Finding the Intrinsic Equation of a Curve from the Curvature or Radius of Curvature

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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 isIfwhenfind the intrinsic equation of the curve.

Hence

so

Example: The curvature of a curve isIfwhenfind the intrinsic equation of the curve.

Rearrangement givesWe 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 isand the task is to minimise the length, hence find the geodesic.

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