Chapter 2: Space Curves
Section 2.7: Frenet-Serret Formalism
If s is arc length, establish the Frenet equation N′s= −κ T+τ B.
The vectors in the TNB-frame form an orthonormal basis, that is, they are linearly independent, mutually orthogonal, and each of unit length. Hence, N′=a T+b N +c B, that is, N′ can be written as a linear combination of the vectors T, N, and B.
Since N is a unit vector, N·N=1 and N·N′=2 N·N′=0, so
0= N·N′=N·a T+b N +c B=a N·T+b N·N+c N·B=a⋅0+b⋅1+c⋅0
which implies b=0 and N′=a T+c B.
From T·N=0 and the following calculation, a= −κ.
=T·a T+c B+κ N·N
=a T·T+c T·B+κ N·N
Hence, N′= −κ T+c B. From B·N=0 and the following calculation, c=τ.
=B·−κ T+c B+N·−τ N
= −κ B·T+c B·B−τ N·N
Consequently, N′= −κ T+τ B.
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