Differential Geometry, Spring 2012.

Instructor: Dmitry Ryabogin

Assignment 6.

1. Problem 1.

Let g(s) be a smooth function on (−1, 1) and let c, d be constants. Define the curve

α(t) = (c ∫_{0}^{t} sin g(s)ds, c ∫_{0}^{t} cos g(s)ds, d t).

Prove that the ratio of the curvature and torsion of α is constant.

2. Problem 2.

a) Let α be an arclength-parametrized plane curve. We create a ”parallel” curve β by taking β = α + ϵN (for a fixed small positive value of ϵ). Explain the terminology and express the curvature of β in terms of ϵ and the curvature of α.

b) Now let α be an arclength-parametrized space curve. Show that we can obtain a ”parallel” curve β by taking β = α + ϵ(cos θN + sin θB) for an appropriate function θ. How many such parallel curves are there?

c) Sketch such a parallel curve for a circular helix α.

3. Problem 3.

a)^{∗} Suppose α is arclength-parametrized and kτ ≠ 0. Show that α has a Bertrand mate if and only if there are constants r and c so that rk + cτ = 1.

b)^{∗} Given α, prove that if there is more than one curve β so that α and β are Bertrand mates, then there are infinitely many such curves and this occur if and only if α is a circular helix.

c) Suppose that α and β are Bertrand mates. Prove that the torsion of α and the torsion of β at corresponding points have constant product.

4. Problem 4.

A convex plane curve with origin in its interior can be determined by its tangent lines x cos θ + y sin θ = h(θ), called support lines. The function h(θ) is called the support function, (here θ is in the polar coordinate, and we assume h(θ) > 0 for all θ ∈ [0, 2π]).

a) Compute h(θ) for an ellipse \{(x, y) : \frac{x^{2}}{16} + \frac{y^{2}}{9} = 1\} and for the l^{p}−ball \{(x, y) : x^{p} + y^{p} = 1}, 1 ≤ p ≤ ∞.

b) Prove that the length of α is given by L = ∫_{0}^{2π}h(θ)dθ. c) Prove that if C is a simple closed (convex) plane curve of constant breadth μ, then length(C) = πμ. d)^{∗} Prove that the line given above is tangent to the curve at the point

α(θ) = (h(θ) cos θ − h′(θ) sin θ, h(θ) sin θ + h′(θ) cos θ).

e)^{∗} Prove that the curvature of the curve at α(θ) is 1/(h′(θ) + h′′(θ)). f)^{∗} Prove that the area enclosed by α is given by A = \frac{1}{2} ∫_{0}^{2π}(h(θ)^{2} − h′(θ)^{2})dθ. g)^{∗∗} Prove the isoperimetric inequality L^{2} ≥ 4πA.