Is Complex Torsion the Key to Understanding Circular Helices Operations?

[Abel Cavaşi]

The following definitions are correct?




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

Definition: It's called complex torsion associated to a circular helix the complex number $$q=\tau+i\kappa$$ , where $$\tau$$ is the torsion of circular helix, $$\kappa$$ is its curvature, and $$i$$ is the imaginary unit.

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

Definition: It's called the sum of the circular helix $$E_1$$ of complex torsion $$\tau_1+i\kappa_1$$ with the circular helix $$E_2$$ of complex torsion $$\tau_2+i\kappa_2$$, and we note $$E=E_1+E_2$$, the circular helix $$E$$ of complex torsion $$q=q_1+q_2=\tau_1+\tau_2+i(\kappa_1+\kappa_2)$$ .

Definition: It's called the product of the circular helix $$E_1$$ of complex torsion $$\tau_1+i\kappa_1$$ with the circular helix $$E_2$$ of complex torsion $$\tau_2+i\kappa_2$$, and we note $$E=E_1\cdot E_2$$, the circular helix $$E$$ of complex torsion $$q=q_1\cdot q_2=\tau_1\tau_2-\kappa_1\kappa_2+i(\tau_1\kappa_2+\tau_2\kappa_1)$$.

With these definitions we have built the so-called field 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.

 

[Mandelbroth]

The following definitions are correct?

Is this a question?

 

[Abel Cavaşi]

Yes, I wanted to look like a question.