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joi, 7 iunie 2012

The recurrence theorem of the Frenet formulas

Studying the Frenet formulas I have concluded that they are recursive. More specifically, using the trigonometric form of the Frenet formulas, we proved the following

Theorem: If there is a right trihedron of the n order
 that satisfies the Frenet formulas of the n order, written in the trigonometric form


 

then there is still a right trihedron of the n+1 order


 

that satisfying, in turn, the Frenet formulas of the n+1 order written also in the trigonometric form



where
  and
  .

Demonstration: Through relations  and
 

we have that

so

  .

We also have

whence


  .

Now, we derive the unit vectors of the trihedron of the n+1 order


 

and we obtain


  .

Replacing   and
  , we obtain


  .

But, from the definition of the unit vectors of the high order, we know that


  ,

so


  .

Because   and   ,

finally result that


  ,

qed.

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