Chapter 2: Space Curves
Section 2.7: Frenet-Serret Formalism
Example 2.7.3
If s is arc length, establish the Frenet equation N′s= −κ T+τ B.
Solution
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.
0
=T·N′
=T·N′+T′·N
=T·a T+c B+κ N·N
=a T·T+c T·B+κ N·N
=a⋅1+c⋅0+κ⋅1
=a+κ
Hence, N′= −κ T+c B. From B·N=0 and the following calculation, c=τ.
=B·N′
=B·N′+N·B′
=B·−κ T+c B+N·−τ N
= −κ B·T+c B·B−τ N·N
= −κ⋅0+c⋅1−τ⋅1
=c−τ
Consequently, N′= −κ T+τ B.
<< Previous Example Section 2.7 Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
