Introduction to Planimetry of Quasi-Elliptic Plane

Transcription

1 Originl scientific pper ccepted Sliepčević, I. Božić Drgun: Introduction to Plnimetry of Qusi-Elliptic Plne N SLIEPČEVIĆ IVN BOŽIĆ DRGUN Introduction to Plnimetry of Qusi-Elliptic Plne Introduction to Plnimetry of Qusi-Elliptic Plne BSTRCT The qusi-elliptic plne is one of nine projective-metric plnes where the metric is induced y the solute figure QE = { j 1, j 2,} consisting of pir of conjugte imginry lines j 1 nd j 2, intersecting t the rel point. Some sic geometric notions, definitions, selected constructions nd theorem in the qusi-elliptic plne will e presented. Key words: qusi-elliptic plne, perpendiculr points, centrl line, qe-conic clssifiction, hyperosculting qe-circle, envelope of the centl lines MSC2010: 5105, 51M10, 51M15 Uvod u plnimetriju kvzieliptičke rvnine SŽETK Kvzieliptičk rvnin jedn je od devet projektivno metričkih rvnin. psolutnu figuru QE = { j 1, j 2,} odred - uju dv imginrn prvc j 1 i j 2 i njihovo relno sjecište. U ovom rdu definirt ćemo osnove pojmove, prikzti odrne konstrukcije i dokzti jedn teorem. Ključne riječi: kvzieliptičk rvnin, okomite točke, centrl, klsifikcij qe-konik, hiperoskulcijsk qekružnic, omotljk centrl 1 Introduction This pper egins the study of the qusi-elliptic plne from the constructive nd synthetic point of view. We will see lthough the geometry denoted s qusi-elliptic is dul to Eucliden geometry it is very rich topic indeed nd there re mny new nd unexpected spects. In this pper some sic nottions concerning the qusielliptic conic nd some selected constructions nd theorem will e presented. It is known tht there exist nine geometries in plne with projective metric on line nd on pencil of lines which re denoted s Cyley-Klein projective metrics nd they hve een studied y severl uthors, such s [2], [3], [4], [8], [9], [10], [13], [14], [15], [16]. The qusi-elliptic geometry, further in text qe-geometry, hs elliptic mesure on line nd prolic mesure on pencil of lines. In the qusi-elliptic plne, further in text qe-plne, the metric is induced y the solute figure QE = { j 1, j 2,}, i.e. pir of conjugte imginry lines j 1 nd j 2, incident with the rel point. The lines j 1 nd j 2 re clled the solute lines, while the point is clled the solute point. In the Cyley-Klein model of the qe-plne only the points, lines nd segments inside of one projective ngle etween the solute lines re oserved. In this pper ll points nd lines of the qe-plne emedded in the rel projective plne P 2 (R) re oserved. It is suitle to otin line s sic element, nd point s pencil of lines (for exmple curve is n envelope of lines; qudrtic trnsformtion in the qe-plne mps pencil of lines into the second clss curve). Using n elliptic involution on the pencil () the solute triple QE = { j 1, j 2,} cn e given s follows: n elliptic involution on the pencil () is determined y two ritrry chosen pirs of corresponding lines 1, 2 ; 1, 2. n elliptic involution () hs the solute lines j 1 nd j 2 for doule lines ([1], p , [6], p.46). Notice tht the solute point cn e finite (igure 1) or t infinity (igure 1c). In this pper the model were involutory pir of corresponding lines re perpendiculr to ech other in Eucliden sense (igure 1) is used in wy tht only the solute point is presented. The solute point is inside the conic k. Pirs of conjugte lines with respect to conic k determine forementioned elliptic involution (). The solute lines j 1 nd j 2 re doule lines for the involution () nd in this cse they re pir of imginry tngent lines to k from the solute point (igure 1d). 16

2 . Sliepčević, I. Božić Drgun: Introduction to Plnimetry of Qusi-Elliptic Plne k 2 2 c d igure 1 2 Bsic nottion nd selected constructions in the qusi-elliptic plne or the points nd the lines in the qe-plne the following re defined: isotropic lines - the lines incident with the solute point, i.e. if they lie on pir of perpendiculr lines in Eucliden sense ([7], p.71-75). 1 isotropic points - the imginry points incident with one of the solute line j 1 or j 2, prllel points - two points incident with the sme isotropic line, perpendiculr lines - if t lest one of two lines is n isotropic line, perpendiculr points - two points, 1 tht lie on pir of corresponding lines, 1 of n elliptic (solute) involution (). Remrk. The perpendiculrity of points in qe-plne is determine y the solute involution, therefore n elliptic involution () is circulr involution in the qe-plne. ([7], p.75) Notice tht the solute point is prllel nd perpendiculr to ech point in the qe-plne. urthermore, in the qe-plne there re no prllel lines. rief review of some sic construction Exmple 1 Let the solute figure QE of the qe-plne e given with the involutory pencil () (igure 1). Let e the point nd p the line which is not incident with the point in the qe-plne (igure 2). Construct the point 1 which is perpendiculr to the point nd incident with the given line p. Points, 1 re perpendiculr if they lie on pir of corresponding lines, 1 of n solute elliptic involution (), p 1 p igure 2 Exmple 2 Let the solute figure QE of the qe-plne e given with the involutory pencil (). Construct the midpoints P i nd the isectors s i of given line segment B (i = 1,2) (igure 3). s 2 P 1 s O 1 s 1 igure 3 O 2 P 2 B 17

