On The Gyrobarycentric Coordinate Representation In Gyrovector Spaces

In1988, Abraham A. Ungar discovered that the Einstein velocity addition law encodes a grouplike algebraic object that became known as a gyrocommu-tative gyrogroup. Furthermore, it was later discovered that Einstein addition admits hyperbolic scalar multiplication, giving rise to the so called gyrovector spaces. The latter turned out to form a nonassociative algebraic setting for the Beltrami-Klein ball model of hyperbolic geometry just as vector spaces form an associative algebraic setting for the standard model of Euclidean geometry. This new discovery will turn to be a powerful way in studying the problems in hyperbolic geometry. Analogy is a good way in our research. We employ our analogies that hyperbolic geometry and relativistic mechanics share with Euclidean geometry and classical mechanics.In2009, Abraham A. Ungar published his newest result on gyrobarycen-tric coordinate representation in the Einstein gyrovector space and Mobius gyrovector space. A lot of expert in hyperbolic geometry focused on his work and did some new result by Ungar’s theorem, such as Milton Ferreira([30,31]), Nilgun Sonmez([31]), Tuval Fogucl ([35]), C.barbu([24,25]). This approach made the study of gyrotriangle centers, in low or high dimension, going up to a new stage. The study of gyrospace also communicates with the development of other fields, such as hyperbolic geometry, relativistic theory, quantum in-formation and Clifford algebra, playing an important role in advancing these fields.In this paper, the author mainly studied the gyrosymmedianpoint, a spe-cial point planted into gyrospace by gyrolanguage, and its interesting prop-erties by the gyrobaryeentric coordinate representation in the Einstein gy-rovector space and Mobius gyrovector space. The paper is divided into three chapters.In chapter1, we simply introduce the background of this paper, such as an introduction to Einstein and Mobius gyrovector space, gyrogroups and their properties, and the relationship between gyrospace and hyperbolic geometry. We, also, give the relation of the two type of the gyrovector spaces and the representations between these two gyrospaces. They are isomorphism. In chapter2, we introduce the gyrobarycentric coordinate representation and its resulting results obtained by Abraham A. Ungar in1988. In order to let us clearly understand this representation, we give the results of gyrocentroid and gyroorthocenter of a gyrotriangle in Einstein gyrovector space. Then, we will translate an important concept, symmedian point, into gyrospace, it is called gyrosymmedian point in the new place. The method is similar to Ungar’s by which he did the gyrobarycentric coordinate research on other im-portant centers in gyrospace. We will study its corresponding gyrobarycentric coordinate representation and its properties, and obtain the following resluts:Theorem2.1(Einstein Gyrotriangle gyrosymmedian point) Let set{A1, A2, A3} be a pointwise independent set of three points in an Einstein gyrovector space (Rsn,(?),(?)), n≥2. The gyrosymmedian point S of gyrotrian-gle A1,A2, A3has the gyrobarycentric coordinate representationwith respect to the set{A1, A2,A3}, with gyrobarycentric coordinate (m1:m2:m3)=(α232:γ232:a132:γ132:α122γ122)(0.12) orwith respect to the set{A1, A2, A3}, with gyrobarycentric coordinate (m1:m2:m3)=(γ232-1:γ132-1:γ122-1)(0.14)For the gyrosymmedian point, we can obtain a lot of good properties, and that is why we researched the gyrosymmedian point. The following properties2.2-2.5will answer the question:whether some good properties of symmedian point in Euclidean Space are still true, or will be true if some conditions are improved. Property2.2:Let S be the gyrosymmedian point of gyrotriangle A1A2A3, and let D1, D2and D3, respectively, be the orthogonal projection of S onto the every gyroside of the gyrotriangle. Then,(γSD1‖SD1‖:γSD2‖SD2‖:-γSD3‖SD3‖)=(γ23α23:γ13α12)(0.15) where β12, α13, α23have the definition at the section2. In order prove Property2.2, we first consider the following Property2.3.Property2.3:Let set{A1, A2,A3} be a pointwise independent set of three points in an Einstein gyrovector space (Rsn,(?),(?)), n≥2. Let Sx be any point between A3and F12on the gyrosymmedian A3F12of the