# Cusp (singularity)

## Point on a curve where motion must move backwards / From Wikipedia, the free encyclopedia

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In mathematics, a **cusp**, sometimes called **spinode** in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of singular point of a curve.

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For a plane curve defined by an analytic, parametric equation

- ${\begin{aligned}x&=f(t)\\y&=g(t),\end{aligned}}$

a cusp is a point where both derivatives of *f* and *g* 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 $\lim(g'(t)/f'(t))$). Cusps are *local singularities* in the sense that they involve only one value of the parameter *t*, in contrast to self-intersection points that involve more than one value. 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

- $F(x,y)=0,$

which is smooth, 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. The theory of Puiseux series implies that, if *F* 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

- ${\begin{aligned}x&=at^{m}\\y&=S(t),\end{aligned}}$

where *a* is a real number, *m* is a positive even integer, and *S*(*t*) is a power series of order *k* (degree of the nonzero term of the lowest degree) larger than *m*. The number *m* is sometimes called the *order* or the *multiplicity* of the cusp, and is equal to the degree of the nonzero part of lowest degree of *F*. In some contexts, the definition of a cusp is restricted to the case of cusps of order two—that is, the case where *m* = 2.

The definitions for plane curves and implicitly-defined curves have been generalized by René Thom and Vladimir Arnold to curves defined by differentiable functions: a curve has a cusp at a point if there is a diffeomorphism of a neighborhood of the point in the ambient space, which maps the curve onto one of the above-defined cusps.