## 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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