A New Polar Representation for Split and Dual Split Quaternions


Atasoy A., ATA E., YAYLI Y., Kemer Y.

ADVANCES IN APPLIED CLIFFORD ALGEBRAS, vol.27, no.3, pp.2307-2319, 2017 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 27 Issue: 3
  • Publication Date: 2017
  • Doi Number: 10.1007/s00006-017-0797-8
  • Journal Name: ADVANCES IN APPLIED CLIFFORD ALGEBRAS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.2307-2319
  • Keywords: Dual split quaternion, Polar form, Split quaternion, Screw operator, KINEMATICS, ROTATIONS
  • Ankara University Affiliated: Yes

Abstract

We present a new different polar representation of split and dual split quaternions inspired by the Cayley-Dickson representation. In this new polar form representation, a split quaternion is represented by a pair of complex numbers, and a dual split quaternion is represented by a pair of dual complex numbers as in the Cayley-Dickson form. Here, in a split quaternion these two complex numbers are a complex modulus and a complex argument while in a dual split quaternion two dual complex numbers are a dual complex modulus and a dual complex argument. The modulus and argument are calculated from an arbitrary split quaternion in Cayley-Dickson form. Also, the dual modulus and dual argument are calculated from an arbitrary dual split quaternion in Cayley-Dickson form. By the help of polar representation for a dual split quaternion, we show that a Lorentzian screw operator can be written as product of two Lorentzian screw operators. One of these operators is in the two-dimensional space produced by 1 and i vectors. The other is in the three-dimensional space generated by 1, j and k vectors. Thus, an operator in a four-dimensional space is expressed by means of two operators in two and three-dimensional spaces. Here, vector 1 is in the intersection of these spaces.