- #1
Abel Cavaşi
- 34
- 2
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.
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.
