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Hyperbolic Cosines and Sines Theorems for the Triangle Formed by Arcs of Intersecting Semicircles on Euclidean Plane

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Hyperbolic Cosines and Sines Theorems for the Triangle Formed by Arcs of Intersecting Semicircles on Euclidean Plane Hindawi Publishing Corporation Journal of Mathematics Volume 2013, Article ID 920528, 10 pages http://dx.doi.org/10.1155/2013/920528 Research Article Hyperbolic Cosines and Sines Theorems for the Triangle Formed by Arcs of Intersecting Semicircles on Euclidean Plane Robert M. Yamaleev1,2 1 Joint Institute for Nuclear Research, Laboratory of Information Technology, Street Joliot Curie 6, CP 141980, P.O. Box 141980, Dubna, Russia 2 Universidad Nacional Autonoma de Mexico, Facultad de Estudios Superiores, Avenida 1 Mayo, 54740 Cuautitlan´ Izcalli, MEX, Mexico Correspondence should be addressed to Robert M. Yamaleev; [email protected] Received 17 December 2012; Accepted 21 February 2013 Academic Editor: Sujit Samanta Copyright © 2013 Robert M. Yamaleev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The hyperbolic cosines and sines theorems for the curvilinear triangle bounded by circular arcs of three intersecting circles are formulated and proved by using the general complex calculus. The method is based on a key formula establishing a relationship between exponential function and the cross-ratio. The proofs are carried out on Euclidean plane. 1. Introduction laws of sines-cosines for that triangle are proved by using properties of the Mobius¨ group and the upper half-plane . Classically, the models of the hyperbolic plane are regarded In this paper we suggest another way of construction of as based on Euclidean geometry. One starts with a piece of a proofs of the sines-cosines theorems of the Poincaremodel.´ Euclidean plane, a half-plane, or a circular disk and then, in The curvilinear triangle formed by circular arcs is the figure that half-plane or disk the notions of points, lines, distances, of the Euclidean plane; consequently, on the Euclidean plane angles are defined as things that could be described in terms we have to find relationships antecedent to the sines-cosines of Euclidean geometry [1–3]. The model of the hyperbolic plane is the half-plane model. The underlying space of this hyperbolic laws. Therefore, first of all, we establish these model is the upper half-plane model in the complex plane relationships by making use of axioms of the Euclidean plane, ,definedtobe only. Secondly, we proove that these relationships can be formulated as the hyperbolic sine-cosine theorems. For that ={∈:Im () >0} . (1) purpose we refer to the general complex calculus and within its framework establish a relationship between exponential (, ) In coordinates thelineelementisdefinedas function and the cross-ratio. In this way the hyperbolic 1 trigonometry emerges on Euclidean plane in a natural way. 2 = (2 +2). 2 (2) The paper is organized as follows. In Section 2 akey formula connecting the hyperbolic calculus with the cross- The geodesics of this space are semicircles centered on the - ratio is established. By employing the key formula and the axis and vertical half-lines. The geometrical properties of the Pythagoras theorem the elements of the right-angled triangle figures on the half-plane are studied by considering quantities are expressed as functions of the hyperbolic trigonometry. In invariant under an action of the general Mobius¨ group,which Section 3 we explore Euclidian properties of the curvilinear consists of compositions of Mobius¨ transformations and triangle bounded by circular arcs of three intersecting semi- reflections [4]. The curvilinear triangle formed by circular circles. Two main relationships connecting three intersecting arcs of three intersecting semicircles is one of the principal semicircles are established. In Section 4,weprovethetheo- figures of the upper half-plane model . The hyperbolic rem of cosines for the curvilinear triangle bounded by the 2 Journal of Mathematics 1 This formula we denominate as the key formula and use it 2 in order to introduce hyperbolic trigonometry on Euclidean II(a) plane. Notice that the argument of the exponential function is proportional to the difference between the numerator and the denominator, 1 −2 =1 −2, and this difference does not depend on the parameter . 2.2. Elements of Right-Angled Triangle as Functions of a Hyper- Figure 1: Motion of the hypothenuse of the right angled triangle. bolic Trigonometry. Let bearight-angledtrianglewith right angle at . Denote the sides by ,,thehypotenuseby , and the angles opposite to , , by , , , correspondingly. circular arcs. In Section 5, the hyperbolic law of sines and Traditionally interrelations between angles and sides of a second hyperbolic law of cosines are derived on the basis of triangle are described by the trigonometry via periodical the main relationships between these semicircles. sine-cosine functions. The periodical functions of the circular angles are defined via the ratios 2. Hyperbolic Trigonometry in sin = , tan = . (10) Euclidean Geometry 2.1. Trigonometry Induced by General Complex Algebra. The Notice that these two ratios are functions of the same angle simplest generalization of the complex algebra, denominated .Onmakinguseofformula(9)weareabletointroducethe as General Complex Algebra, is defined by unique generator e hyperbolic trigonometry besides the circular trigonometry. satisfying the quadratic equation [5] Define the following relationships: + e2 −e + =0, = (2) . 