Lecture Notes for 120 - UCLA Department of Mathematics

1.3. CURVATURE 23
Proof. The first formula follows directly from the definition of T. For the
second we note that
a = dv
dt
= d✓ dv
dt d✓
= d✓
dt
= d✓
dt
✓ d |v|
◆
dT
T + |v|
d✓
d✓
✓ ◆
d |v|
d✓ T + |v| N
= d |v|
dt T + d✓
dt |v| N
This shows that a is a linear combination of T, N. It also shows that
a · N = d✓
dt |v|
= d✓ ds
ds dt |v|
= apple |v| 2
So we obtain the second equation. The last formula then follows from the fact that
N is the direction of the normal component of the acceleration.
⇤
Definition 1.3.5. An involute of a curve q (t) is a curve q ⇤ (t) that lies on the
corresponding tangent lines to q (t) and intersects these tangent lines orthogonally.
We can always construct involutes to regular curves. First of all
q ⇤ (t) =q (t)+u (t) T (t)
as it is forced to lie on the tangent lines to q. Secondly the velocity v ⇤ must be
parallel to N. As
dq ⇤
= dq
dt dt + du
dt T + uappleN
this forces us to select u so that
du
dt =
Thus
q ⇤ (t) =q (t) s (t) T (t)
where s is any arclength parametrization of q. Note that s is only determined up
to a constant so we always get infinitely many involutes to a given curve.
Example 1.3.6. If we strip a length of masking tape glued to a curve keeping
it taut while doing so we obtain an involute.
Assume the original curve is unit speed q (s). The process of stripping the tape
from the curve forces the endpoint of the tape to have an equation of the form
ds
dt
q ⇤ (s) =q (s)+u (s) T (s)
since for each value of s the tape has two parts, the first being the curve up to q (s)
and the second the line segment from q (s) to q (s)+u (s) T (s). The length of this
is up to a constant given by
s + u (s)