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On the Geometry of a Triangle in the Elliptic and in the Extended Hyperbolic Plane 3

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On the Geometry of a Triangle in the Elliptic and in the Extended Hyperbolic Plane 3 ON THE GEOMETRY OF A TRIANGLE IN THE ELLIPTIC AND IN THE EXTENDED HYPERBOLIC PLANE MANFRED EVERS ABSTRACT. We investigate several topics of triangle geometry in the elliptic and in the extended hyperbolic plane. For both planes a uniform metric is used. The concept for this metric was developed by C. Vörös at the beginning of the 20th century and subsequently described by Horváth in 2014. INTRODUCTION This paper is a continuation of a previous work [5]. Whilst the previous paper was re- stricted to triangles in the elliptic plane, these investigations also deal with triangles in the extendedhyperbolic plane. In addition, both papersdiffer in the selection of centers, central lines, central conics and cubics. The first section gives an introduction to the metric used on the projective plane. We assume that the reader is familiar with euclidean triangle geometry, but in order to intro- duce the terminologyand fix notations, we give some basic definitions, rules and theorems. There are many introductory books on euclidean triangle geometry; we refer to presen- tations of Yiu [30] and Douillet [4]. For geometry on the sphere / elliptic geometry we recommend the book [24] by Todhunterand Leathem. Several publications appeared in the last decade dealing with triangle geometry in the hyperbolicplane, to name Ungar [25, 26], Wildberger [28], Wildberger and Alkhaldi [29], Horváth [13, 14], Vigara [27] and Russell [21]. The topics of the second section are: centers based on orthogonality, centers related to circumcircles and incircles, radical centers and centers of similitude, orthology, Kiepert perspectors and related objects, Tucker circles, isoptics, and substitutes for the Euler line. Remark: A detailed proof is not given for every statement, but quite often the statement can be checked by simple computation (supported by a CAS-program). arXiv:1908.11134v1 [math.MG] 29 Aug 2019 1. METRIC GEOMETRY IN THE PROJECTIVE PLANE 1.1. The projective plane, its points and its lines Let V be thethreedimensionalvectorspace ℝ3, equippedwiththe canonicaldot product p ⋅ q =(p1,p2,p3) ⋅ (q1, q2, q3)= p1q1 + p2q2 + p3q3, and let denote the projective plane ℝ× (V −{0})∕ . The image of a non-zero vector p = (p1,p2,p3) ∈ V under the canonical → projection Π∶ V will be denoted by (p1∶p2∶p3) and will be regarded as a point in this plane. Given two different points P and Q in this projective plane, there exists exactly one line incident with these two points. It is called the join P ∨ Q of P and Q. If p = (p1,p2,p3), q =(q1, q2, q3) are two non zero vectors with Π(p)= P and Π(q)= Q, then the line P ∨ Q through P and Q is the set of points Π(sp + tq) with s,t ∈ ℝ. One can find linear forms Date: August 30, 2019. 1 2 MANFRED EVERS l ∈ V ∗−{0∗} with ker(l)= span(p, q). A suitable l is, for example, l =∗(p×q)=(p×q)∗, where × stands for the canonical cross product in V = ℝ3 and ∗ for the isomorphism V → V ∗, ∗(r) = r⋅(.). The linear form l is uniquely determined up to a nonzero real factor, so there is a 1∶1-correspondence between the lines in the projective plane and the elements of ∗ = (V ∗−{0∗})∕ℝ×. We identify the line l = P ∨ Q with the element ∗ ∗ (p2q3 − p3q2 ∶ p3q1 − p1q3 ∶ p1q3 − p2q1) ∈ . ∗ ∗ In the projective plane, two different lines k = (k1∶k2∶k3) , l = (l1∶l2∶l3) always meet at one point k ∧ l = Π((k1, k2, k3)×(l1, l2, l3)), the so-called meet of these lines. 1.2. Collineations, correlations and polarities The automorphism group Aut() of can be identified with the projective linear group PGL(V ) = GL(V )∕ℝ×. These automorphisms on are called collineations because the image of a line under an automorphismis again a line. A correlation is a bijective mapping ∗ ∗ c∶ ∪ → ∪ with the following property: The restriction c of c to is a point-to-line transformation that maps collinear points to concurrent lines, while c ∗ is a line-to-point transformation that maps concurrent lines to collinear points.