DOCSLIB.ORG

Quaternions and Biquaternions for Symmetric Markov-Chain System Analysis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quaternions and Biquaternions for Symmetric Markov-Chain System Analysis 15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3-7, 2007, copyright by EURASIP QUATERNIONS AND BIQUATERNIONS FOR SYMMETRIC MARKOV-CHAIN SYSTEM ANALYSIS Soo-Chang Pei, Jian-Jiun Ding Department of Electrical Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., 10617, Taipei, Taiwan, R.O.C TEL: 886-2-23635251-321, Fax: 886-2-23671909, Email: [email protected], [email protected] ABSTRACT In this condition, j2 = 1, k2 = −1, ij = ji, ik = ki, and jk = kj. In this conference, we use quaternions, biquaternions, and The Clifford biquaternion and the complex Tessarine have their related algebras similar to them to model symmetric idempotent elements e1 and e2: Markov systems. With these algebras, the original N×N eI1 = (1+ ) / 2 , eI2 = (1− ) / 2 , (6) Markov system can be reduced into an (N/2)×(N/2) or 2 2 e1 = e1, e2 = e2, e1e2 = 0. (N/4)×(N/4) system. It makes the system easier for imple- Idempotent elements are helpful for fast computation. If q menting and analysis. In addition to Markov chains, the is defined in (3) and p is defined as proposed idea is also helpful for simplifying the complexi- pp=+⋅+⋅+⋅ pipjpk ties of other symmetric systems whose interactions between 0 ij k , (7) two objects are determined by their distance. + pIiIjIkIIpiIpjIpkI+⋅+⋅+⋅ then p and q can be expressed as: 1. INTRODUCTION p = pe+ pe , qqeqe= + , (8) 11 2 2 11 2 2 The quaternion [1][2] is a generalization of the complex where algebra. Its application in signal and image processing were p110= pe=+⋅+⋅+⋅( p pij i p j p k k noticed in recent years [2][3][4]. The quaternion has been , p22= pe . (9) ++ppipjpk ⋅+⋅+⋅)/2 applied for filter design, feature extraction, pattern recogni- IiIjIkI tion, rotation analysis, differential equation analysis, and Similarly, SVD decomposition, etc. In this paper, we propose another qqe11= and qqe22= . (10) application of the quaternion and its related algebras, i.e., Then, the product of p and q can be expressed as: the Markov chain analysis. pqpepeqeqepqepqe= ()()++=+. (11) 11 22 11 22 111 2 22 2. QUATERNIONS AND RELATED ALGEBRAS In addition to the quaternion and the Clifford biquater- nion, there are also other types of 4-D and 8-D algebras. Quaternions [1][2] are generalizations of complex numbers. The algebras used in this paper are: A quaternion has 4 components: real, i-, j-, and k-parts: [2-D algebras] • Complex algebra. q = qr + qi ⋅i + q j ⋅ j + qk ⋅k , (1) • Tessarines (also called as split-complex numbers, and and i, j, k obey the rules as below: double complex numbers) [5][6]: 2 2 2 i = j = k = −1, ij⋅= k, jk⋅= i, ki⋅= j, 2 qa= rI+ Ia, where I = 1. (12) ji⋅=− k, kj⋅=− i, ik⋅=− j. (2) [4-D algebras] Quaternions can be used for the color image analysis, filter • Quaternions (defined in (1) and (2)) design, and 3-D object analysis. • Complex Tessarines (defined in (5)). The definition of the Clifford biquaternion [6][10] is • Reduced biquaternion: See [3][7]. It is also called the similar to that of the quaternion, but there are 8 elements: bicomplex number (C×C). Its definition is the same as the 2 qq=+⋅+⋅+⋅0 qiqjqkij k reduced biquaternion in (5) but I = −1. , (3) ++⋅+⋅+⋅qI q iI q jI q kI • 4-D purely hyperbolic complex numbers: IiIjIkI qq= ++ qIqJq +⋅ IJ, I2 = J2 = 1, IJ = JI. (13) where i, j, and k satisfy (2) and rI J IJ [8-D algebras] I2 = 1, iI⋅=⋅ Ii, jI⋅=⋅ I j, kI⋅=⋅ Ik. (4) • Octonions (standard definition [1][9]). It has 7 imaginary The Tessarine with complex coefficients [5] (we call it the parts and the multiplicative rule is shown in [6]. complex Tessarine) is defined in the similar form. • Clifford biquaternions (see (3)). qq=+⋅++⋅ri qiqIqiI I iI. (5) • 8-D Tessarines (4-D hyperbolic complex numbers with complex coefficients) ©2007 EURASIP 1337 15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3-7, 2007, copyright by EURASIP JJG JJG q=+ ari ia + Ja J + iJa iJI + Ia + iIa iIJIiJI + JIa + iJIa , (b) If pqrs= − , then the interactions between p and q 2 2 2 where i = −1, I = J = 1, iI = Ii, iJ = Ji, and IJ = JI. (14) (denoted by hp,q) and r and s (hr,s)) has the relation of • Hamilton's biquaternions [1][3]: Its definition is the same hp,q = −hr,s as the biquaternion in (3) but I2 = −1. then • Tri-complex numbers x112[naxnbxn+=1] [ ][] − , (23) q=+ ari ia + Ia I + iIa iIJ + Ja + iJa iJK + Ka + iKa iK, x212[nbxnaxn+=1] [ ][] + , (24) where I2 = J2 = K2 = −1, Ii = iI, Ji = iJ, then, we can define a function of Ki = iK, IJ = JI, IK = KI, and JK = KJ. (15) x[nxnIxn] =+12[ ] [ ] (25) 3. 2×2 AND 4×4 SYMMETRIC SYSTEM where I2 = −1, iI = I i. (26) In this case, we can also prove that For many systems, the distance between two points will x nhxn+=⋅1 , where ha=+ι b. (27) determine the mutual effect. For example, in physics, the [ ] [ ] 2 gravitation between two objects is proportional to 1/R If x1[n] and x2[n] are complex functions, then the system is where R is the distance between two objects. Intensities of modeled by the following algebra: the electric and magnetic fields also depend on R. In social qq= 01+⋅++⋅ qiq 4ι qi 5ι . (28) science, the interaction between two cities can also be It is just the same as the reduced biquaternion algebra. modeled by a function of the distance between two points. Here, we suppose that there are N points in a Markov 1 3 system and the field in each point is denoted by xk[n], where k = 1, 2, …, N. Suppose that the interaction between two dots is determined by their distance: N xmkmk[1]nxna+=∑ [], , where afRmk,,= ( mk) , (16) 2 4 k =1 Rm,k is the distance between m and. Fig. 2 A doubly symmetric 4 × 4 rectangular system. We find that in the system is symmetric, we can model The idea can be extended to any symmetric Markov