry t the Grz University of Tehnology. Her reserh interests regrd NonEuliden Geometry nd Grphis Edutions. She n e rehed y emil: or y mil: Grz University of Tehnology, Institute of Geometry, Kopernikusgsse 24, Grz, ustri. 6
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