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MA3D9 Example Sheet 2 1. Let α(s), s ∈ [0, l] be a closed convex plane curve positively oriented (i.e. κs > 0). The curve β(s) = α(s) − rn(s), where r is a positive constant, is called a parallel curve of α. Show that a. The length l(β) = l(α) + 2πr. b. The area A(β) = A(α) + rl + πr2 . c. κβ = κα . 1+rκα (s) Solution: a.) Assume α(s) is unit-speed. We have β 0 (s) = α0 (s) − rn0 = (1 + rκα (s))α0 (s). So Z l(α) Z κα ds = l(α) + 2πr (1 + rκα (s))ds = l(α) + r l(β) = l(α) 0 0 by Hopf’s Umlaufsatz. R R b.) A(β) = 21 ||β × β 0 ||ds = 12 ||(α − rn) × (t − rn0 )||ds. It is 1 2 Z l 0 l Z ||α × α ||ds + r 0 ||b||ds + r 0 2 Z l κα ds = A(α) + rl + πr2 . 0 An easier way is to integrate the formula of (a) from 0 to r, which gives Z r A(β) − A(α) = (l(α + 2πρ)dρ = A(α) + rl + πr2 . 0 c.) β 0 (s) = (1 + rκα (s))α0 (s), so ||β 0 (s)|| = 1 + rκα (s). Since α and β have the same normal n(s), d d ds κα κβ (s) = || n(s)|| = || n(s) || = . dsβ ds dsβ 1 + rκα (s) 2. Let α(s) be a regular plane curve with unit-speed parametrization. Assume that κs is nonzero everywhere. The curve 1 β(s) = α(s) + ns κs is called the evolute of α. a. Show that the tangent at s0 of the evolute of α is the normal to α at s0 . b. Show that a plane curve α has a vertex at s0 if and only if the evolute β of α has a singular point at s0 . Solution: a.) β 0 (s) = α0 (s) + 1 (−κs t) κs + ( κ1s )0 ns = ( κ1s )0 ns , which is normal to α. b.) β 0 (s0 ) = 0 if and only if ( κ1s )0 (s0 ) = 0, if and only if κ0s (s0 ) = 0. 3. Let γ(s), s ∈ [0, l], be a plane simple closed curve. Assume that the curvature κ(s) ≤ c, where c is a constant. Prove that 2π . l(γ) ≥ c Solution: Assume γ(s) is unit-speed. By Hopf’s Umlaufsatz, Rl 0 κ(s)ds = 2π. Since κ(s) ≤ c, 2π ≤ cl(γ). 4. Find all the vertices for the ellipse γ(t) = (p cos t, q sin t) when p 6= q. Solution: κs = y 00 x0 − x00 y 0 (x02 + y 02 ) 3 2 = pq 3 (p2 sin2 t + q 2 cos2 t) 2 3pq(p2 − q 2 ) sin t cos t dκs =− 5 dt (p2 sin2 t + q 2 cos2 t) 2 So dκs dt = 0 if and only if sin t cos t = 0 if and only if t = 0, π2 , π, 3π , i.e. at (p, 0), (0, q), (−p, 0), (0, −q). 2 2