DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§2. Local Theory: Frenet Frame 19Remark. It is an amusing exercise to give a and b (in our formula for the circular helix)in terms of κ 0 and τ 0 .*11. Proceed as in the derivation of Proposition 2.2 to show thatτ = α′ · (α ′′ × α ′′′ )‖α ′ × α ′′ ‖ 2 .12. Let α be a C 4 arclength-parametrized curve with κ ≠0. Prove that α is a generalized helix ifand only if α ′′ · (α ′′′ × α (iv) )=0. (Here α (iv) denotes the fourth derivative of α.)13. Suppose κτ ≠0atP .Ofall the planes containing the tangent line to α at P , show that α lieslocally on both sides only of the osculating plane.14. Let α be a regular curve with κ ≠0atP . Prove that the planar curve obtained by projectingα into its osculating plane at P has the same curvature at P as α.15. A closed, planar curve C is said to have constant breadth µ if the distance between paralleltangent lines to C is always µ. (No, C needn’t be a circle. See Figure 2.5.) Assume for therest of this problem that the curve is C 2 and κ ≠0.(the Wankel engine design)Figure 2.5a. Let’s call two points with parallel tangent lines opposite. Prove that if C has constantbreadth µ, then the chord joining opposite points is normal to the curve at both points.(Hint: If β(s) isopposite α(s), then β(s) =α(s)+λ(s)T(s)+µN(s). First explain whythe coefficient of N is µ; then show that λ = 0.)b. Prove that the sum of the reciprocals of the curvature at opposite points is equal to µ.(Warning: If α is arclength-parametrized, β is quite unlikely to be.)16. Let α and β be two regular curves defined on [a, b]. We say β is an involute of α if, for eacht ∈ [a, b],(i) β(t) lies on the tangent line to α at α(t), and(ii) the tangent vectors to α and β at α(t) and β(t), respectively, are perpendicular.Reciprocally, we also refer to α as an evolute of β.a. Suppose α is arclength-parametrized. Show that β is an involute of α if and only ifβ(s) =α(s)+(c − s)T(s) for some constant c (here T(s) =α ′ (s)). We will normally refer