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MTH3110 Differential Geometry

Assignment 2

Semester 1, 2022

Question 1. For each of the following sets of conditions, either give an example of a unit speed closed curve in R2 satisfying the conditions, or show that none exists. (Examples may be given via a formula or a picture, but you must explain why the conditions are satisfied. Your curve need not be C∞ smooth, but must be smooth enough to have the required total curvatures well defined.)

(a) A simple curve with total signed curvature −2π and total curvature 12π . (b) A simple curve with total signed curvature −4π and total curvature 8π .

(d) A curve with total signed curvature −6π and total curvature 5π .

[2+2+2+2 = 8 marks]

Question 2. The osculating circle. A unit speed parametrisation of a circle in R3 may be written as γ : R −→ R3 where

γ(s) = c + r cos v1 + r sin v2

where r > 0 is a constant real number, c, v1 , v2 are constant vectors and vi · vj = δij . Let β : I −→ R3 be a unit speed curve with κ(0) > 0. Show that there is precisely

one unit speed circle γ that agrees with β to second order at β(0), i.e. such that β(0) = γ(0), β˙(0) = γ˙ (0) and β¨(0) = (0).

Show that this circle has radius 1/κ(0). The circle γ is called the osculating circle and c the centre of curvature of β at β(0).

[8 marks]

Question 3. For the curve γ : R −→ R3 , γ(t) = (2t,t2,t3/3),

(a) Compute the Frenet frame T,N,B, curvature κ and torsion τ . (b) Find the limiting values of T,N,B as t → −∞ and as t → ∞ .

Page 1 of2

[6+2 = 8 marks]

Question 4. Ada is learning differential geometry and has previously studied some physics. Upon learning about the Frenet frame, she says

Aha! The Frenet frame allows me to put some physics together! When I accelerate, the component in the T direction contributes to increas- ing my speed; and the rest is my centripetal acceleration, which must be in the N direction. I learned in physics that centripetal accelera- tion is v2/r. Here we write ||γ˙ || instead of v, and 1/r is like curvature

= ||γ˙ || T + ||γ˙ ||2κN.

Is Ada right? Explain why or why not, proving or disproving the above equation for an appropriate family of curves. (You may use any results or formulas from lectures or lecture notes. You only need to comment on the mathematical equation, not the physics.)

[8 marks]

Question 5. In school you drew many graphs of the form y = f(x). We will now examine the curvature of such graphs. Let γ : R −→ R2 be a regular parametrisation of such a graph, given by γ(t) = (t,f(t)).

(a) On a previous problem set you saw that the signed curvature of γ is given by the formula

f′′ (t)

κs(t) =

Let’s first derive this directly from κs = , without reference to other cur- vature formulas.