 In some contexts, the condition on the directional derivative may be omitted, although, in this case, the singularity may look like a regular point.For a curve defined by an implicit equation 
which is smooth, cusps are points where the terms of lowest degree of the Taylor expansion of are a power of a linear polynomial; however, not all singular points that have this property are cusps.The theory of Puiseux series implies that, if is an analytic function (for example a polynomial), a linear change of coordinates allows the curve to be parametrized, in a neighborhood of the cusp, as
where is a real number, is a positive even integer, and is a power series of order (degree of the nonzero term of the lowest degree) larger than .