Original sientifi paper Aepted 20. 12. 2013. ANA SLIEPČEVIĆ IVANA BOŽIĆ HELENA HALAS Introdution to the Planimetry of the Quasi-Hyperboli Plane Introdution to the Planimetry of the Quasi- Hyperboli Plane ABSTRACT The quasi-hyperboli plane is one of nine projetive-metri planes where the absolute figure is the ordered triple {,,}, onsisting of a pair of real lines and through thereal point. In thispapersome basi geometri notions of the quasi-hyperboli plane are introdued. Also the lassifiation of qh-onis in the quasi-hyperboli plane with respet to their position to the absolute figure is given. The notions onerning the qh-oni are introdued and some seleted onstrutions for qh-onis are presented. Key words: quasi-hyperboli plane, perpendiular points, entral line, qh-onis lassifiation, osulating qh-irle MSC2010: 51A05, 51M10, 51M15 In the seond half of the 19th entury. Klein opened a new field for geometers with his famous Erlangen program whih is the study of the properties of a spae whih are invariant under a given group of transformations. Klein was influened by some earlier researh of A. Cayley, so today it is known that there exist nine geometries in plane with projetive metri on a line and on a penil of lines whih are denoted as Cayley-Klein projetive metris. Hene, these plane geometries differ aording to the type of the measure of distane between points and measure of angles whih an be paraboli, hyperboli, or ellipti. urthermore, eah of these geometries an be embedded in the real projetive plane P 2 (R) where an absolute figure is given as non-degenerated or degenerated oni [4], [5], [12] (for spae and n-dimension see [11]). In this artile the geometry, denoted as quasi-hyperboli, with hyperboli measure of distane and paraboli measure of angle will be presented. Uvod u planimetriju kvazi-hiperboličke ravnine SAŽETAK Kvazihiperbolička ravnina je jedna od devet projektivno metričkih ravnina kojoj je apsolutna figura ured - ena trojka {,,}, gdje su i realni pravi koji se sijeku u realnoj točki. U ovom članku uvodimo neke osnovne pojmove za kvazihiperboličku ravninu, te dajemo klasifikaiju konika u odnosu na njihov položaj prema apsolutnoj figuri. Nadalje, uvesti ćemo pojmove vezane uz konike u kvazihiperboličkoj ravnini i pokazati nekoliko izabranih konstrukija vezanih uz konike. Ključne riječi: kvazihiperbolička ravnina, okomite točke, entrala, klasifikaija qh-konika, oskulaijske qh-kružnie 1 Introdution 2 Basi notation in the quasi-hyperboli plane In the quasi-hyperboli plane (further in text qh-plane) the metri is indued by a real degenerated oni i.e. a pair of real lines and inidental with the real point. The lines and are alled the absolute lines, while the point is alled the absolute point. In the Cayley-Klein model of the qh-plane only the points, lines and segments inside of one projetive angle between the absolute lines are observed. In this artile all points and lines of the qh-plane embedded in the real projetive planep 2 (R) are observed. There are three different positions for the absolute triple {,,}: neither of the absolute elements are at infinity, only the absolute point is at infinity and the absolute point and one absolute line are at infinity (see ig. 1). The first position of the absolute triple is used for onstrutions in this artile. 58
3 Qh-onis lassifiation igure 1 or the points and the lines in the qh-plane the following terms are defined: isotropi lines - the lines inidental with the absolute point, isotropi points - the points inidental with one of the absolute lines or, parallel lines - two lines whih interset at an isotropi point, parallel points - two points inidental with an isotropi line, perpendiular lines - if at least one of two lines is an isotropi line, perpendiular points - two points (A and B) that lie on a pair of isotropi lines (a and b) that are in harmoni relation with the absolute lines and. There are nine types of regular qh-onis lassified aording to their position with respet to the absolute figure: qh-hyperbola - a qh-oni whih has a pair of real tangent lines from the absolute point, - hyperbola of type 1 ( ) - intersets eah