Intrinsic coordinates are another way of defining the position of a point P on a line (compare the cartesian and polar systems). Often the intrinsic equivalent of a cartesian form is much more complicated- however, intrinsic forms have the advantage of easy calculation of radius of curvature.

Consider a point P on a curve y=f(x). In cartesian coordinates, this would be the point P(x,y). From a fixed point A the length of the arc AP is s and the tangent at P makes an angle ψ (called the gradient angle) with the positive x axis. These are combined to give the intrinsic coordinates, P(s,ψ).

The intrinsic form s=g(ψ) of a cartesian line y=f(x) can be found using the facts that:

  • dy/dx = tan ψ
  • dx/ds = cos ψ
  • dy/ds = sin ψ

Having found the intrinsic form, the radius of curvature is then simply ds/dψ.