3 . Sliepčević, I. Božić Drgun: Introduction to Plnimetry of Qusi-Elliptic Plne The midpoint of segment in the qe-plne is dul to n ngle isector in the Eucliden plne, consequently segment in qe-plne hs two perpendiculr midpoints P 1 nd P 2 tht re in hrmonic reltion with the points nd B. line segment B in the qe-plne hs two isotropic isectors s 1 nd s 2 tht re common pir of corresponding lines of two involutions () with the center, denoted s I 1, I 2. In order to construct the midpoints nd isectors we oserve forementioned involutions (), circulr involution I 1 is determined y perpendiculr corresponding lines in Eucliden sense nd the second hyperolic involution I 2 is determined y isotropic lines =, = B s its doule lines. The construction is sed on the Steiner s construction ([6], p.26, [7], p.74-75). These two pencils will e supplemented y the sme Steiner s conic s, which is n ritrry chosen conic through. The involutions I 1 nd I 2 determine two involutions on the conic s. Let the points O 1 nd O 2 e denoted s the centers of these involutions, respectively. The line O 1 O 2 intersects the conic s t two points. Isotropic lines s 1 nd s 2 through these points re common pir of these two involutions (). The intersection points P 1 nd P 2 of isectors s 1 nd s 2 with the line B re midpoints of the line segment B. Exmple 3 Let the solute figure QE of the qe-plne e given with the involutory pencil (). Let two non-isotropic lines, e given. Construct n ngle isector etween given rys, (igure 4). The ngle isector in the qe-plne is dul to midpoint of segment in the Eucliden plne. Let V e the vertex of n ngle (,). Let the isotropic line V e denoted s f. The ngle isector s is line in pencil (V ) tht is in hrmonic reltion with triple (,, f ). The isotropic line f is n isotropic isector. f s 2 4 determine trilterl BC with the vertices, B, C. Construct the ortocentr line of the given trilterl (igure 5). The orthocentr line o of the trilterl in the qe-plne is dul to the orthocenter of tringle in the Eucliden plne. The points 1, B 1, C 1 re incident with lines,, c nd perpendiculr to the opposite vertices, B, C, respectively. The points 1, B 1, C 1 re colliner nd determine unique ortocentr line. c C 1 B 1 igure 5 Exmple 5 Let the solute figure QE of the qe-plne e given with the involutory pencil (). Let the lines,, c determine trilterl BC with the vertices, B, C. Construct the centroid line of trilterl (igure 6). The centroid line o of trilterl in the qe-plne is dul to the centroid of tringle in the Eucliden plne. The ngel isectors s, s, s c of trilterl intersect opposite sides,, c of the trilterl t the points S, S B, S C, respectively. The points S, S B, S C re colliner nd determine unique centroid line. c s c s t S C S B B 1 C B o s V C igure 4 S Exmple 4 Let the solute figure QE of the qe-plne e given with the involutory pencil (). Let the lines,, c igure 6 18