sys- the system by the complex Tessarine. tem. For the system in Fig. 2, suppose that the values at the For example, for the system of Fig. 1 points 1, 2, 3, and 4 are 1 2 x1[n], x2[n], x3[n], and x4[n] (29) and the interactions between the points m, n are denoted by hm,n. Therefore, Fig. 1 A symmetric 2 × 2 system. ⎡ x11[1]nxn+ ⎤⎡⎤⎡⎤hhhh1,1 1,2 1,3 1,4 [] ⎢⎥ If the interaction is determined by the instance between two ⎢x [1]nxn+ ⎥⎢⎥hhhh [] dots, it can be modeled by the following Markov chain: ⎢ 22⎥⎢⎥= ⎢⎥2,1 2,2 2,3 2,4 . (30) ⎢x [1]nxn+ ⎥⎢⎥⎢⎥hhhh [] x naxnbxn+=1 + , (17) 333,1 3,2 3,3 1,4 112[ ] [] [] ⎢ ⎥⎢⎥⎢⎥ hhhh ⎣x44[1]nxn+ ⎦⎣⎦⎣⎦⎢⎥4,1 4,2 4,3 4,4 [] x212[nbxnaxn+=1 ] [] + []. (18) First, note that Then, we can define a function as hm,n = hn,m. (31) x[]nxnIxn=+12 [] [], (19) Moreover, since interactions are determined by distances, 2 where I = 1. If we define h1,1 = h2,2 = h3,3 = h4,4 = a h1,2 = h2,1 = h3,4 = h4,3 = b, haIb=+ , (20) h1,3 = h2,4 = h3,1 = h4,2 = c, h1,4 = h2,3 = h3,2 = h4,1 = d. (32) then Then (30) can be rewritten as: x [1]nxn+ abcd [] h⋅= x[] n ax12 [] n + bx [] n + I() ax 21 [] n + bx [] n , (21) ⎡ 11⎤⎡⎤⎡⎤ ⎢ ⎥⎢⎥⎢⎥ i.e., . (22) x22[1]nxn+ badc [] x[]nhxn+=⋅1 [] ⎢ ⎥⎢⎥= ⎢⎥ (33) If x1[n] and x2[n] are complex functions, then (22) is a ⎢x33[1]nxn+ ⎥⎢⎥⎢⎥cdab [] complex Tessarine operation. That is, the system 1 can be ⎢ ⎥⎢⎥⎢⎥ ⎣x44[1]nxn+ ⎦⎣⎦⎣⎦dcba [] modeled by a complex Tessarine operation. If x [n] and 1 If we set that x2[n] are quaternion functions, Fig. 1 can be modeled by a Clifford biquaternion operation. yn11[ ] =+ xn[ ] Ixn 2[ ] , yn23[]=+ xn [] Ixn 4[ ] , (34) If the effect of the system is determined by the only the haIb= + , hcId= + . distance but also the direction, that is, 1 2 Then (33) can be rewritten as (a) The interaction between JJGthe pointsJJG p and q and that be- ⎡ yn1121[1]+ ⎤⎡ h h ⎤⎡ yn [] ⎤ tween r and s are the same pq and rs have the same dis- ⎢ ⎥⎢= ⎥⎢ ⎥. (35) tance and direction. ⎣ yn2212[1]+ ⎦⎣ h h ⎦⎣ yn [] ⎦ ©2007 EURASIP 1338 15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3-7, 2007, copyright by EURASIP That is, after converting into the complex Tessarine form, Table 1 2×2 and 4×4 symmetric system simplification by quater- the 4 × 4 system become a 2×2 system. nion and octonion. Moreover, notice that (35) is also a symmetric system. original system algebra reduced system We can further reduce it into the 1 × 1 form. We can apply 2×2 symmetric complex 1-to-1 system the 8-D Tessarines algebra. We can set that system Tessarine 2×2 directional reduced 1-to-1 system yn[ ] =+ x123[ n] Jxn[ ] + Ixn[ ] + IJxn 4[ ] , (36) symmetric system biquaternion Then 2×2 symmetric Tessarine 1-to-1 system yn+=1 hyn where h=+ a Jb + Ic + IJd . (37) [ ] [ ] system, real input That is, after applying the complex Tessarines, the 4 × 4 2×2 symmetric sys- Clifford 1-to-1 system symmetric rectangular system becomes a 2 × 2 symmetric tem, quaternion input biquaternion system.
