 A cusp is thus a type of singular point of a curve.For a plane curve defined by an analytic, parametric equation
a cusp is a point where both derivatives of and are zero, and the directional derivative, in the direction of the tangent, changes sign (the direction of the tangent is the direction of the slope formula_2).Cusps are "local singularities" in the sense that they involve only one value of the parameter , in contrast to self-intersection points that involve more than one value.