Cusp (singularity)

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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.

For a plane curve defined by an analytic, parametric equation

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

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 ${\displaystyle \lim(g'(t)/f'(t))}$). 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

${\displaystyle F(x,y)=0,}$

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

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

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.