ð Thus there are → ∗ ∗ → for each correlation c uniquely determined linear mappings c1∶ , c2∶ ð with c1 = c and c2 = c ∗ . If c2 is the inverse of c1, this correlation is an involution. ∗ In this case, c1, c2 are symmetric mappings since and have even dimensions, and the correlation cðis a polarity. ð 1.3. Elliptic and hyperbolic metric structures on . We now fix a nonzero real number . By we denote the polarity which maps a point ∗ ∗ P = (p1∶p2∶p3) to its dual line P = (p1∶p2∶p3) and a line l = (l1∶l2∶l3) to its dual point l =(l1∶l2∶l3). ∗ 1.3.1. We make use of to introduce orthogonality. A line k = (k1∶k2∶k3) is orthog- ∗ onal (or perpendicular) to a line l = (l1∶l2∶l3) when the dual k of k is a point on l; this is precisely when k1l1 + k2l2 + k3l3 = 0. Obviously, if k is orthogonal to l, then l is orthogonal to k. Two points P = (p1∶p2∶p3) and Q = (q1∶q2∶q3) are orthogonal to each other if p1q1 + p2q2 + p3q3 = 0. This is the case exactly when their dual lines P and Q are orthogonal. Self-orthogonal points and lines are called isotropic. The isotropic points form the so-called absolute conic abs. This is either the empty set, in which case the geometry is called elliptic, or it is a properconic and the geometryis called hyperbolic. Each symmetric real 3x3-matrix M with det M ≠ 0 determines a scalar product (sym- metric bilinear form) [M]∶ V ×V → ℝ by w1 M M (v1, v2, v3) [ ] (w1, w2, w3) = (v1, v2, v3) w2 . ⎛w3⎞ The orthogonality of points and lines can be expressed with the help⎜ ⎟ of the scalar products S S−1 S ⎜ ⎟ [ ] and [ ] where = diag(, 1, 1). Two points P = (p1∶p2∶⎝p3)⎠and Q = (q1∶q2∶q3) ∗ are orthogonal precisely when (p1,p2,p3) [S](q1,q2,q3)=0, two lines k =(k1∶k2∶k3) and ∗ −1 l =(l1∶l2∶l3) are orthogonal precisely when (k1,k2,k3) [S ] (l1,l2,l3)=0. ∗ 1.3.2. Consider some point P =(p1∶p2∶p3) and some line l =(l1∶l2∶l3) with l ≠ P . The perpendicular from P to l is the line ∗ perp(l, P ) ∶= P ∨ l =(p2l3−p3l2 ∶ (p3l1−p1l3) ∶ (p1l2−p2l1)) , and the point on l orthogonal to P is perp(P , l) ∶= l ∧ P =((p2l3−p3l2) ∶ p3l1−p1l3 ∶(p1l2−p2l1)). ON THE GEOMETRY OF A TRIANGLE IN THE ELLIPTIC AND IN THE EXTENDED HYPERBOLIC PLANE 3 The line through P parallel to l is par(l, P ) ∶= perp(perp(l, P ), P ). The point ped(P , l) ∶= perp(perp(P , l), l), is the orthogonal projection of P on l, also called the pedal of P on l. 1.3.3. Line segments and angles. Define the function ∶ V → {−1, 0, 1} by 0, if (p1,p2,p3)=(0, 0, 0) , (p0,p1,p2)= 1, if (p1,p2,p3) > (0, 0, 0) with respect to the lexicographic order, ⎧−1, if (p ,p ,p ) < (0, 0, 0) with respect to the lexicographic order. ⎪ 1 2 3 For an anisotropic⎨ point P =(p ∶p ∶p ) let P ◦ ∈ V be the vector ⎪ 1 2 3 ⎩ ◦ (p1,p2,p3) (p1,p2,p3) P ∶= (p1,p2,p3)= (p1,p2,p3). (p1,p2,p3) [S] (p1,p2,p3) p2 p2 p2 1 + 2 + 3 ◦ P is obviously√ uniquely determined by P . t ð ð For any two different points P and Q we introduce two lineð segments [ðP,Q]+, [P,Q]−, thesearetheclosuresof thetwo connectedcomponentsof the set P ∨Q−{P,Q}. If R, S are ◦ ◦ ℝ two different anisotropic points in [P,Q]+, then [R, S]+ = {Π(sR + tS ) s,t ∈ ,st ≥ 0} is a subset of [P,Q]+, and if R, S are two different anisotropic points in [P,Q]−, then ◦ ◦ ℝ [R, S]− = {Π(sR + tS ) s,t ∈ ,st ≤ 0} is a subset of [P,Q]−. ð We define angles as subsets of the pencil of lines through a point, which is the vertex of this angle: Given three noncollinearð points Q,R,S, put ∠+QSR ∶= {S ∨ P P ∈[Q, R]+} , ∠−QSR ∶= {S ∨ P P ∈[Q, R]−} . The union of these two angles is the complete pencil of lines through S. ð ð 1.3.4. The length of a segment and the measure of an angle. We introduce the measure of segments and angles using concepts that were developed by C. Vörös in Hungary at the beginning of the 20th century and more recently (2014) described by Horváth in two papers [13, 14], where he applied these to various configurations in hyperbolic geometry. To each line segment s and to each angle is assigned its measure (s) resp. () which is a complex number with a real part in ℝ̄ = ℝ ∪ {−∞, ∞} and an imaginary part in the interval [0, ]. Define for each anisotropic point P =(p1∶p2∶p3) a number "P ∈ {1, −i} by ◦ ◦ 1) "P ∶= 1∕ P [S]P . First we describe the measure ([P,Q]±) of√ line segments with anisotropic endpoints P,Q, P ≠ Q. (The remaining cases will be treated afterwards.) Here, the function satisfies the following rules: (1) ([P,Q]+) and ([P,Q]−) are finite complex numbers with imaginary parts in the interval [0, ] satisfying ([P,Q]+)+ ([P,Q]−)= (P ∨ Q)= i. ◦ ◦ (2) cosh([P,Q]+)= "P "QP [S] Q . (3) If R is an anisotropic inner point of [P,Q]+, 1) We follow the convention r exp(i) = r exp( 1 i) for r ∈ ℝ+and 0 ≤ < 2.
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