HOMOTHETIC MOTIONS AND BICOMPLEX NUMBERS

In this study, one of the concepts of conjugate which is de…ned [1] for bicomplex numbers is investigated. In this case, the metric, in four dimensional semi-Euclidean space E2, has been de…ned by the help of the concept of the conjugate. We de…ne a motion in E2 with the help of the metric in bicomplex numbers. We show that the motions de…ned by a curve lying on a hypersurface M of E2 are homothetic motions . Furthermore, it is shown that the motion de…ned by a regular curve of order r and derivations of the curve on the hypersurface M has only one acceleration centre of order (r-1) at every tinstant. 1. Introduction In 2006, Dominic Rochon and S.Tremblay , presented a paper based on bicomplex quantum mechanics : II. The Hilbert Space [1, 2]. Bicomplex (hyperbolic) numbers are given in this paper from a number of di¤erent points of view of Hilbert Space for quantum mechanics. In this study , a new operator similar to Hamilton operator [3] has been given for bicomplex numbers [4] , homothetic motion has been de…ned by the help of the components of the hyper surface and di¤erent theorems have been given . It is shown that this study can be repeated for bicomplex numbers,which is a homothetic motion in four-dimensional semi-Euclidean spaces and this homothetic motion satis…es all of the properties [5]. 2. Bicomplex Numbers Bicomplex numbers are de…ned by [1, 2, 6] T = fz1 + z2 i2 : z1; z2 2 C(i1)g where the imaginary units i1; i2 and j are governed by the rules: i1 = i 2 2 = 1 ; j = 1 Received by the editors March 16, 2009, Accepted: June. 05, 2009. 2000 Mathematics Subject Classi…cation. 53A05, 53A17. Key words and phrases. Bicomplex number, Homothetic motion, Hypersurface, Pole points, Semiorthogonal matrix. c 2009 Ankara University 23 24 FAIK BABADA 1⁄4 G, YUSUF YAYLI AND NEJAT EKMEKCI i1:i2 = i2:i1 = j : i1:j = j:i1 = i2 : i2:j = j:i2 = i1 where we de…ne C(ik)= x+ y ik : i 2 k = 1 and x; y 2 R for k = 1; 2: Hence it is easy to see that the multiplication of two bicomplex numbers is commutative . It is also convenient to write the set of bicomplex numbers as T = fw j w = w1 + w2 i1 + w3 i2 + w4 j j (w1; w2; w3; w4) 2 R:g Complex conjugation plays an important role both for algebraic and geometric properties of complex numbers [1, 2]. w = z1 + z2i2 = z1 z2i2 = w1 + w2 i1 w3 i2 w4 j where w:w = w 1 w 2+w 3 w 4 + 2i2 (w1w2+w3w4) and w1w2+w3w4 = 0 i.e w:w = w 1 w 2 + w 3 w 4 2 R The system fT; ; R;+; :; ; g is a commutative algebra. It is referred as the bicomplex number algebra and shown with T , brie‡y one of the bases of this algebra is f1; i1; i2; jg and the dimension is 4. 2.1. Multiplication Operation. The operation : T T ! T (u;w) ! u w = w u is de…ned with the following multiplication u w = (u1 + i1u2 + i2u3 + ju4) (w1 + i1w2 + i2w3 + jw4) (2) = (u1w1 u2w2 u3w3 + u4w4) + i1(u1w2 + u2w1 u3w4 u4w3) +i2(u1w3 u2w4 + u3w1 u4w2) + j(u1w4 + u2w3 + u3w2 + u4w1) It is possible to give the production in T similar to the Hamilton operators which has been given in [3]. Because it is not a quaternion commutative matrix, there are two di¤erent matrixes for each of the right and left-multiplications. However, here only one matrix is obtained. Because it is similar to Hamilton operators. (for Hamilton operators see [3, 5]). If w = w1 + w2 i1 + w3 i2 + w4 j is a bicomplex number ,then N = N = N is de…ned as N(w) = 664 w1 w2 w3 w4 w2 w1 w4 w3 w3 w4 w1 w2 w4 w3 w2 w1 775 If w = z1 + z2 i2 then N(w) = N(z1) N(z2) N(z2) N(z1) HOMOTHETIC MOTIONS AND BICOMPLEX NUMBERS 25 Using the de…nition of N , the multiplication of two bicomplex numbers x and y is given by w u = N(w):u : u;w 2 T and detN(w) = w 1 w 2 + w 3 w 4 + 2(w1w2 + w3w4) 2 3. Homothetic Motions at E2 Let, M = fw = (w1; w2; w3; w4) j w1w2 + w3w4 = 0g be a hyper surface S 2 = w = (w1; w2; w3; w4) j w 1 w 2 + w 3 w 4 = 1 be a unit sphere , K = w = (w1; w2; w3; w4) j w 1 w 2 + w 3 w 4 = 0 be a null cone in E 2 . Let us consider the following curve: : I R ! M E 2 de…ned by (t) = [w1(t); w2(t); w3(t); w4(t)] for every t 2 I: We suppose that (t) is a di¤erentiable curve of order r. The operator B, corresponding to (t) is de…ned by B = N(w) = 664 w1 w2 w3 w4 w2 w1 w4 w3 w3 w4 w1 w2 w4 w3 w2 w1 775 (3) Let (t) be a unit velocity curve. The matrix can be represent as B = h 664 w1 h w2 h w3 h w4 h w2 h w1 h w4 h w3 h w3 h w4 h w1 h w2 h w4 h w3 h w2 h w1 h 775 (4)