Assignment 5, due 30 May - Institut für Mathematik

Summer 2016
Technische Universität Berlin
Institut für Mathematik
Pinkall/Lam
Due: 30 May 2016
http://www3.math.tu-berlin.de/geometrie/Lehre/SS16/DGI
Differentialgeometrie I: Kurven und Flächen
Homework 5
Problem 1
(4 points)
Determine all space curves with constant curvature and torsion. What happens if
the Torsion approaches to infinity while the curvature is a fixed constant?
Problem 2
(4 points)
3
Let γ : [a, b] → R be a parametrized Frenet curve (not necessarily arc length
parametrized). Show that the curvature κ and the torsion τ of its Frenet frame
satisfies:
0
00 |
0 ,γ 00 ,γ 000 )
κ = |γ|γ×γ
τ = det(γ
.
0 |3 ,
|γ 0 ×γ 00 |2
Problem 3
3
Let γ : (−1, 1) → R be a curve given via γ(t) :=
1
(1
3
3
2
+ t) ,
1
(1
3
(4 points)
− t) , √t2 . Find
3
2
a) its Frenet frame, curvature and torsion.
b) its parallel frame and curvature.
Problem 4
(4 points)
3
Let γ : [a, b] → R be a Frenet curve with curvature κ und torsion τ (not simultaneously zero). The curve γ is called Bertrand curve if there exists another curve
γ̃ : [a, b] → R3 such that for all t ∈ [a, b] the principal normal line of γ at γ(t)
coincides with the principal normal line of γ̃ at γ̃(t). Show:
a) There exists a constant λ ∈ R such that γ̃ = γ + λN .
b) A curve γ : [a, b] → R3 is Bertrand if and only if there exists v, w ∈ R \ {0} such
that vκ + wτ = 1.

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