Differential Geometry/Frenet-Serret Formulae

From Wikibooks, open books for an open world
Jump to: navigation, search

The derivatives of the vectors t, p, and b can be expressed as a linear combination of these vectors. The formulae for these expressions are called the Frenet-Serret Formulae. This is natural because t, p, and b form an orthogonal basis for a three-dimensional vector space.

Of course, we know already that \frac{dt}{ds}=\kappa p and \frac{db}{ds}=-\tau p so it remains to find \frac{dp}{ds}. First, we differentiate p \sdot p = 1 to obtain \frac{dp}{ds}\sdot p = 0 so it takes on the form \frac{dp}{ds}=at+cb. We take the dot product of this with t to obtain a=\frac{dp}{ds}\sdot t. Taking the derivative of p\sdot t=0, we get \frac{dp}{ds}\sdot t + p \sdot \frac{dt}{ds} = 0 or \frac{dp}{ds}\sdot t=-p \sdot \frac{dt}{ds} = -\kappa p \sdot p = -\kappa. Also, taking the dot product of \frac{dp}{ds}=at+cb with b, we obtain c=\frac{dp}{ds}\sdot p. Taking the derivative of p\sdot b=0, we get \frac{dp}{ds}\sdot b=-p\sdot\frac{db}{ds}=\tau. Thus, we arrive at the following expression for \frac{dp}{dt}:

\frac{dp}{dt}=-\kappa t+\tau b.

This formula, combined with the previous two formulae, are together called the Frenet-Serret Formulae and they can be represented by a skew-symmetric matrix.