The field of circular helices

In summary, the complex torsion of a circular helix is a complex number q=\tau+i\kappa where \tau is the torsion and \kappa is the curvature. Operations can be performed on circular helices, such as addition and multiplication, similar to complex numbers. This allows for the construction of a field of circular helices, which can be useful for studying the movement of a mobile on a curve with changing complex torsion over time.
  • #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 [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 called the sum of 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 called the product of 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-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.
 
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  • #2
Abel Cavaşi said:
The following definitions are correct?
Is this a question?
 
  • #3
Yes, I wanted to look like a question.
 

1. What is a circular helix?

A circular helix is a three-dimensional curve that resembles a coiled spiral. It is formed by a combination of circular and helical motions, resulting in a shape that continuously curves around a central axis.

2. What are some real-world applications of circular helices?

Circular helices are commonly found in nature, such as in the structure of DNA and the shape of certain plants and shells. They are also used in engineering and design, such as in the construction of springs, screws, and spiral staircases.

3. How is the curvature of a circular helix calculated?

The curvature of a circular helix can be calculated using the formula k = (2πr) / (h2 + 4π2), where k is the curvature, r is the radius of the circular portion, and h is the pitch or height of the helix.

4. What is the relationship between circular helices and screw threads?

Circular helices and screw threads are closely related as they both follow the same spiral shape. In fact, screw threads are essentially circular helices that have been cut into a solid object, such as a bolt or screw.

5. Are there any practical limitations for the use of circular helices?

One practical limitation of circular helices is that they can only be formed on surfaces with a constant radius. This means that they cannot be created on a flat surface, and the curvature of the helix will change if the radius of the surface changes.

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