Chapter 2: Space Curves
Section 2.7: Frenet-Serret Formalism
If s is arc length, establish the Frenet equation B′s= −τ N.
By definition, the torsion is τ= −B′s. Since B is a unit vector, B·B=1, and B·B′=2 B·B′=0, so B′ is orthogonal to B. But B is orthogonal to the plane containing T and N, so B′ must lie in that plane.
Now B is orthogonal to T, so B·T=0. Differentiating gives B′·T+B·T′=0. Rearranging gives
= −B·κ N
where T′=κ N is the basis for the definition of the curvature κ.
Thus, B′, already in the osculating plane, is now orthogonal to T. Hence, it must be proportional to N, with the constant of proportionality taken as −τ, that is, as ∥B′∥
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