Lopez, Diffrential geometry of curves and surfaces in Lorentz–Minkowski space. O’Neill, Semi-Riemannian Geometry (New York: Academic Press Inc) (1983) Goriely, Surface growth kinematics via local curve evolution. (Cambridge University Press, Cambridge, 1890)ĭ.E. Tait, An Elementary Treatise on Quaternions, 3rd edn. Hanson, Visualizing Quaternions (Amsterdam: Elsevier) Kuipers J B 1999 Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality (Princeton University Press, Princeton, 2006) Chanyal, A new development in quantum field equations of dyons. Chanyal, A relativistic quantum theory of dyons wave propagation. Pathak, Quaternionic approach to dual magnetohydrodynamics of dynos cold plasma. Roulstone, Quaternions and particle dynamics in the Euler fluid equations. Gibbon, Orthonormal quaternion frames, Lagrangian evolution equations and the three-dimensional Euler equations. Gibbon, A quaternionic structure in the three-dimensional Eulerand ideal magneto-hydrodynamics equations. Holm, Lagrangian analysis of alignment dynamics for isentropic compressible magnetohydrodynamics. Holm, Lagrangian particle paths & ortho-normal quaternion frames. Russer, Differential forms and electromagnetic field theory. Haralick, Biquaternion Generalized Maxwell Equations: Longitudinal and Scalar Waves (2017) Ryutova, Physics of Magnetic Flux Tubes (Springer, Basel, 2018)

Bruno, Twisted magnetic flux tubes in the solar wind. Neukirch, Magnetic pinching of hyperbolic flux tubes. Özdemir, A geometrical and physical interpretation of quaternionic generalized magnetic flux tubes. Abbett, Magnetic flux tubes inside the sun, Phys. Field, The topological properties of magnetic helicity. Sun, Analytic and algebraic properties of canal surfaces. İlarslan, New types of canal surfaces in Minkowski 3-space. Yaylı, Circular surfaces with split quaternionic representations in minkowski 3-space. Yu, Analysis and applications of pipe surfaces. Yaylı, A new representation of canal surfaces with split quaternions inMinkowski 3-space. Gök, Quaternionic approach of canal surfaces constructed by some new ideas. Gross, Analyzing generalized tubes, in Proceedings of SPIE-The International Society for Optical Engineering, pp. Yaylı, The construction of circular surfaces with quaternions, Asian Eur. Yaylı, Split quaternions and canal surfaces in Minkowski 3-space. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd edn.

Cockle, On systems of algebra involving more than one imaginary. Shoemake, Animating rotation with quaternion curves, in Proceedings of the 12th Annual Conference on Computer Graphics and Interactive Techniques (SIG-GRAPH-85), vol. Hamilton, On quaternions or on a new system of imagniaries in algebra. Then, the analytical solutions of the kinematic equations are obtained and some examples related to the theory are visualized. Furthermore, characterizations of the MFTs through the stretch factor are presented. This allows us to determine the magnetic field lines and kinematic equations of magnetic tube surfaces. The Frenet frame is attached to each point of the generating curve so that the magnetic field components are derived in terms of the local coordinate directions. Then, we show that the flux surfaces can be generated through the magnetic vector field components at each point on a generating curve. Moreover, we obtain the magnetic field components by using the split quaternionic expression of an MFT. For this, we prove that an MFT can be completely characterized by a split quaternion product or a homothetic motion. The aim of the paper is to present a new geometric viewpoint for magnetic flux tubes (MFTs) in the three-dimensional Lorentzian space through split quaternion algebra.

Magnetic flux equations