Suppose that and are smooth curves in which pass through a common point . We say that and have zeroth order contact if their tangents at are distinct.

Suppose that and are tangent at . We may then set up a coordinate system in which is the origin and the axis is tangent to both curves. By the implicit function theorem, there will be a neighborhood of such that can be described parametrically as with and can be described parametrically as with . We then define the order of contact of and at to be the largest integer such that all partial derivatives of and of order not greater than at are equal.