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Some Comments on Isometries of Sphere 핊ퟐ Годишњак Педагошког факултета у Врању, књига VIII, 1/2017. Milena BOGDANOVIĆ УДК 514.113.6 511.84 Marko STANKOVIĆ - оригинални научни рад - Marija JORDANOVIĆ Pedagogical faculty in Vranje University of Niš SOME COMMENTS ON ISOMETRIES OF SPHERE 핊ퟐ Abstract: The aim of the paper is to show that isometries of sphere are restrictions of isometries in ℝ3, and it is a consequence of the reflections of large cir- cles. This paper presents a modified proof of the theorem of three reflections in 핊2 and the classification of isometries in 핊2. The group of isometries of sphere 핊2 is ”very large”; that confirms the theorem of 2-transitivity of sphere. The proof of this theorem is given in the paper. The group of rotations of sphere is especially emphasi- zed and it is shown that the product of any two rotations in 핊2 is rotation, too. At the end, the paper contains interesting examples of application of isometries of sphere. Key words: Isometries of sphere 핊2, plane reflection, 2-transitivity of sphe- re, group of rotations of sphere. Introduction The unit sphere in ℝ3 consists of all points on the unit distance from point 푂; these are all points (푥, 푦, 푧) that satisfy the equation 푥2 + 푦2 + 푧2 = 1. This sur-face is also called 2-sphere, or 핊2, because the points of sphere may be represented by two coordinates - width and height, for example. That geometry is essentially two-dimensional, as the Euclidean plane ℝ2 or the real projective plane ℝℙ2, and indeed, the basic objects of spherical geometry are ”points” (regular point on the sphere) and ”lines” (big circles on the sphere). A distance 푋푌⏜ between points of the sphere 푋 and 푌 is defined as 푋푌⏜ = 훼, cos 훼 = 〈푂푋⃗⃗⃗⃗⃗ , 푂푌⃗⃗⃗⃗⃗ 〉, 0 ≤ 훼 ≤ 휋. The next theorem from (Blažić, Bokan, Lučić & Rakić, 2003) holds in spherical geometry: Theorem 1 The function ⏜. : 핊2 × 핊2 → [0, 휋] is inner metric on 핊2. If 푋, 푌 are points of sphere 핊2, such that 푋 ≠ 푌, than there is the unique shortest curve that connects 푋 and 푌. It is a shorter arc of the great circle, that passes through the points 푋 and 푌. If 푋 = −푌, the shortest curve from 푋 to 푌 is great circle with ends 푋, −푋. 65 The isometries First we introduce definition of isometric transformation in the same manner as (Lopandić, 2011). Definition 1 The isometric transformation or geometric motion of the space 퐸푛 (푛 = 1,2,3) is a bijective transformation 푓: 퐸푛 → 퐸푛so that for each two points 푋, 푌 ∈ 퐸푛 and their images 푋′, 푌′ ∈ 퐸푛 the following relation is valid (푋, 푌) ≅ (푋′, 푌′). Therefore, if 푓 is an isometry that fixes point 푂, than 푓 maps all of the points on distance 1 from 푂 into other points on the same distance from 푂. In other words, isometry 푓 in ℝ3 that fixes 푂 maps 핊2, into itself. A restriction of isometry 푓 in 핊2, is isometry in 핊2, because 푓 keeps the distance on 핊2, as in ℝ3. Isometries in 핊2, are functions of 핊2, into itself that keep the distance on the great circle. The simplest isometries in ℝ3 with a fixed point 푂 are reflections in a plane through 푂. Corresponding isometries in 핊2 are reflections in the great cir- cles. The product of reflections 휋1 and 휋2 is rotation about an axis 푙 (for dou- ble angle between 휋1 and 휋2), wherein 푙 is an intersection of the planes 휋1 and 2 휋2. This situation is completely analogous to that in ℝ where the product of reflections through 푂 is rotation (for double angle between lines). Finally, those are products of three reflections in planes which are different from products of one or two reflections in planes. One of those isome-tries is anti- podal map that maps each point (푥, 푦, 푧) into itself antipodal point (−푥, −푦, −푧). The following theorem shows why all isometries in 핊2 are restrictions of isometries in ℝ3. Namely, this is true because of the reflections in great circles. Reflection in a great circle in 핊2 corresponds to reflection in a plane in ℝ3. Theorem 2 (Three Reflections Theorem in 핊2) Any isometry of 핊2 is the com-position of one, two or three reflections in great circles. Proof 1 Let 푓 be an arbitrary isometry of sphere and 퐴, 퐵, 퐶 three ”non-collinear” points on sphere, and let 퐴1, 퐵1, 퐶1 be images of those points in isometry 푓, respectively. We distinguish the following cases. Case 1 If 퐴 = 퐴1, 퐵 = 퐵1, 퐶 = 퐶1, 푓 is identity so, if 푎 is an arbitrary line, then 푓 = 푎푎. Therefore, suppose that points 퐴, 퐵, 퐶, are not invariant in isometry 푓 but one of them, for example 퐴, maps by isometry 푓 into some point on the sphere differente from 퐴. If 1 is reflection which maps 퐴 into 퐴1, then 퐴 is invariant point in composition 1푓. If 퐵, 퐶, are invariant points in that com- position, then 푓 = 1. Case 2 Suppose that in isometry 1푓 at least one of the points 퐵, 퐶, for example 퐵, isn’t invariant but 퐵 maps into 퐵2, and 퐶 into 퐶2. Let 2 be the reflection in a line which maps 퐵 into 퐵2. In that case (퐴, 퐵) ≅ (퐴1, 퐵1) ≅ (퐴, 퐵2), so point 퐴 belongs to basis 푎2 of the reflection 2. Therefore, points 퐴 and 퐵 are invariant in composition 21푓. If 퐶 is invariant point, that composition is identity so 푓 = 12. 