gyrotriangle AiA2A3, SxD31be the gyroaltitude of gyrotriangle A3SxA2drawn from vertex Sx; to its foot D31on its opposite side A2A3or its extension part. Similar, we can define point D32on side A1A3or its extension. Then we will have the following relationshipFor Property2.2and Property2.3, the converse is also true. We consider the converse of Property2.3first, then, the proof of the converse of Property2.2will be obvious.Property2.4(The Converse of Property2.3):Let set{A1, A2, A3} be a pointwise independent set of three points in an Einstein gyrovector space (Rs,(?),(?)), n≥2. Let Px be any point inside the gyrotriangleA1A2A3that satisfies the proportionwhere PxH31be the gyroaltitude of gyrotriangle A3PxA2drawn from vertex Px to its foot H31on its opposite side A2A3or its extension part. Similar, we can define point H32on side A1A3or its extension. Then, Px must lie on the gyrosymmedian A3F12.We can immediately give the results on the points at the other two gy-rosymmedians. We can describe them as in the Property2.4’.Property2.4’Let set{A1, A2,A3} be a pointwisc independent set of three points in an Einstein gyrovector space (Rs,(?)(?)), n≥2. Let Py (Pz) be any point inside the gyrotriangle.A1A2.A3that satisfies the proportion in (0.17)((0.18)), where PyH23(PzH12) be the gyroaltitude of gyrotriangle A1PzA2(A1PzA3) drawn from vertex Py (Pz) to its foot H23(H12) on its opposite side A1A2(A1A3) or its extension part. Similar, we can define point H21(H13) on side A2A3(A1A2)or its extension. Then, Py (Pz) must lie on the gyrosymmedian A2F13(A1F23).Property2.5(The Converse of Property2.2):Let set{A1,A2,A3} be a pointwise independent set of three points in an Einstein gyrovector space (Rsn,(?),(?)), n≥2. Let P be any point inside of the gyrotriangleA1A2A3, and let H1,H2and H3, respectively, be the orthogonal projection of P onto the every gyroside of the gyrotriangle. If P satisfies the proportion(γPH1‖PH1‖:γpH2‖PH2‖:γPH3‖PH3‖)=(γ23α23:γ13α13:712α12)(0.20) Then, P must be the gyrosymmedian point.In chapter3, we continue to study the gyrosymmedian point in Mobius gyrovector space.Theorem3.1:Let A1A2A3be a gyrotriangle in a Mobius gyrovector space (Vs,(?),(?))-The gyrosymmedian point S of the gyrotriangle is given by the equation where the homogeneous gyrobarycentric coordinates (m1:m2:m3) are given by and where γij=γ(?)MAi(?)MAij (0.23) i.j=1,2,3, and i<j.In chapter4, we will continue to study the gyrosymmedian point in the3-gyrosimplex, our research method in this chapter is similar to that in chapter2and3. Lemma4.1:In an Einstein gyrovector space(Rsn,(?),(?)),n≥3,a3-gyrosimplex has a gyrosymmedian point if and only if it satisfies the following condition (γ232-1)(γ142-1)=(γ122-1)(γ342-1)=(γ242-1)(γ132-1), or α232γ232α142γ142=α122γ122α342γ342=α132γ132α242γ242, where the definitions of all gamma factors above Can be found in the proof of chapter4.Theorem4.2:Let set {A1,A2,A3,A4}be a pointwise independent set of four points in an Einstein gyrovector space(Rsn,(?),(?)),n≥3.The gyrosym-median point of the3-gyrosimplex A1A2A3A4that is formed by A1,A2,A3,A4has the gyrobarycentric coordinate representation,under the condition of Lem-ma4.1. where(m1:m2:m3:m4)can be written as α132γ132α242γ242:α132γ132α142γ142:α122γ122α142γ142:α122γ122α132γ132, which is the gyrobarycentric coordinates of the gyrosymmedian point with respect the set{A1,A2,A3,A4},it can also be written as ((γ132-)(γ242-1):(γ132-1)(γ142-1):(γ122-1)(γ142-1):(γ122-1)(γ132-1)), according to the condition in Lemma4.1,we can obtain other forms,but the proportion will not change.Theorem4.3:Let A1A2A3A4be a3-gyrosimplex in a Mobius gyrovector space(Vs,(?),(?))which satisfies the following condition γ132γ242(γ132-1)(γ242-1)=γ142γ232(γ142-1)(γ232-1)=γ122γ342(γ122-1)(γ242-1), then there is a gyrosymmedian point inside of the simplex,and its gyrobarycon-tric coordinate representation,with respect the set A1,A2,A3,A4,can be writ-t,en as where S is the gyrosymmedian point of3-gyrosimplex A1A2A3A4in the Mobius gyrovector space(Vs,(?),(?)),(m1:m2:m3:m4)is its gyrobarycentric coordi-nates as following m1=16γ232γ242(γ232-1)(γ242-1), In Mobius gyrovector space, there are many other forms that can repre-sent the gyrobarycentric coordinates for the gyrosymmedian point. Of course, they are equivalence.