Hi!(adsbygoogle = window.adsbygoogle || []).push({});

Most of you know that http://en.wikipedia.org/wiki/Frenet_formulas" [Broken] written for the trihedron

[tex](\vec{T},\;\vec{N},\;\vec{B})[/tex]

is

[tex]\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[/tex]

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

[tex]\vec{\omega}=v(\tau\vec{T}+\kappa\vec{B})[/tex]

(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 trihedronformed 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

[tex](\vec{\Omega}=\frac{\vec{\omega}}{\omega},\;\vec{N},\;\vec{D}=\vec{\Omega}\times\vec{N})[/tex] .

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 http://www.astronomy.ro/forum/viewtopic.php?t=1322" , 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.

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Consequences of the Frenet formulas

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads - Consequences Frenet formulas | Date |
---|---|

Frenet-Serret Formulas | Sep 5, 2013 |

Inverting Consequences of Uniform Convergence | Jan 17, 2010 |

How do you prove this consequence of the triangle inequality? | Apr 16, 2009 |

TNB Frenet Frames | Sep 11, 2008 |

Frenet formulas | Mar 25, 2006 |

**Physics Forums - The Fusion of Science and Community**