## Intrinsic Coordinates

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 as 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.

Example:The radius of curvature of a curve is If when find the intrinsic equation of the curve.  Hence  so The curvature of a curve is If when find the intrinsic equation of the curve. Rearrangement gives We integrate this: Now put to 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. 