ABSTRACT
Let a space curve y be given by its position vector r = r(s) viewed as a vector-valued function of the arc length S. We will denote by T the unit tangent vector r l ( s ) , by v the principal normal, by 0 the binormal of y. Three vectors T , v, P depend on the parameter S, hence we consider these vectors to be vector-valued functions of S. At any point of y the vectors T , v, p are mutually orthogonal and form a basis of Euclidean space. We say that T , v, P.form the natural frame. The derivatives of these vectors are decomposed into linear combinations with respect to the natural frame at the corresponding point:
Let us find the coefficients of the decomposition. It follows from the definition of the principal normal that
Therefore a1 = 0, a2 = k, a3 = 0. When we proved the theorem about torsion (see chapter 7), we proved the formula
Using the decomposition of v,: and the found values of al, c2 we obtain
Thus
These decompositions of the vector ri, v.:, /3: are called Frenetformulas. They have a very important significance in the differential geometry of curves.
