so
if we have an expression for the curvature
or
the radius of curvature
we
can find an the intrinsic equation of the curve in one or other form
by integrating. Rearrangement of
gives
or
-
which one we use depends on which is easier to integrate.
A Level Maths Notes: FP2 – Finding the Intrinsic Equation of a Curve from the Curvature or Radius of Curvature
Intrinsic coordinates label a point on a curve by the length along the curve from a fixed point, often the origin.
The curvature of a point on
the curve can be written asso
if we have an expression for the curvatureor
the radius of curvaturewe
can find an the intrinsic equation of the curve in one or other form
by integrating. Rearrangement of
givesor
-
which one we use depends on which is easier to integrate.
Example: The radius of
curvature of a curve is
If
when
find
the intrinsic equation of the curve.
Hence
so
Example: The curvature of a
curve is
If
when
find
the intrinsic equation of the curve.
Rearrangement gives
We
integrate this:
Now putto
obtain
Intrinsic coordinates and
equations are very important in differential geometry and general
relativity, since in the absence of absolute space, we can establish
a reference frame relative to a moving body, which follows a
'geodesic' in curved space time. In general relativity, the length of
a geodesic is
and
the task is to minimise the length, hence find the geodesic.