absolute line in a pair of real and distint points, - hyperbola of type 2 (h 2 ) - intersets one absolute line in a pair of real and distint points and another absolute line in a pair of imaginary points, - hyperbola of type 3 ( ) - intersets eah absolute line in a pair of imaginary points, - speial hyperbola of type 1 (h s1 ) - one absolute line is a tangent line and another absolute line intersets the qh-oni in a pair of real and distint points, - speial hyperbola of type 2 (h s2 ) - one absolute line is a tangent line and another absolute line intersets the qh-oni in a pair of imaginary points, qh-ellipse (e) - a qh-oni (imaginary or real) whih has a pair of imaginary tangent lines from the absolute point, qh-parabola (p) - a qh-oni passing through the absolute point i.e. both isotropi tangent lines oinide, - speial parabola (p s ) - a qh-parabola whose isotropi tangent is an absolute line, qh-irle (k) - a qh-oni for whih the tangents from the absolute point are the absolute lines. In the projetive model of the qh-plane every type of a qhoni an be represented with the Eulidean irle without loss of generality (see ig. 2). This fat simplifies the onstrutions in the qh-plane. urthermore, an involution of penil of lines () having the absolute lines for double lines is alled the absolute involution, denoted as I QH. This is a hyperboli involution on the penil () where every pair of orresponding lines is in a harmoni relation with the double lines and ([1], p.244-245, [6], p.46). Notie that every pair of perpendiular points lie on a pair of I QH orresponding lines. Hene, the perpendiularity of points in qh-plane is determined by the absolute involution, therefore I QH is a irular involution in the qh-plane ([7], p.75). h 2 p e p s h s1 k Remark. Any two isotropi points on the same absolute line are perpendiular and parallel. Any two lines from a penil() are perpendiular and parallel. igure 2 h s2 59
urthermore every qh-oni q, exept qh-parabolae, indues an involution φ q on the penil () where the double lines are the isotropi tangents of the qh-oni q, and the orresponding lines of the involution φ q are alled onjugate lines. Notie that every qh-ellipse indues an ellipti involution, every qh-hyperbola indues a hyperboli involution and every qh-irle indues an involution that oinides with the absolute involution I QH. f 3 1 3 f 2 f 1 2 4 Remark. A qh-oni is alled equiform if the isotropi tangent lines of the qh-oni are in harmoni relation with the absolute lines and. In terms of the above mentioned involutions a qh-oni q is equiform if the absolute involution I QH is ommutative with the involution φ q indued by the qh-oni q. Notie that only qh-ellipses, qhhyperbolae of type 2 and qh-irles an be equiform [2], [3]. In the following some basi notions related to a qh-oni in the qh-plane are defined: The polar line of the absolute point with respet to a qh-oni is alled the entral line or the major diameter of the qh-oni (see ig. 3). All qh-onis, exept qhparabolas, have a non-isotropi entral line. The entral line of a qh-parabola is its isotropi tangent line, while for the speial parabola it is an absolute line. igure 4 The pole of the diretrix with respet to a qh-oni is alled a fous of the qh-oni. The number of foi i, i {1,2,3,4}, is equal to the number of diretries (see ig. 4). The lines that are inident with the opposite foi are alled isotropi diameters of a qh-oni (see ig. 5). Espeially for the qh-irles, whih have one fous, the isotropi diameters are the lines of the penil (). Hene a qh-oni an have none, one, two or infinitely many isotropi diameters o i, i {1,2}. f 4 The qh-enters of a qh-oni are the points of intersetion of the isotropi diameters and the entral line of the qh-oni. A qh-oni an have none, one, two or infinitely many qh-enters S i, i {1,2} (see ig. 5). The intersetion points of a qh-oni with its isotropi diameters are alled verties of the qh-oni (see ig. 5). A qh-oni an have four, two, one or none verties T i, i {1,2,3,4}. 4 3 S 2 o 1 igure 3 1 T 1 S 1 T 2 2 o 2 The diretries of a qh-oni are (non-absolute) lines inident with the isotropi points of the qh-oni, i.e. lines inidental with the intersetion points of the qh-oni with the absolute lines and. A qh-oni an have none, one, two or four diretries f i, i {1,2,3,4} (see ig. 4). igure 5 60
