|
In mathematics and mechanics, the set of dual quaternions is a Clifford algebra that can be used to represent spatial rigid body displacements.〔A.T. Yang, ''Application of Quaternion Algebra and Dual Numbers to the Analysis of Spatial Mechanisms'', Ph.D thesis, Columbia University, 1963.〕 A dual quaternion is an ordered pair of quaternions and therefore is constructed from eight real parameters. Because rigid body displacements are defined by six parameters, dual quaternion parameters include two algebraic constraints. In ring theory, dual quaternions are a ring constructed in the same way as the quaternions, except using dual numbers instead of real numbers as coefficients. A dual quaternion can be represented in the form ''p'' + ε ''q'' where ''p'' and ''q'' are ordinary quaternions and ε is the dual unit (εε = 0) and commutes with every element of the algebra. Unlike quaternions they do not form a division ring. Similar to the way that rotations in 3D space can be represented by quaternions of unit length, rigid motions in 3D space can be represented by dual quaternions of unit length. This fact is used in theoretical kinematics (see McCarthy〔), and in applications to 3D computer graphics, robotics and computer vision.〔(A. Torsello, E. Rodolà and A. Albarelli, ''Multiview Registration via Graph Diffusion of Dual Quaternions'', Proc. of the XXIV IEEE Conference on Computer Vision and Pattern Recognition, pp. 2441-2448, June 2011. )〕 ==History== W. R. Hamilton introduced quaternions〔W. R. Hamilton, "On quaternions, or on a new system of imaginaries in algebra," Philos. Mag. 18, installments July 1844 – April 1850, ed. by D. E. Wilkins (2000)〕〔(W. R. Hamilton, ''Elements of Quaternions, Longmans, Green & Co., London, 1866 )〕 in 1843, and by 1873 W. K. Clifford obtained a broad generalization of these numbers that he called ''biquaternions'',〔W. K. Clifford, "Preliminary sketch of bi-quaternions, Proc. London Math. Soc. Vol. 4 (1873) pp. 381–395〕〔W. K. Clifford, ''Mathematical Papers'', (ed. R. Tucker), London: Macmillan, 1882.〕 which is an example of what is now called a Clifford algebra.〔(J. M. McCarthy, ''An Introduction to Theoretical Kinematics'', pp. 62–5, MIT Press 1990. )〕 At the turn of the 20th century, Aleksandr Kotelnikov〔A. P. Kotelnikov, ''Screw calculus and some applications to geometry and mechanics'', Annal. Imp. Univ. Kazan (1895)〕 and E. Study〔E. Study, ''Geometrie der Dynamen'', Teubner, Leipzig, 1901.〕 developed dual vectors and dual quaternions for use in the study of mechanics. In 1891 Eduard Study realized that this associative algebra was ideal for describing the group of motions of three-dimensional space. He further developed the idea in ''Geometrie der Dynamen'' in 1901. B. L. van der Waerden called the structure "Study biquaternions", one of three eight-dimensional algebras referred to as biquaternions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「dual quaternion」の詳細全文を読む スポンサード リンク
|