1 0 (3) − exp (11) where coefficients 0,1 are given by real positive numbers 2 From this equation it follows that and −40 >0. 1 The following Euler formula holds true: = () , = () . tanh sinh (12) exp (e) =0 () + e1 () . (4) In this way we arrive to the following interrelations between Denote by 1,2 ∈rootsofthequadraticequation(3). circular and hyperbolic functions: The Euler formula (4) is decomposed into two independent 1 equations: cos = = tanh () , tan = = . (13) sinh () exp () =0 () +1 () , =1,2, (5) Now let us introduce the following geometrical motion; namely, change position of the point along the line from which one may find explicit formulae for -functions. These functions are linear combinations of the exponential (see, Figure 1). Then, the sides and will change while the side will not change. Since the length is a constant of this functions exp(), =1,2. Geometrical interpretation of the general complex algebra is done in [6]. evolution, in agreement with key formula (9)werewrite(11) Form the following ratio as follows: + = exp (2) , (14) 2 −() − exp ((2 −1))= , (6) 1 −() that is, the argument of exponential function is proportional = where to : ,where is a parameter of the evolution. Formulae in (13) are rewritten as 0 () ()=− . (7) 1 1 () cos = = tanh () , tan = = . (15) sinh () Introduce a pair of variables 1, 2 by From these formulas it is derived that () = −(), =1,2. (8) cot =sinh () , (16) Then, (6) is written as follows: or, () (( − ))= 2 . exp 2 1 (9) tan = exp (−) . (17) 1 () 2 Journal of Mathematics 3 2 1 1 2 1 2 Figure 2: Semicircle and lines tangent to the semicircle. Install the triangle in such a way that the side = In this way, we have established connection of formula (17) lies on the line ,andtheside=is perpendicular to with the main formula of hyperbolic geometry (19). this line at the point In Figure 1 the line movesinsuchawaythatcutsline 2.3. Rotational Motion of a Line Tangent to the Semicircle. The at points , , , and so forth. Now, let us recall the concept of the circular angle in Euclidean plane is intimately problemofparallellinesingeometry[7]. Let the ray tend relatedtothefigureofacircleandtomotionofapointalong to a definite limiting position; in Figure 1 this position is given the circumference. The hyperbolic angle is also related to the by line 1.Asthepoint moves along line away from point circle because of a motion along the circumference coherence there are two possibilities to consider. with the motion along the hyperbola [9]. C (1) In Euclidean geometry, the angle between lines 1 and Consider semicircle (Figure 2) with end-points and is equal to right angle. the center on -axis. Denote by the center and by 1, 2 the end-points of the semicircle. Through end-points of the (2) The hypothesis of hyperbolic geometry is that this semicircle, 1, 2 erect the lines parallel to vertical axis, - angleislessthantherightangle. axis. Draw a line tangent to the semicircle at the point ; The most fundamental formula of the hyperbolic geom- this line crosses -axis at the point and intersects with the Π() etry is the formula connecting the angle of parallelism vertical lines at points 1 and 2. Draw a line parallel to -axis and the length of the perpendicular from the given point from the center which crosses C at the top point . to the given line. In order to establish those relationships Denote by radius of the circle, so that ==and the concept of horocycles, some circles with center and axis =12.Denoteby the angle ∠.Thetriangle at infinity, was introduced [8]. The great theorem which is a right triangle, so that enables one to introduce the circular functions, sines, and ()2 − ()2 =2. cosinesofanangleisthatthegeometryofshortestlines (21) (horocycles) traced on horosphere is the same as plane Consider rotational motion of the line tangent to the semi- Euclidean geometry. The function connecting the angle of circle at the point C.InFigure 2,twopositionsofthislineare parallelism with the distance is given by given by lines 2 and 1. When the point runs from Π () end point 1 to the top-point ,thepoint runs along - (− )= . (18) exp tan 2 axis from point 1 to infinity. During this motion the triangle K =1/ formed by the line tangent to the semicircle, the -axis and Now, we introduce the value inverse to by and the radius of the circle remains to be right angled. This is rewrite (17)intheform exactly the case considered in Section 2.2,consequently,we can apply formula (13). According to (13)wewrite exp (− )=tan . (19) K 2 = coth () = ,1 = =tan . Let the point run away from the point to infinity. The cos sinh () following two cases can be considered.
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