Derivative transformation of the T,U,N-frame

When we presented the theory of curves in three-space, we introduced the Frenet frame and found that problems were often simplified if we decomposed objects related to a curve into their components in the Frenet frame. For curves on surfaces in three-space, since many properties of these curves are related to the surface, the T,U,N-frame becomes more appropriate. The T,U,N-frame is of course the orthonormal frame formed by the ordered triple (T,U ,N) .

When we studied the Frenet frame , we found it necessary to determine the derivatives of each of the vectors in the frame. For this purpose, we found it useful to write the transformation in terms of a matrix. We give this transformation without any calculations:

(T^'U^'N^')=s^'(0k_gk_n-k_g0t_g-k_n-t_g0)(TUN)

It is quite difficult to determine these coefficients and a long calculation would not be appropriate here. You may notice, however, that we have already encountered the first line. For the acceleration of a general space curve we have

X^'
'=s^'' T+(s^')²kP.

Recall, though, how we defined k_g and k_n for a curve on a surface: kP=k_g U+k_nN . Hence we have for the acceleration of a curve on a surface:

X^'
'=s^'' T+s^'T^'
=s^''T +(s^')²(k_gU +k_nN)

Hence: T^' =s^'(k_g U+k_nN) .