4 KoG Sliepc evic, I. Boz ic Drgun: Introduction to Plnimetry of Qusi-Elliptic Plne Qe-conic clssifiction There re four types of the second clss curves clssified ccording to their position with respect to the solute figure (igure 7): qe-hyperol (h) - curve of the second clss tht hs pir of rel nd distinct isotropic lines. Equilterl qe-hyperol (heq ) - curve of the second clss tht hs isotropic lines s corresponding lines for the solute involution (). qe-ellipse (e) - curve of the second clss tht hs pir of imginry isotropic lines. qe-prol (p) - curve of the second clss where oth imginry isotropic lines coincide. qe-circle (k) - is specil type of qe-ellipse for which the isotropic lines coincide with the solute lines j1 nd j2. In model of n solute figure tht is used in this pper ech qe-conic tht hs n solute point s its Eucliden foci is qe-circle. In the projective model of the qe-plne every type of qeconic cn e represented with every type of Eucliden conics without loss of generlity. e h k p igure 7 The polr line of the solute point with respect to qe-conic is clled the centrl line or the mjor dimeter of the second clss conic in the qe-plne. The centrl line of conic in the qe-plne is dul to center of conic in the Eucliden plne. ll conics in the qe-plne, except qe-prols, hve rel non-isotropic centrl line. The centrl line of qe-prol is isotropic tngent line t the point. Dul to the Eucliden dimeter of conic is the point on the centrl line tht is the pole of the isotropic line with respect to qe-conic. pir of points incident with the centrl line tht re perpendiculr nd conjugte with respect to qe-conic re clled the qe-centers of the qe-conic. Qecenters re dul to n xis of Eucliden conic. qe-ellipse nd qe-hyperol hve two rel nd distinct qe-centers, while oth qe-center of qe-prol coincide with the solute point. Ech pir of conjugte points incident with the centrl line with respect to qe-circle re perpendiculr, consequently qe-circle hs infinitely mny pirs of qe-centers. The isotropic (the minor) dimeters re the lines joining qe-center to the solute point. qe-ellipse nd qehyperol hve two isotropic dimeters. The lines incident with qe-centers of qe-conic re clled the vertices lines of qe-conic in the qe-plne. qehyperol hs two rel vertices lines, while qe-ellipse hs four rel vertices lines. hyperosculting qe-circle of qe-conic cn e constructed only t the vertices lines of qe-conic. The intersection points of qe-conic nd vertices lines re clled co-vertices points. 4 Some construction ssignments Exercise 1 Construct qe-circle k determined with the given centrl line c nd the line p (igure 8). In order to construct the qe-circle s line envelope, perspective collinetion tht mps ritrry chosen qe-circle k1 into qe-circle k is used. The construction is crried out in the following steps: The solute point is selected for the center of the collinetion. Let k1 e n ritrry chosen qe-circle with the center. polr line c1 of, is the centrl line for chosen qe-circle k1. Notice tht c1 is the line t infinity. The lines c nd c1 re corresponding lines for the perspective collinetion with the center. Let the point S e the intersection point of the lines p nd c. To determine n xis o of the perspective collinetion, the point R tht is perpendiculr to the point S nd incident with the line p is constructed. ry R of the collinetion intersect the qe-circle k1 t the points R1 nd R2. Let the line p1 touch the qe-circle k1 t point R1. The lines p nd p1 re corresponding lines for the perspective collinetion with center. The xis o psses through the intersection point S1 of the lines p1 nd p, nd it is prllel to c. igure 8 19

5 KoG Sliepc evic, I. Boz ic Drgun: Introduction to Plnimetry of Qusi-Elliptic Plne Exercise 2 Construct the hyperosculting qe-circle of qe-hyperol h1 (igure 9). Let the qe-hyperol h1 e given nd its centrl line e denoted s c. hyperosculting qe-circle of the qe-hyperol h1 cn e constructed only t the vertices lines. qehyperol h1 hs two rel vertices lines t1 nd t2. Let the points T1 nd T2 e co-vertices points. Let the line t2 nd the point T2 e oserved. In order to construct hyperosculting qe-circle, the point S2 tht is perpendiculr to T2, nd incident with the line t2 is constructed. The centrl line ch of hyperosculting qe-circle is incident with S2. In order to construct ch, let the line y1 of the qe-hyperol h1 e ritrry chosen. The intersection point of h1 nd the line y1 is denoted s Y1. The intersection point of lines t2 nd y1 is denoted s K. The point K1 is perpendiculr to K nd incident with joining line T2Y1. The line S2 K1, denoted s ch, is centrl line of hyperosculting qe-circle. The centrl line ch nd the line t2 determine hyperosculting qe-circle nd to construct it the sme principle s in Exercise 1 is used. igure 9 Theorem 1 Let the lines {,, c, d} e the se of pencil of qe-conics in qe-plne (igure 10). Then, the envelope of the centrl lines of ll qe-conics in the pencil is curve of the second clss. In the given pencil of qe-conics there re three qe-conics degenerted into three pirs of points, denoted s (1, 10 ), (2, 20 ), (3, 30 ). Let the involution () e determined with n isotropic lines of ny two degenerted qe-conics in pencil i.e. (1, 10 ), (3, 30 ). The pencil of qe-conics contins two, one or none rel qe-prol. rom the viewpoint of qe-geometry, the envelope δ1 is qe-hyperol if the pencil contins two qe-prols. The centrl lines of these qe-prols denoted s, p1 nd p2 re doule lines for the involution () nd they coincide with the isotropic lines of the envelope δ1. The envelope is determinte with five lines; the lines p1, p2, nd centrl lines of three degenerted qe-conics c1, c2, c3. The envelope δ1 is qe-prol if the pencil contin one qe-prol. The envelope δ1 is n qe-ellipse if the pencil does not contin qe-prols (igure 10). Doule lines for the elliptic involution () re imginry lines. Pencil will e supplemented y the Steiner s conic s, which is n ritrry chosen conic through. Let the point O e denoted s center of the involution (). If the point O is outside the conic s, involutory pencil () contins rel doule lines, nd the envelope δ1 is qehyperol. If the point O is on the conic s, doule lines of involution () coincide, nd the envelope δ1 is qeprol. If the point O is inside the conic s, involutory pencil () contins imginry doule lines, nd the envelope δ1 is n qe-ellipse (igure 10). If the point O coincides with the center of conic s, doule lines of involution () coincides with the solute lines j1 nd j2, nd the envelope δ1 is qe-circle (circulr involution). If one of the se lines in pencil is isotropic line, the pencil of qe-conics contins qe-hyperols nd one qeprol, the envelope δ1 is qe-prol. If two of the se lines in pencil re isotropic lines, the envelope δ1 degenertes into point. Proof: It is known tht the envelope of polr lines of conics in pencil of conics with respect to common pole P is curve of the second clss ([6]). Consequently, in the qe-plne, if common pole P coincides with the solute point, thn the envelope of its polr lines coincides with the envelope of the centrl lines in the given pencil. In order to construct the envelope of the centrl lines of ll qe-conics in the pencil of qe-conics, denoted s δ1, we oserve involutory pencil () of pirs of isotropic lines of ll qe-conics in pencil. Ech qe-conic in pencil of qe-conics hs two rel or imginry isotropic lines. 20 igure10: Qe-ellipse - n envelope of the centrl lines

