Abstract
Quaternions, which are found in many fields, have been studied for a long time. The interest in dual quaternions has also increased after real quaternions. Nagaraj and Bharathi developed the basic theories of these studies. The Serret–Frenet Formulae for dual quaternion-valued functions of one real variable are derived. In this paper, by making use of the results of some previous studies, helixes and harmonic curvature concepts in Q D 3 and Q D 4 are considered and a characterization for a dual harmonic curve to be a helix is given.
Highlights
Some fundamental definitions and concepts related to the algebra of dual quaternions are given
A dual quaternion is defined as q = q + eq∗, ε2 = 0
The norm of a dual quaternion pis defined by k pk2 = h( p, p) = p × α por k pk2 = D2 + A2 + B2 + C2, where k pk2 is a dual number
Summary
Some fundamental definitions and concepts related to the algebra of dual quaternions are given. Let q and q∗ be two real quaternions. The norm of a dual quaternion pis defined by k pk2 = h( p, p) = p × α por k pk2 = D2 + A2 + B2 + C2 , where k pk is a dual number. A dual spatial-quaternion may be considered as a dual vector in D3. Let pand qbe two unit dual spatial-quaternions. A dual quaternion valued function of a single real variable is called a dual quaternionic curve. With { T, N1 , N2 } being the Frenet frame field along β, the Serret–Frenet Formulae of dual spatial-quaternionic curve β is given by. Where ( T, Ñ1 , Ñ2 , Ñ3 , K, K, R − K ) is the Frenet Apparatus for the curve βsuch that K and R are principal curvature and torsion of the dual spatial-quaternionic curve β, respectively [5]
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