# Consequences of the Frenet formulas

1. Dec 1, 2007

### Abel Cavaşi

Hi!
Most of you know that the Frenet formulas written for the trihedron

$$(\vec{T},\;\vec{N},\;\vec{B})$$

is

$$\left\{\begin{matrix}\dot{\vec{T}}=v\kappa\vec{N}=\vec{\omega}\times\vec{T}\hfill\null\\\dot{\vec{N}}=v(-\kappa\vec{T}+\tau\vec{B})=\vec{\omega}\times\vec{N}\hfill\null\\\dot{\vec{B}}=-v\tau\vec{N}=\vec{\omega}\times\vec{B}\hfill\null\end{matrix}\right$$

and these formulas lead to the fact that the angular rotation speed of the Frenet trihedron, given by the relation

$$\vec{\omega}=v(\tau\vec{T}+\kappa\vec{B})$$

(so, because of not having a component on the normal vector), is always perpendicularly on the normal vector. Taking into consideration this fact, it can be created another trihedron formed by the angular speed's versor, by the normal vector, and by the cross product between the angular speed versor and the normal vector. In other words, we are able to create the trihedron

$$(\vec{\Omega}=\frac{\vec{\omega}}{\omega},\;\vec{N},\;\vec{D}=\vec{\Omega}\times\vec{N})$$ .

I called this trihedron „the complementary Frenet trihedron”.

Well, the most interesting fact is that, according to my calculations (which I have detalied on the romanian site www.astronomy.ro , but which I can present here too if you consider it oppourtune) the complementary Frenet trihedron also satisfy the Frenet formulas.

Obviously, this process is recursive, I mean that if the complementary Frenet trihedron also satisfy the Frenet formulas, then we can find as many complementary Frenet trihedrons as everytime we can derive the broken number between the curvature and torsion.

What can you say, have you ever read about this somewhere else, because I want to study deeper this subject. How important and general do you think this subject would be? I think that these properties can explain why are the microscopic physical sizes quantizing.