6 . Sliepčević, I. Božić Drgun: Introduction to Plnimetry of Qusi-Elliptic Plne Corollry 1 Let ny two degenerted qe-conics in pencil of qe-conics e given s pir of perpendiculr points i.e. the pencil of equilterl qe-hyperols. Thn the envelope of the centrl line is qe-circle. References [1] H.S.M. COXETER, Introduction to geometry, John Wiley & Sons, Inc, Toronto, 1969; [2] N. KOVČEVIĆ, E. JURKIN, Circulr cuics nd qurtics in pseudo-eucliden plne otined y inversion, Mth. Pnnon. 22/1 (2011), 1-20; [3] N. KOVČEVIĆ, V. SZIROVICZ, Inversion in Minkowskischer geometrie, Mth. Pnnon. 21/1 (2010), ; [4] N.M. MKROV, On the projective metrics in plne, Učenye zp. Mos. Gos. Ped. in-t, 243 (1965), (Russin); [5] M. D. MILOJEVIĆ, Certin comprtive exmintions of plne geometries ccording to Cyley-Klein, Novi Sd J.Mth., 29/3, (1999), [6] V. NIČE, Uvod u sintetičku geometriju, Školsk knjig, Zgre, 1956.; [7] D. PLMN, Projektivne konstrukcije, Element, Zgre, 2005; [8]. SLIEPČEVIĆ, I. BOŽIĆ, Clssifiction of perspective collinetions nd ppliction to conic, KoG 15, (2011), 63-66; [9]. SLIEPČEVIĆ, M. KTIĆ ŽLEPLO, Pedl curves of conics in pseudo-eucliden plne, Mth. Pnnon. 23/1 (2012), 75-84; [10]. SLIEPČEVIĆ, N. KOVČEVIĆ, Hyperosculting circles of conics in the pseudo-eucliden plne, Mnuscript; [11] D.M.Y SOMMERVILLE, Clssifiction of geometries with projective metric, Proc. Ediurgh Mth. Soc. 28 (1910), 25-41; [12] I.M. YGLOM, B.. ROZENELD, E.U. YSIN- SKY, Projective metrics, Russ. Mth Surreys, Vol. 19/5, No. 5, (1964), ; [13] G. WEISS,. SLIEPČEVIĆ, Osculting circles of conics in Cyley-Klein plnes, KoG 13, (2009), 7-13; [14]. SLIEPČEVIĆ, I. BOŽIĆ, H. HLS, Introduction to the plnimetry of the qusi-hyperolic plne, KoG 17, (2013), 58-64; [15] M. KTIĆ ŽLEPLO, Curves of centers of conic pencils in pseudo-eucliden plne, Proceedings of the 16th Interntionl Conference on Geometry nd Grphics, (2014); [16]. SLIEPČEVIĆ, I. BOŽIĆ, The nlogue of theorems relted to Wllce-Simson s line in qusihyperolic plne, Proceedings of the 16th Interntionl Conference on Geometry nd Grphics, (2014); Ivn Božić Drgun e-mil: ivn.ozic@tvz.hr University of pplied Sciences Zgre, venij V. Holjevc 15, Zgre, Croti n Sliepčević emil: nsliepcevic@gmil.com culty of Civil Engineering, University of Zgre, Kčićev 26, Zgre, Croti 21