Chapter 2: Space Curves
Section 2.7: Frenet-Serret Formalism
Verify the prescription in Table 2.7.2; that is, show the resulting position vector R defines a curve for which T=R′s satisfies T=1 and T′s=κs.
If a curve C is defined by
R=a+∫cscosθu ⅆu i+b+∫cssinθu ⅆu j
R′=T=cosθs i+sinθs j
and clearly, ∥R′∥=1. Also,
T′= −sinθs i+cosθs j θ′s
where θs=∫csκt dt, and θ′s=κs by the Fundamental Theorem of Calculus.
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