The absolute involution I QH an be observed as a point range involution on any non-isotropi line, hene it an be observed on the entral line of a qh-oni, exept for qhparabolae. Also the involution φ q on a penil () indued by a qh-oni q an be observed as the involution ϕ q of a point range on the entral line of the qh-oni q, and two orresponding points of involution ϕ q are alled onjugate points. Therefore, the qh-enters for the qh-ellipses and qh-hyperbolae an be found as a pair of perpendiular and onjugate points on the entral line, and the isotropi diameters as the perpendiular and onjugate lines of the penil (). The onstrution will be shown later. Notie that beause the involution indued by a qh-irle oinides with the absolute involution all pairs of onjugate points on the entral line of the qh-irle are perpendiular points. Hene any point on the entral line is its enter and every line of the penil() is its isotropi diameter. Aforementioned qh-onis and notions an be summarized in the following table: Qh-Coni Diretrix ous Isotropi diameter Center Vertex Ellipse 4 real 4 real 2 real 2 real 4 real e Hyperbola 4 real 4 real 2 real 2 real 2 real+ 2 imaginary Hyperbola 4 imaginary 4 imaginary 2 imaginary 2 imaginary 4 imaginary h 2 Hyperbola 4 imaginary 4 imaginary 2 real 2 real 2 real+ 2 imaginary Parabola 2 real 2 real 1 real 1 real 2 real p Speial 0 0 0 0 0 parabola p s Speial 2 real 2 real 1 real 1 real 1 real hyperbola h s1 Speial 2 imaginary 2 imaginary 1 real 1 real 1 real hyperbola h s2 Cirle 1 real 1 real infinite infinite 0 Table 1 or parabolae and speial hyperbolae see figure 6. =o 1 h s1 =f 1 f 2 p 1 = 1 =S 1 =T 1 T 2 S 1 =T 1 2 o 1 f 1 f 2 2 igure 6 61
Remark. The qh-plane is dual to the pseudo-eulidean (Minkowski) plane where the metri is indued by a real line and two real points inident with it. Therefore the notions defined above an be explained as duals of the Minkowski plane. The onis in pseudo-eulidean plane (pe-plane) are lassified in nine subtypes, hene the lassifiation of qhonis was based on [3], [9], [10]. urthermore, the aforementioned elements for qh-onis an be presented as follows: the entral line is a dual of the enter of a oni in the pe-plane, the diretries are a dual of the foi of a oni in the pe-plane, the foi are a dual of the diretries of a oni in the pe-plane, the qh-enters are dual to the axes of a oni in the peplane. The dual of the isotropi diameters are the intersetions of the axes with the absolute line, but they were not of speial interest in the pe-plane. Also the dual of the verties in qh-plane are the tangents to the oni in pe-plane from the above mentioned intersetions. It should be emphasized that the dual of the verties in pe-plane are tangents to the qh-plane from the qh-enters. Sine the axes in pe-plane and qh-enters in qh-plane are dual, therefore it was not hosen in this artile to observe the verties of a qh-oni as a line. urthermore, the pairs of onjugate points on the entral line of the involution ϕ q indued by a qh-oni q in the qh-plane are dual to the pairs of lines on whih lie the onjugate diameters of a oni in the pe-plane. Consequently, the aforementioned property of qh-enters for a qh-irle is dual to the fat that all pairs of onjugate diameters of a pseudo-eulidean irle are perpendiular. 4 Some onstrution assignments qh-oni q we observe the involution φ q indued by the qhoni q and the absolute involution I QH. These penils will be supplemented by the same Steiner s oni s, whih is an arbitrary hosen oni through. Let a pair of isotropi lines n and n 1 be the double lines of the involution φ q. The involutions I QH and φ q determine two involutions on the oni s. Let the points O 1 and O 2 be denoted as the enters of these involutions. The line O 1 O 2 intersets the oni s at two points I 1 and I 2. Isotropi lines (o 1 = I 1, o 2 = I 2 ) through these points are a ommon pair of these two involutions. Hene, lines o 1 and o 2 are isotropi diameters for the