# Cusp (singularity)

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:
*x*^{2} − *y*^{2k+1} = 0, where *k* ≥ 1 is an integer.

## Classification in differential geometry

Consider a smooth real-valued function of two variables, say *f*(*x*, *y*) where *x* and *y* are real numbers. So *f* is a function from the plane to the line. The space of all such smooth functions is acted upon by the group of diffeomorphisms of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of coordinate in both the source and the target. This action splits the whole function space up into equivalence classes, i.e. orbits of the group action.

One such family of equivalence classes is denoted by *A _{k}*

^{±}, where

*k*is a non-negative integer. This notation was introduced by V. I. Arnold. A function

*f*is said to be of type

*A*

_{k}^{±}if it lies in the orbit of

*x*

^{2}±

*y*

^{k+1}, i.e. there exists a diffeomorphic change of coordinate in source and target which takes

*f*into one of these forms. These simple forms

*x*

^{2}±

*y*

^{k+1}are said to give normal forms for the type

*A*k

_{}^{±}-singularities. Notice that the

*A*

_{2n}

^{+}are the same as the

*A*

_{2n}

^{−}since the diffeomorphic change of coordinate (

*x*,

*y*) → (

*x*, −

*y*) in the source takes

*x*

^{2}+

*y*

^{2n+1}to

*x*

^{2}−

*y*

^{2n+1}. So we can drop the ± from

*A*

_{2n}

^{±}notation.

The cusps are then given by the zero-level-sets of the representatives of the *A*_{2n} equivalence classes, where *n* ≥ 1 is an integer.

## Examples

- An
**ordinary cusp**is given by*x*^{2}−*y*^{3}= 0, i.e. the zero-level-set of a type*A*_{2}-singularity. Let*f*(*x*,*y*) be a smooth function of*x*and*y*and assume, for simplicity, that*f*(0,0) = 0. Then a type*A*_{2}-singularity of*f*at (0,0) can be characterised by:

- Having a degenerate quadratic part, i.e. the quadratic terms in the Taylor series of
*f*form a perfect square, say*L*(*x*,*y*)^{2}, where*L*(*x*,*y*) is linear in*x*and*y*,*and* -
*L*(*x*,*y*) does not divide the cubic terms in the Taylor series of*f*(*x*,*y*).

- A
**rhamphoid cusp**(coming from the Greek meaning beak-like) is given by*x*^{2}–*y*^{5}= 0, i.e. the zero-level-set of a type*A*_{4}-singularity. These cusps are non-generic as caustics and wavefronts. The rhamphoid cusp and the ordinary cusp are non-diffeomorphic.

For a type *A*_{4}-singularity we need *f* to have a degenerate quadratic part (this gives type *A*_{≥2}), that *L* *does* divide the cubic terms (this gives type *A*_{≥3}), another divisibility condition (giving type *A*_{≥4}), and a final non-divisibility condition (giving type exactly *A*_{4}).

To see where these extra divisibility conditions come from, assume that *f* has a degenerate quadratic part *L*^{2} and that *L* divides the cubic terms. It follows that the third order taylor series of *f* is given by *L*^{2} ± *LQ* where *Q* is quadratic in *x* and *y*. We can complete the square to show that *L*^{2} ± *LQ* = (*L* ± ½*Q*)^{2} – ¼*Q*^{4}. We can now make a diffeomorphic change of variable (in this case we simply substitute polynomials with linearly independent linear parts) so that (*L* ± ½*Q*)^{2} − ¼*Q*^{4} → *x*_{1}^{2} + *P*_{1} where *P*_{1} is quartic (order four) in *x*_{1} and *y*_{1}. The divisibility condition for type *A*_{≥4} is that *x*_{1} divides *P*_{1}. If *x*_{1} does not divide *P*_{1} then we have type exactly *A*_{3} (the zero-level-set here is a tacnode). If *x*_{1} divides *P*_{1} we complete the square on *x*_{1}^{2} + *P*_{1} and change coordinates so that we have *x*_{2}^{2} + *P*_{2} where *P*_{2} is quintic (order five) in *x*_{2} and *y*_{2}. If *x*_{2} does not divide *P*_{2} then we have exactly type *A*_{4}, i.e. the zero-level-set will be a rhamphoid cusp.

## Applications

Cusps appear naturally when projecting into a plane a smooth curve in the three dimensional Euclidean space. In general, such a projection is a curve whose singularities are self-crossing points and ordinary cusps. Self-crossing points appear when two different points of the curves have the same projection. Ordinary cusps appear when the tangent to the curve is parallel to the direction of projection (that is when the tangent projects on a single point). More complicated singularities occur when several phenomena occurs simultaneously. For example, rhamphoid cusps occur for inflection points (and for undulation points) for which the tangent is parallel to the direction of projection.

In many cases, and typically in computer vision and computer graphics, the curve that is projected is the curve of the critical points of the restriction to a (smooth) spatial object of the projection. A cusp appears thus as a singularity of the contour of the image of the object (vision) or of its shadow (computer graphics).

Caustics and wave fronts are other examples of curves having cusps that are visible in the real world.

## See also

## References

- Bruce, J. W.; Giblin, Peter (1984).
*Curves and Singularities*. Cambridge University Press. ISBN 0-521-42999-4. - Porteous, Ian (1994).
*Geometric Differentiation*. Cambridge University Press. ISBN 0-521-39063-X.