Differential Geometry, Spring 2012.

Instructor: Dmitry Ryabogin

Assignment 5.

1. Problem 1.

a) Suppose kτ ≠ 0 at P. Of all the planes containing the tangent line to α at P, show that α lies locally on both sides only of the osculating plane.

b) Let α be a regular curve with k ≠ 0 at P. Prove that the planar curve obtained by projecting α into its osculating plane at P has the same curvature at P as α.

2. Problem 2.

A closed, planar curve C is said to have constant breadth (or width) μ if the distance between parallel tangent lines to C is always μ, (C need not be a circle). Assume for the rest of the problem that the curve is C^{2} and k ≠ 0.

a) Let’s call two points with parallel tangent lines opposite. Prove that if C has constant width μ, then the chord joining opposite points is normal to the curve at both points.

Hint: If β is opposite to α, then β(s) = α(s) + λ(s)T (s) + μN(s). First explain why the 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).

3. Problem 3.

Let α and β be two regular curves defined on [a, b]. We say β is an involute of α if, for each t ∈ [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 if β(s) = α(s) + (c − s)T (s) for some constant c. We will normally refer to the curve β at s = 0 as the involute of α. If you were to wrap a string around the curve α, starting at s = 0, the involute is the path the end of the string follows as you unwrap it, always pulling the string taut.

b) Show that the involute of a helix is a plane curve.

c) Show that the involute of a catenary (the graph of f(x) = d cosh(x/d) for some constant d > 0 is a tractrix

α(θ) = (cos θ + ln tan(θ/2), sin θ), π/2 < θ < π,

(or α(t) = (t − tanh t, sech t), t > 0).

d) If α is an arclength-parametrized plane curve, prove that the curve β given by

β(s) = α(s) + \frac{1}{k(s)}N(s)

is the unique evolute of α lying in the plane of α. Prove, moreover, that this curve is regular if k′ ≠ 0.