Load more

Recommended publications
  • Arxiv:1001.0240V1 [Math.RA]
    Fundamental representations and algebraic properties of biquaternions or complexified quaternions Stephen J. Sangwine∗ School of Computer Science and Electronic Engineering, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, United Kingdom. Email: [email protected] Todd A. Ell† 5620 Oak View Court, Savage, MN 55378-4695, USA. Email: [email protected] Nicolas Le Bihan GIPSA-Lab D´epartement Images et Signal 961 Rue de la Houille Blanche, Domaine Universitaire BP 46, 38402 Saint Martin d’H`eres cedex, France. Email: [email protected] October 22, 2018 Abstract The fundamental properties of biquaternions (complexified quaternions) are presented including several different representations, some of them new, and definitions of fundamental operations such as the scalar and vector parts, conjugates, semi-norms, polar forms, and inner and outer products. The notation is consistent throughout, even between representations, providing a clear account of the many ways in which the component parts of a biquaternion may be manipulated algebraically. 1 Introduction It is typical of quaternion formulae that, though they be difficult to find, once found they are immediately verifiable. J. L. Synge (1972) [43, p34] arXiv:1001.0240v1 [math.RA] 1 Jan 2010 The quaternions are relatively well-known but the quaternions with complex components (complexified quaternions, or biquaternions1) are less so. This paper aims to set out the fundamental definitions of biquaternions and some elementary results, which, although elementary, are often not trivial. The emphasis in this paper is on the biquaternions as an applied algebra – that is, a tool for the manipulation ∗This paper was started in 2005 at the Laboratoire des Images et des Signaux (now part of the GIPSA-Lab), Grenoble, France with financial support from the Royal Academy of Engineering of the United Kingdom and the Centre National de la Recherche Scientifique (CNRS).
    [Show full text]
  • Hypercomplex Algebras and Their Application to the Mathematical
    Hypercomplex Algebras and their application to the mathematical formulation of Quantum Theory Torsten Hertig I1, Philip H¨ohmann II2, Ralf Otte I3 I tecData AG Bahnhofsstrasse 114, CH-9240 Uzwil, Schweiz 1 [email protected] 3 [email protected] II info-key GmbH & Co. KG Heinz-Fangman-Straße 2, DE-42287 Wuppertal, Deutschland 2 [email protected] March 31, 2014 Abstract Quantum theory (QT) which is one of the basic theories of physics, namely in terms of ERWIN SCHRODINGER¨ ’s 1926 wave functions in general requires the field C of the complex numbers to be formulated. However, even the complex-valued description soon turned out to be insufficient. Incorporating EINSTEIN’s theory of Special Relativity (SR) (SCHRODINGER¨ , OSKAR KLEIN, WALTER GORDON, 1926, PAUL DIRAC 1928) leads to an equation which requires some coefficients which can neither be real nor complex but rather must be hypercomplex. It is conventional to write down the DIRAC equation using pairwise anti-commuting matrices. However, a unitary ring of square matrices is a hypercomplex algebra by definition, namely an associative one. However, it is the algebraic properties of the elements and their relations to one another, rather than their precise form as matrices which is important. This encourages us to replace the matrix formulation by a more symbolic one of the single elements as linear combinations of some basis elements. In the case of the DIRAC equation, these elements are called biquaternions, also known as quaternions over the complex numbers. As an algebra over R, the biquaternions are eight-dimensional; as subalgebras, this algebra contains the division ring H of the quaternions at one hand and the algebra C ⊗ C of the bicomplex numbers at the other, the latter being commutative in contrast to H.
    [Show full text]
  • Gauging the Octonion Algebra
    UM-P-92/60_» Gauging the octonion algebra A.K. Waldron and G.C. Joshi Research Centre for High Energy Physics, University of Melbourne, Parkville, Victoria 8052, Australia By considering representation theory for non-associative algebras we construct the fundamental and adjoint representations of the octonion algebra. We then show how these representations by associative matrices allow a consistent octonionic gauge theory to be realized. We find that non-associativity implies the existence of new terms in the transformation laws of fields and the kinetic term of an octonionic Lagrangian. PACS numbers: 11.30.Ly, 12.10.Dm, 12.40.-y. Typeset Using REVTEX 1 L INTRODUCTION The aim of this work is to genuinely gauge the octonion algebra as opposed to relating properties of this algebra back to the well known theory of Lie Groups and fibre bundles. Typically most attempts to utilise the octonion symmetry in physics have revolved around considerations of the automorphism group G2 of the octonions and Jordan matrix representations of the octonions [1]. Our approach is more simple since we provide a spinorial approach to the octonion symmetry. Previous to this work there were already several indications that this should be possible. To begin with the statement of the gauge principle itself uno theory shall depend on the labelling of the internal symmetry space coordinates" seems to be independent of the exact nature of the gauge algebra and so should apply equally to non-associative algebras. The octonion algebra is an alternative algebra (the associator {x-1,y,i} = 0 always) X -1 so that the transformation law for a gauge field TM —• T^, = UY^U~ — ^(c^C/)(/ is well defined for octonionic transformations U.
