Thread: Normal & Geodesic Curvatures...

Hi everyone! :-D

This is a Differential Geometry question - I wasn't quite sure which forum to post this in - please move this if needs be. :-)

My question is as follows:

When dealing with a curve parametrised by arc length we set up the Serret-Frenet frame using the following vectors: the unit tangent T, the unit normal N & TxN. (the binormal.)

But when dealing with a curve on a surface we set up the Darboux frame: the unit tangent T, the unit normal, N & NxT.

We can then use this frame to say that the 2nd derivative of a curve A(s) is in the plane spanned by N & NxT (as it is orthogonal to A), and hence that is equals Kg(NxT) + KnN. We call Kg the geodesic curvature & Kn the normal curvature.

My question though is why to we cross the 2 vectors in different orders for the Serret-Frenet frame and the Darboux frame?

Many thanks in advance. :-) x

The general curve does not enjoy a 'natural' choice of tangent level
in the manner a surface curve does.