Lecture Notes for 120 - UCLA Department of Mathematics

3.1. THE FUNDAMENTAL EQUATIONS 61
(10) Show that B is regular when |⌧| > 0 and that in this case the curvature
of B is given by
r
⇣ apple
⌘ 2
1+
⌧
(11) Show that N is regular when apple 2 +⌧ 2 > 0 and that in this case the curvature
of N is given by
v
u
t apple d⌧
1+
ds
⌧ dapple 2
ds
(apple 2 + ⌧ 2 ) 3
(12) Define ⇢ = p apple 2 + ⌧ 2 and by
apple = ⇢ cos
, ⌧ = ⇢ sin
Show that ⇢ = |D| and that is the natural arclength parameter for the
unit field 1 ⇢ D.
(13) Show that a space curve is part of a line if all its tangent lines pass through
afixedpoint.
(14) Let Q (t) be a vector associated to a curve q (t) such that
d ⇥ T
dt N
⇤ ds B =
dt Q ⇥ ⇥ T N B ⇤
Show that Q = D.
(15) Let q (s) be a unit speed space curve with non-vanishing curvature and
torsion. Show that
d
ds
✓ 1
⌧
✓
d 1 d 2 ◆◆
q
ds apple ds 2 + d ✓ apple
ds ⌧
◆
dq
+ ⌧ d 2 q
ds apple ds 2 =0
(16) For a regular space curve q (t) we say that a normal field X is parallel
along q if X · T =0and dX
dt
is parallel to T.
(a) Show that for a fixed t 0 and X (t 0 ) ? T (s 0 ) there is a unique parallel
field X that is X (t 0 ) at t 0 .
(b) A Bishop frame consists of an orthonormal frame T, N 1 , N 2 along
the curve so that N 1 , N 2 are both parallel along q. Forsuchaframe
show that
2
3
d ⇥ ⇤ ds ⇥ ⇤ 0 apple 1 apple 2
T N1 N 2 = T N1 N 2
4 apple 1 0 0 5
dt
dt
apple 2 0 0
Note that such frames always exist, even when the space curve doesn’t
have positive curvature everywhere.
(c) Show further that for such a frame
apple 2 = apple 2 1 + apple 2 2
(d) Show that if q has positive curvature so that N is well-defined, then
where
N = cos N 1 +sin N 2
d
dt = ds
dt ⌧
apple 1 = apple cos , apple 2 = apple sin