    [Show full text]
  • Hypercomplex Numbers
    Hypercomplex numbers Johanna R¨am¨o Queen Mary, University of London [email protected] We have gradually expanded the set of numbers we use: first from finger counting to the whole set of positive integers, then to positive rationals, ir- rational reals, negatives and finally to complex numbers. It has not always been easy to accept new numbers. Negative numbers were rejected for cen- turies, and complex numbers, the square roots of negative numbers, were considered impossible. Complex numbers behave like ordinary numbers. You can add, subtract, multiply and divide them, and on top of that, do some nice things which you cannot do with real numbers. Complex numbers are now accepted, and have many important applications in mathematics and physics. Scientists could not live without complex numbers. What if we take the next step? What comes after the complex numbers? Is there a bigger set of numbers that has the same nice properties as the real numbers and the complex numbers? The answer is yes. In fact, there are two (and only two) bigger number systems that resemble real and complex numbers, and their discovery has been almost as dramatic as the discovery of complex numbers was. 1 Complex numbers Complex numbers where discovered in the 15th century when Italian math- ematicians tried to find a general solution to the cubic equation x3 + ax2 + bx + c = 0: At that time, mathematicians did not publish their results but kept them secret. They made their living by challenging each other to public contests of 1 problem solving in which the winner got money and fame.
    [Show full text]
  • Split Quaternions and Spacelike Constant Slope Surfaces in Minkowski 3- Space
    Split Quaternions and Spacelike Constant Slope Surfaces in Minkowski 3- Space Murat Babaarslan and Yusuf Yayli Abstract. A spacelike surface in the Minkowski 3-space is called a constant slope surface if its position vector makes a constant angle with the normal at each point on the surface. These surfaces completely classified in [J. Math. Anal. Appl. 385 (1) (2012) 208-220]. In this study, we give some relations between split quaternions and spacelike constant slope surfaces in Minkowski 3-space. We show that spacelike constant slope surfaces can be reparametrized by using rotation matrices corresponding to unit timelike quaternions with the spacelike vector parts and homothetic motions. Subsequently we give some examples to illustrate our main results. Mathematics Subject Classification (2010). Primary 53A05; Secondary 53A17, 53A35. Key words: Spacelike constant slope surface, split quaternion, homothetic motion. 1. Introduction Quaternions were discovered by Sir William Rowan Hamilton as an extension to the complex number in 1843. The most important property of quaternions is that every unit quaternion represents a rotation and this plays a special role in the study of rotations in three- dimensional spaces. Also quaternions are an efficient way understanding many aspects of physics and kinematics. Many physical laws in classical, relativistic and quantum mechanics can be written nicely using them. Today they are used especially in the area of computer vision, computer graphics, animations, aerospace applications, flight simulators, navigation systems and to solve optimization problems involving the estimation of rigid body transformations. Ozdemir and Ergin [9] showed that a unit timelike quaternion represents a rotation in Minkowski 3-space.
