In mathematics a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point on the curve must start to move backward. A typical example is given in the figure. A cusp is thus a type of singular point of a curve. For a plane curve defined by a differentiable parametric equation a cusp is a point where both derivatives of f and g are zero, and at least one of them changes sign. Cusps are local singularities in the sense that they involve only one value of the parameter t, contrarily to self-intersection points that involve several values. For a curve defined by an implicit equation cusps are points where the terms of lowest degree of the Taylor expansion of F are a power of a linear polynomial; however not all singular points that have this property are cusps. In some contexts, and in the remainder of this article, one restricts the definition of a cusp to the case where the non-zero part of lowest degree of the Taylor expansion of F has degree two. A plane curve cusp may be put in one of the following forms by a diffeomorphism of the plane: x2 − y2k+1 = 0, where k ≥ 1 is an integer.

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