Chapter 2: Space Curves
Section 2.7: Frenet-Serret Formalism
Example 2.7.16
Prove that T′×T″=κ2d.
Solution
Since T′=κ N, T″=κ N′=κ′N+κ N′, and N′= −κ T+τ B,
T′×T″
=κ N×κ′N+κ N′
=κ N×κ′N+κ −κ T+τ B
=κ κ′N×N+κ2 N×−κ T+τ B
=0−κ3N×T+κ2τN×B
=−κ3−B+κ2τ T
=κ2 τ T+κ B
=κ2d
Setting U=N,V=T,W=N in the vector identity A×B×C=A·CB−A·BC, Example 1.4.2(e), written as U×V×W=U·WV−U·VW, is used to establish the identity N×B=T. Indeed,
N×B=N×T×N=N·NT−N·TN=1⋅T−0⋅N=T
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