the
radius of the circle is
and
the angle subtended by the arc is
then
and
the radius
The
curvature of the circle is
We
can generalise this idea to find the curvature of a curve at any
point on the curve by taking the limit as
and
A Level Maths Notes: FP2 – Derivation of the Curvature Formulae
If we have an arc of a
circle, the length of the arc isthe
radius of the circle isand
the angle subtended by the arc isthenand
the radiusThe
curvature of the circle isWe
can generalise this idea to find the curvature of a curve at any
point on the curve by taking the limit asand
I f a curve is the graph of
a twice differentiable function
then
the curvature can be calculated from the formula
Proof:
so
Differentiate with respect
to
using
the chain rule:
(1)
If
is
the length of a small piece of curve then
Substitute this into (1) to
obtain
after
some rearrangement. Take the magnitude of both sides to obtain
For a curve given in
parametric coordinates the curvature is given by
