Lecture Notes for 120 - UCLA Department of Mathematics

3.3. CLOSED SPACE CURVES 66
(12) Prove that a unit speed curve q with non-zero curvature and torsion lies
on a sphere if there are constants a, b such that
✓ ✓ˆ ◆ ✓ˆ ◆◆
apple a cos ⌧ds + b sin ⌧ds =1
Hint: Show
✓ ◆
1 d 1
⌧ ds apple
✓ˆ
= a sin
◆ ✓ˆ
⌧ds + b cos
◆
⌧ds
and
⌧
apple = d ✓ ✓ˆ ◆ ✓ˆ ◆◆
a sin ⌧ds + b cos ⌧ds
ds
and use the previous exercise.
(13) Show that if a curve with constant curvature lies on a sphere then it is
part of a circle, i.e., it is forced to be planar.
(14) Show that
q ⇤ (s) =q + 1 apple N + 1 ✓ˆ ◆
apple cot ⌧ds B
defines an envelope for q.
(15) Show that a planar curve has infinitely many Bertrand mates.
(16) Let q, q ⇤ be two Bertrand mates.
(a) (Schell) Show that
⌧⌧ ⇤ = sin2 ✓
r 2
(b) (Mannheim) Show that
(1 rapple)(1+rapple ⇤ ) = cos 2 ✓
(17) Consider a curve q (s) parametrized by arclength with positive curvature
and non-vanishing torsion such that
appler + ⌧rcot ✓ =1
i.e., there is a Bertrand mate.
(a) Show that the Bertrand mate is uniquely determined by r.
(b) Show that if q has two different Bertrand mates then it must be a
generalized helix.
(c) Show that if a generalized helix has a Bertrand mate then its curvature
and torsion are constant, consequently it is a circular helix.
(18) Investigate properties of a pair of curves that have the same normal planes
at corresponding points, i.e., their tangent lines are parallel.
(19) Investigate properties of a pair of curves that have the same binormal lines
at corresponding points.
3.3. Closed Space Curves
We start by studying spherical curves. In fact any regular space curve generates
a natural spherical curve, the unit tangent. We studied this for planar curves where
the unit tangent became a curve on a circle. In that case the length of the unit
tangent curve can be interpreted as an integral of the curvature and it also measures
how much the curve turns. When the planar curve is closed this turning necessarily
has to be a multiple of 2⇡.