    [Show full text]
  • K-Theory and Algebraic Geometry
    http://dx.doi.org/10.1090/pspum/058.2 Recent Titles in This Series 58 Bill Jacob and Alex Rosenberg, editors, ^-theory and algebraic geometry: Connections with quadratic forms and division algebras (University of California, Santa Barbara) 57 Michael C. Cranston and Mark A. Pinsky, editors, Stochastic analysis (Cornell University, Ithaca) 56 William J. Haboush and Brian J. Parshall, editors, Algebraic groups and their generalizations (Pennsylvania State University, University Park, July 1991) 55 Uwe Jannsen, Steven L. Kleiman, and Jean-Pierre Serre, editors, Motives (University of Washington, Seattle, July/August 1991) 54 Robert Greene and S. T. Yau, editors, Differential geometry (University of California, Los Angeles, July 1990) 53 James A. Carlson, C. Herbert Clemens, and David R. Morrison, editors, Complex geometry and Lie theory (Sundance, Utah, May 1989) 52 Eric Bedford, John P. D'Angelo, Robert E. Greene, and Steven G. Krantz, editors, Several complex variables and complex geometry (University of California, Santa Cruz, July 1989) 51 William B. Arveson and Ronald G. Douglas, editors, Operator theory/operator algebras and applications (University of New Hampshire, July 1988) 50 James Glimm, John Impagliazzo, and Isadore Singer, editors, The legacy of John von Neumann (Hofstra University, Hempstead, New York, May/June 1988) 49 Robert C. Gunning and Leon Ehrenpreis, editors, Theta functions - Bowdoin 1987 (Bowdoin College, Brunswick, Maine, July 1987) 48 R. O. Wells, Jr., editor, The mathematical heritage of Hermann Weyl (Duke University, Durham, May 1987) 47 Paul Fong, editor, The Areata conference on representations of finite groups (Humboldt State University, Areata, California, July 1986) 46 Spencer J. Bloch, editor, Algebraic geometry - Bowdoin 1985 (Bowdoin College, Brunswick, Maine, July 1985) 45 Felix E.
    [Show full text]
  • Split Octonions
    Split Octonions Benjamin Prather Florida State University Department of Mathematics November 15, 2017 Group Algebras Split Octonions Let R be a ring (with unity). Prather Let G be a group. Loop Algebras Octonions Denition Moufang Loops An element of group algebra R[G] is the formal sum: Split-Octonions Analysis X Malcev Algebras rngn Summary gn2G Addition is component wise (as a free module). Multiplication follows from the products in R and G, distributivity and commutativity between R and G. Note: If G is innite only nitely many ri are non-zero. Group Algebras Split Octonions Prather Loop Algebras A group algebra is itself a ring. Octonions In particular, a group under multiplication. Moufang Loops Split-Octonions Analysis A set M with a binary operation is called a magma. Malcev Algebras Summary This process can be generalized to magmas. The resulting algebra often inherit the properties of M. Divisibility is a notable exception. Magmas Split Octonions Prather Loop Algebras Octonions Moufang Loops Split-Octonions Analysis Malcev Algebras Summary Loops Split Octonions Prather Loop Algebras In particular, loops are not associative. Octonions It is useful to dene some weaker properties. Moufang Loops power associative: the sub-algebra generated by Split-Octonions Analysis any one element is associative. Malcev Algebras diassociative: the sub-algebra generated by any Summary two elements is associative. associative: the sub-algebra generated by any three elements is associative. Loops Split Octonions Prather Loop Algebras Octonions Power associativity gives us (xx)x = x(xx). Moufang Loops This allows x n to be well dened. Split-Octonions Analysis −1 Malcev Algebras Diassociative loops have two sided inverses, x .
