 Any surface with an isolated umbilic point at the origin can be expressed as a Monge form parameterisation formula_18, where formula_19 is the unique principal curvature.The type of umbilic is classified by the cubic form from the cubic part and corresponding Jacobian cubic form.Whilst principal directions are not uniquely defined at an umbilic the limits of the principal directions when following a ridge on the surface can be found and these correspond to the root-lines of the cubic form.