Assignment 2, due 9 May - Institut für Mathematik

Technische Universität Berlin
Institut für Mathematik
Pinkall/Lam
Summer 2016
Due: 9 May 2016
http://www3.math.tu-berlin.de/geometrie/Lehre/SS16/DGI
Differentialgeometrie I: Kurven und Flächen
Homework 2
Problem 1
(4 points)
Show that a regular plane curve has constant curvature κ 6= 0 if and only if it is a
circular arc.
Problem 2
(4 points)
2
Let γ : [a, b] → R be a regular curve with non-vanishing curvature κ(t). The curve
φ := γ − κ1 N is the evolute of γ, which is the locus of centers of curvature.
Compute the evolute of the following curves γ : R → R2 .
(a) Logarithmic spiral: γ(t) = eat (cos t, sin t) for some a ∈ R.
(b) Cycloid: γ(t) = (t − sin t, 1 − cos t).
(Hint: The logarithmic spiral and the cycloid are respectively congruent to their
evolutes. Be careful of the sign convention that we use. Here N = −JT and
J(x, y) = (−y, x).)
Problem 3
(4+4 Points)
(a) Consider the trace of a fixed point on a circle of radius r > 0, which rolls
around the inside of another circle of radius R > r. The resulting curve is a
hypotrochoid. Show that it is closed if and only if Rr ∈ Q.
(b) Let > 0 and m, n ∈ N so that m
= Rr . A closed curve (γ, 2πn) is parametrized
n
by
m m t 7→ (R − r) cos (t) + (r + ε) cos
t , (R − r) sin (t) + (r + ε) sin
t .
n
n
Show that (γ, 2πn) is regularly homotopic to the m-fold unit circle by explicitly
finding a regular homotopy.

Download Report