Chapter 2: Space Curves
Section 2.4: Curvature
Show the equivalence of the definition κ = T′s, and the formula κ=R′p×R″p/ρ3.
From the definition κ=T′s, obtain κ=dTdpdpds = ∥T′p∥ρ by the chain rule.
Of course, the astute reader realizes that the prime is being used to designate differentiation with respect to both s and t. However, the explicit display of the argument of T clarifies the differentiation variable.
In the following analysis of the formula κ=R′p×R″p/ρ3, the prime will represent differentiation with respect to p.
Since R′=ρ T, R″=ρ′T+ρ T′. Then,
=ρ T×ρ′T+ρ T′
=ρ ρ′ T×T+ρ2 T×T′
=ρ ρ′⋅0+ρ2 T×T′
where T×T=0 because T is collinear with itself. Next, obtain
= ρ2 T ∥T′∥
= ρ2⋅1⋅ ∥T′∥
= ρ2 ∥T′∥
where ∥T×T′∥ = T ∥T′∥ because T and T′ are orthogonal, and where T=1 because T is a unit vector. Finally, then,
∥R′×R″∥ρ3=ρ2 ∥T′∥ρ3 = ∥T′∥ρ
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