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We associate to a circular helix a complex numbers called complex torsion defined as follows:

Definition: It's calledcomplex torsionassociated to a circular helix the complex number [tex]q=\tau+i\kappa[/tex] , where [tex]\tau[/tex] is the torsion of circular helix, [tex]\kappa[/tex] is its curvature, and [tex]i[/tex] is the imaginary unit.

Next, we define operations with circular helices, as follows:

Definition: It's calledthe sumof the circular helix [tex]E_1[/tex] of complex torsion [tex]\tau_1+i\kappa_1[/tex] with the circular helix [tex]E_2[/tex] of complex torsion [tex]\tau_2+i\kappa_2[/tex], and we note [tex]E=E_1+E_2[/tex], the circular helix [tex]E[/tex] of complex torsion [tex]q=q_1+q_2=\tau_1+\tau_2+i(\kappa_1+\kappa_2)[/tex] .

Definition: It's calledthe productof the circular helix [tex]E_1[/tex] of complex torsion [tex]\tau_1+i\kappa_1[/tex] with the circular helix [tex]E_2[/tex] of complex torsion [tex]\tau_2+i\kappa_2[/tex], and we note [tex]E=E_1\cdot E_2[/tex], the circular helix [tex]E[/tex] of complex torsion [tex]q=q_1\cdot q_2=\tau_1\tau_2-\kappa_1\kappa_2+i(\tau_1\kappa_2+\tau_2\kappa_1)[/tex].

With these definitions we have built the so-calledfield of circular helices. Thereby, we can add and multiply circular helix like we can operate as complex numbers, and such a construction can be useful by using complex analysis to study the movement of a mobile on a certain curve whose complex torsion depends on time.

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# The field of circular helices

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