    [Show full text]
  • An Octonion Model for Physics
    An octonion model for physics Paul C. Kainen¤ Department of Mathematics Georgetown University Washington, DC 20057 USA [email protected] Abstract The no-zero-divisor division algebra of highest possible dimension over the reals is taken as a model for various physical and mathematical phenomena mostly related to the Four Color Conjecture. A geometric form of associativity is the common thread. Keywords: Geometric algebra, the Four Color Conjecture, rooted cu- bic plane trees, Catalan numbers, quaternions, octaves, quantum alge- bra, gravity, waves, associativity 1 Introduction We explore some consequences of octonion arithmetic for a hidden variables model of quantum theory and ideas regarding the propagation of gravity. This approach may also have utility for number theory and combinatorial topology. Octonion models are currently the focus of much work in the physics com- munity. See, e.g., Okubo [29], Gursey [14], Ward [36] and Dixon [11]. Yaglom [37, pp. 94, 107] referred to an \octonion boom" and it seems to be acceler- ating. Historically speaking, the inventors/discoverers of the quaternions, oc- tonions and related algebras (Hamilton, Cayley, Graves, Grassmann, Jordan, Cli®ord and others) were working from a physical point-of-view and wanted ¤4th Conference on Emergence, Coherence, Hierarchy, and Organization (ECHO 4), Odense, Denmark, 2000 1 their abstractions to be helpful in solving natural problems [37]. Thus, a con- nection between physics and octonions is a reasonable though not yet fully justi¯ed suspicion. It is easy to see the allure of octaves since there are many phenomena in the elementary particle realm which have 8-fold symmetries.
    [Show full text]
  • Quaternion Algebra and Calculus
    Quaternion Algebra and Calculus David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA. Created: March 2, 1999 Last Modified: August 18, 2010 Contents 1 Quaternion Algebra 2 2 Relationship of Quaternions to Rotations3 3 Quaternion Calculus 5 4 Spherical Linear Interpolation6 5 Spherical Cubic Interpolation7 6 Spline Interpolation of Quaternions8 1 This document provides a mathematical summary of quaternion algebra and calculus and how they relate to rotations and interpolation of rotations. The ideas are based on the article [1]. 1 Quaternion Algebra A quaternion is given by q = w + xi + yj + zk where w, x, y, and z are real numbers. Define qn = wn + xni + ynj + znk (n = 0; 1). Addition and subtraction of quaternions is defined by q0 ± q1 = (w0 + x0i + y0j + z0k) ± (w1 + x1i + y1j + z1k) (1) = (w0 ± w1) + (x0 ± x1)i + (y0 ± y1)j + (z0 ± z1)k: Multiplication for the primitive elements i, j, and k is defined by i2 = j2 = k2 = −1, ij = −ji = k, jk = −kj = i, and ki = −ik = j. Multiplication of quaternions is defined by q0q1 = (w0 + x0i + y0j + z0k)(w1 + x1i + y1j + z1k) = (w0w1 − x0x1 − y0y1 − z0z1)+ (w0x1 + x0w1 + y0z1 − z0y1)i+ (2) (w0y1 − x0z1 + y0w1 + z0x1)j+ (w0z1 + x0y1 − y0x1 + z0w1)k: Multiplication is not commutative in that the products q0q1 and q1q0 are not necessarily equal.
    [Show full text]
  • Hyperbolicity of Hermitian Forms Over Biquaternion Algebras
    HYPERBOLICITY OF HERMITIAN FORMS OVER BIQUATERNION ALGEBRAS NIKITA A. KARPENKO Abstract. We show that a non-hyperbolic hermitian form over a biquaternion algebra over a field of characteristic 6= 2 remains non-hyperbolic over a generic splitting field of the algebra. Contents 1. Introduction 1 2. Notation 2 3. Krull-Schmidt principle 3 4. Splitting off a motivic summand 5 5. Rost correspondences 7 6. Motivic decompositions of some isotropic varieties 12 7. Proof of Main Theorem 14 References 16 1. Introduction Throughout this note (besides of x3 and x4) F is a field of characteristic 6= 2. The basic reference for the staff related to involutions on central simple algebras is [12].p The degree deg A of a (finite dimensional) central simple F -algebra A is the integer dimF A; the index ind A of A is the degree of a central division algebra Brauer-equivalent to A. Conjecture 1.1. Let A be a central simple F -algebra endowed with an orthogonal invo- lution σ. If σ becomes hyperbolic over the function field of the Severi-Brauer variety of A, then σ is hyperbolic (over F ). In a stronger version of Conjecture 1.1, each of two words \hyperbolic" is replaced by \isotropic", cf. [10, Conjecture 5.2]. Here is the complete list of indices ind A and coindices coind A = deg A= ind A of A for which Conjecture 1.1 is known (over an arbitrary field of characteristic 6= 2), given in the chronological order: • ind A = 1 | trivial; Date: January 2008. Key words and phrases.
