Differential Geometry I: Worksheet 1 - Freie Universität Berlin

Freie Universität Berlin
Institut für Mathematik WS 09/10
Prof. Dr. K. Polthier / K. Hildebrandt / Dr. C. Lange Due: 30.10.09
http://geom.mi.fu-berlin.de/teaching/ws0910/di¤geo1/index.html
Di¤erential Geometry I: Worksheet 1
(Curves in R n )
Exercise 1. (2 Points)
An epicycloid is the path traced out by a point on the edge of a circle of
radius r rolling on the outside of a circle of radius R. Find a parametrization
of the epicycloid and plot the curve for r = 1 and R = 3.
Exercise 2. (2 Points)
The trace of the parametrized curve
c : [0; 1) ! R 2
t 7! e t (cos t; sin t)
lies in a logarithmic spiral. Show that the arc length of the trace of c is …nite.
Exercise 3. (2 Points)
The map
helixr;a : R ! R 3 ;
t 7! (r cos t; r sin t; a t)
parametrizes a helix with radius r and slant a. For …xed r and a …nd an
arc length parametrization of the helix and compute the curvature and the
torsion.
Exercise 4. (3 Points)
Let c : I ! R 2 be an arc length parametrized C 2 -curve, and let s0; s 2 I
satisfy s0 < s. Show that the limit
](c
lim
s!s0
0 (s0); c0 (s))
s s0
exists and equals the curvature of c at the point s0.
Exercise 5. (3 Points)
Let c(t) = (x(t); y(t)) be a regular parametrized curve.

1. Show that for the curvature the following formula holds
(t) =
_x(t)y(t) x(t) _y(t)
:
( _x(t) 2 + _y(t) 2 ) 3
2
2. Let R be an orientation preserving rigid motion of R 2 ; i.e.
R(x; y) = M x
y
+ a
b
where M 2 SO(2) and a; b 2 R. Show that R c has the same curvature
as c.
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