given qh-oni q. The intersetion points S 1 and S 2 of o 1 and o 2 with the entral line are qh-enters of the given qh-oni. igure 7 shows the desribed onstrution for hyperbola of type 3. The onstrution is based on the Steiner s onstrution ([6], p.26, [7], p.74-75). Notie that for the hyperbola of type 2 the line O 1 O 2 in the onstrution will not interset the oni s, and therefore it has a pair of imaginary isotropi diameters. In general, two involutions on a same penil (line) have a ommon pair of real orresponding lines (points) if at least one of them is an ellipti involution. If both of the involutions are hyperboli then they have a ommon pair of real orresponding lines (points) if both double lines of one involution are between the double lines (points) of the other involution. In the other ase the ommon pair is a pair of imaginary lines ([6], p.60). O 1 O 2 I 1 s I2 S 1 o 2 o 1 n 4.1 Qh-enters and isotropi diameters of the qhellipses and qh-hyperbolae n 1 Let a qh-oni q be given, that is not a qh-parabola. As already mentioned, a pair of onjugate and perpendiular points on the entral line will be qh-enters of a qhoni. In order to onstrut these qh-enters for the given S 2 igure 7 62
4.2 Osulating qh-irle of a qh-oni Generally, it is know that two arbitrary onis have four ommon tangents, therefore the same applies for a oni and a irle. urthermore, if three of this ommon tangents oinide then the irle is alled a osulating irle of the oni at the point whih is the point of tangeny of the triple tangent. Hene, there is an osulating irle at any point of a oni. Let a qh-oni be given, and a tangent t A at an arbitrary point A of the qh-oni. igure 8 shows the onstrution of the qh-irle osulating a qh-oni at the point A by using the elation (C,t A,D 1,D 1 ) [8]. Let points J 1 and J 2 be the isotropi points of the tangent t A. The tangents d 1 and d 2 from the points J 1 and J 2, respetively, to the given qhoni interset at the point whih orresponds to the absolute point. The ray intersets the tangent t A, whih is the axis of the elation, at the enter C of the elation. Hene the tangent lines and (absolute lines) of the osulating qh-irle orrespond to the tangent lines d 1 and d 2 of a given qh-oni. Let the points of tangeny of a qh-irle and, be denoted as D 1 and D 2, respetively. Let the point of tangeny of a qh-oni and d 1, d 2 be denoted as D 1 and D 2. Therefore D 1, D 1 and D 2, D 2 are the pairs of orresponding points of the elation. Similar onstrution priniple is given in [13]. d 1 D 1 D 1 J 1 C k J 2 J 1 A igure 9 4.3 Hyperosulating qh-irle of qh-onis A hyperosulating irle of a oni has a ommon quadruple tangent with the oni, hene it an be onstruted only at the verties of a oni. The similar onstrution priniple as for the osulating irle an be performed to onstrut the hyperosulating qh-irle at the vertex of a qhoni. Let the hyperbola be given. The intersetion points T 1 and T 2 of the qh-oni with its isotropi diameter are the verties of the hyperbola. The hyperosulating qh-irle at the vertex T 2 is ompletely determined with the elation (T 2,t 2,D i,d i ) (i=1,2) where T 2 is the enter and tangent t 2 at T 2 its axis. The tangent lines and of the hyperosulating qh-irle orrespond to the tangent lines d 1 and d 2 of the. Let the point of tangeny of a qh-oni and d 1, d 2 be denoted as D 1 and D 2, respetively. Let the points of tangeny of a qh-irle and, be denoted as D 1 and D 2, respetively. D 1, D 1 and D 2, D 2 are the pairs of orresponding points of the elation (see ig. 10). A D 2 D 2 A J 2 t 2 d 2 t A D 1 D 1 J 1 T 1 igure 8 D 2 T 2 Remark. It should be emphasized that in a qh-plane it is possible to onstrut infinitely many osulating qh-irles at the isotropi tangeny point if the given qh-oni is a qh-irle. The qh-irle osulating the given qh-irle k at its isotropi point J i,(i=1,2) an be onstruted by using the elation (, j i,a,a ), (i=1,2). The point is the enter of the elation, the absolute line j i its axis, A an arbitrary hosen point on qh-irle and A an arbitrary hosen point on the ray A (see ig. 9). D 2 J 2 igure 10 d 1 d 2 63
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