    [Show full text]
  • Some Notes on the Real Numbers and Other Division Algebras
    Some Notes on the Real Numbers and other Division Algebras August 21, 2018 1 The Real Numbers Here is the standard classification of the real numbers (which I will denote by R). Q Z N Irrationals R 1 I personally think of the natural numbers as N = f0; 1; 2; 3; · · · g: Of course, not everyone agrees with this particular definition. From a modern perspective, 0 is the \most natural" number since all other numbers can be built out of it using machinery from set theory. Also, I have never used (or even seen) a symbol for the whole numbers in any mathematical paper. No one disagrees on the definition of the integers Z = f0; ±1; ±2; ±3; · · · g: The fancy Z stands for the German word \Zahlen" which means \numbers." To avoid the controversy over what exactly constitutes the natural numbers, many mathematicians will use Z≥0 to stand for the non-negative integers and Z>0 to stand for the positive integers. The rational numbers are given the fancy symbol Q (for Quotient): n a o = a; b 2 ; b > 0; a and b share no common prime factors : Q b Z The set of irrationals is not given any special symbol that I know of and is rather difficult to define rigorously. The usual notion that a number is irrational if it has a non-terminating, non-repeating decimal expansion is the easiest characterization, but that actually entails a lot of baggage about representations of numbers. For example, if a number has a non-terminating, non-repeating decimal expansion, will its expansion in base 7 also be non- terminating and non-repeating? The answer is yes, but it takes some work to prove that.
    [Show full text]
  • Biquaternions Lie Algebra and Complex-Projective Spaces
    Caspian Journal of Mathematical Sciences (CJMS) University of Mazandaran, Iran http://cjms.journals.umz.ac.ir ISSN: 1735-0611 CJMS. 4(2)(2015), 227-240 Biquaternions Lie Algebra and Complex-Projective Spaces Murat Bekar 1 and Yusuf Yayli 2 1 Department of Mathematics and Computer Sciences, Konya Necmettin Erbakan University, 42090 Konya, Turkey 2 Department of Mathematics, Ankara University, 06100 Ankara, Turkey Abstract. In this paper, Lie group and Lie algebra structures of unit complex 3-sphere S 3 are studied. In order to do this, adjoint C representation of unit biquaternions (complexified quaternions) is obtained. Also, a correspondence between the elements of S 3 and C the special bicomplex unitary matrices SU C2 (2) is given by express- 2 ing biquaternions as 2-dimensional bicomplex numbers C2. The 3 3 3 relation SO(R ) =∼ S ={±1g = RP among the special orthogonal 3 3 group SO(R ), the quotient group of unit real quaternions S ={±1g 3 and the projective space RP is known as the Euclidean-projective space [1]. This relation is generalized to the Complex-projective space and is obtained as SO( 3) ∼ S 3 ={±1g = P3. C = C C Keywords: Bicomplex numbers, real quaternions, biquaternions (complexified quaternions), lie algebra, complex-projective space. 2000 Mathematics subject classification: 20G20, 32C15, 32M05. 1. Introduction It is known that the special orthogonal group SO(3) forms the set of 3 all the rotations in 3-dimensional Euclidean space E , which preserves lenght and orientation [2]. Thus, Lie algebra of the Lie group SO(3) 3 corresponds to the 3-dimensional Euclidean space R , with the cross 1 Corresponding author: [email protected] Received: 07 August 2013 Revised: 21 November 2013 Accepted: 12 February 2014 227 228 Murat Bekar, Yusuf Yayli product operation.
    [Show full text]