Lecture Notes for 120 - UCLA Department of Mathematics

1.3. CURVATURE 21
of how fast it changes and in turn how far the curve is from being a line. We let
✓ be the arclength parameter for T. The relative change between the arclength
parameters for the unit tangent and the curve is by definition the curvature
and with a general parametrization
apple = d✓
ds
apple = dt d✓
ds dt
We shall see that the curvature is related to the part of the acceleration that is
orthogonal to the unit tangent vector. Note that apple 0 as ✓ increases with s.
Proposition 1.3.1. A regular curve is part of a line if and only if its curvature
vanishes.
Proof. The unit tangent of a line is clearly stationary. Conversely if the
curvature vanishes then the unit tangent is stationary. This means that if the
curve is parametrized by arclength then it will be a straight line.
⇤
Next we show how the curvature can be calculated with general parametrizations.
Proposition 1.3.2. The curvature of a regular curve is given by
Proof. We calculate
apple =
=
|v||a (a · T) T|
|v| 3
area of parallelogram (v, a)
|v| 3
apple = d✓
ds
= d✓ dt
dt ds
= dT 1
|v|
dt
d v
= |v|
dt |v|
=
=
=
1
a v (a · v)
|v| 2 |v| 4
1
|v| 2 a (a · v) v
|v| 2
1
2
|a (a · T) T|
|v|
The area of the parallelogram spanned by v and a is given by the product of the
length of the base represented by v and the height represented by the component