66 Case 3 Suppose that in isometry 21푓 point 퐶 isn’t invariant, but it maps into 퐶3. Let 3 be reflection which maps 퐶 into 퐶3. In that case (퐴, 퐶) ≅(퐴1,퐶1)≅ (퐴, 퐶2) ≅ (퐴, 퐶3) and (퐵, 퐶) ≅(퐵1,퐶1)≅(퐵2, 퐶2 )≅ (퐵, 퐶3), so points 퐴 and 퐵 belong to the basis 푎3 of the reflection 3. Therefore, points 퐴, 퐵, 퐶 are invariant in composition 321푓, so 푓 = 123. The proof of the theorem is completed. The importance of reflections is reflected in the fact that each isometry of a line, plane, or space can be expressed as composition of finite number of reflections. The following theorems from (Lučić, 1997) are held in ℝ3: Theorem 3 Each isometry of a line can be expressed as composition of maximum two reflections in lines. Theorem 4 Each isometry of a plane can be expressed as composition of maximum three reflections in lines. Theorem 5 Each isometry of a space can be expressed as composition of maxi-mum four reflections in planes. All of isometries of space can be divided onto direct (composition of even number of reflections) and indirect (composition of odd number of reflec- tions). Direct isometries of space are: coincidence, translation, rotation about an axis, screw displacement. Indirect isometries of space are: reflection in a plane, improper rotation, glide reflection. Based on the previous, we can perform the classification of isometries of sphere. Direct isometries of sphere 핊2) are: – coincidence (because 푓(푂) = 푂 where 푂 is center of sphere, and 푓 is coincidence) – rotation about an axis (composition of two reflections in planes that contain center of sphere 푂. Whereas point 푂 belongs to both planes it follows that 푂 belongs to their intersection i.e. point 푂 belongs to the axis of rotation so 푓(푂) = 푂). Translation and screw displacement don’t have fixed points so these isometries aren’t isometries of sphere. Indirect isometries of sphere 핊2 are: – reflection in a plane (plane contains the center of sphere 푂, so 푓(푂) = 푂) – improper reflection (the composition of three reflections in planes; each of the planes contains the center of sphere O. In the special case when all planes are mutually perpendicular, their intersection is cen- ter of sphere and that isometry is symmetry of the space). The glide reflection has no fixed points so this isometry isn’t isometry of sphere. Theorem 6 If an isometry in ℝ2 has more than one invariant point, it must be either the identity or a reflection. The previous theorem is taken from (Coxeter, 1969). Considering the fact that the geometry of sphere is two-dimensional, analogous theorem holds in 핊2. 67 The transitivity of group of isometries The previous facts raise the question of ”size of the group of isometries of sphere 핊2”. A suitable notion for this question is the transitivity of group of transformations. We will say that a group of isometries Isom(ℳ), of some set ℳ, is transitive, if for two arbitrary points 퐴 and 퐵 from ℳ there is an isome- try 푓: ℳ → ℳ, so that 푓(퐴) = 퐵. For example, the group of isometries of sphere 핊2 is transitive, because for any two points of sphere 퐴 and 퐵, there is a relation of space which axis of rotation is perpendicular to the plane determined by points 푂, 퐴, and 퐵 and which translates point 퐴 into point 퐵. Therefore, this rotation is actually rotation in the plane determined by points 푂, 퐴, and 퐵 beca- use two points of axis of rotation are fixed in this rotation. Analogously we also introduce the following notion. We say that group of isometries of set ℳ is 2-transitive if for each two pairs of points 퐴, 퐵 and 퐴1, 퐵1 which are on the same distances (i.e. if 푑(퐴, 퐵) = 푑(퐴1, 퐵1)), there is an isometry 푓: ℳ → ℳ, which maps 퐴 into 퐴1, and 퐵 into 퐵1. For example, affine space is one example of the spaces with 2-transitive group of isometries.
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