Cusp (singularity)

An ordinary cusp on the curve x3y2=0

In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve.

The plane curve cusps are all diffeomorphic to one of the following forms: x2 − y2k+1 = 0, where k ≥ 1 is an integer.

More general background

Consider a smooth real-valued function of two variables, say f(xy) 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 Ak±, where k is a non-negative integer. This notation was introduced by V. I. Arnold. A function f is said to be of type Ak± if it lies in the orbit of x2 ± yk+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 x2 ± yk+1 are said to give normal forms for the type Ak±-singularities. Notice that the A2n+ are the same as the A2n since the diffeomorphic change of coordinate (x,y) → (x, −y) in the source takes x2 + y2n+1 to x2 − y2n+1. So we can drop the ± from A2n± notation.

The cusps are then given by the zero-level-sets of the representatives of the A2n equivalence classes, where n ≥ 1 is an integer.

Examples

An ordinary cusp occurring as the caustic of light rays in the bottom of a teacup.
• An ordinary cusp is given by x2 − y3 = 0, i.e. the zero-level-set of a type A2-singularity. Let f(xy) be a smooth function of x and y and assume, for simplicity, that f(0,0) = 0. Then a type A2-singularity of f at (0,0) can be characterised by:
1. Having a degenerate quadratic part, i.e. the quadratic terms in the Taylor series of f form a perfect square, say L(xy)2, where L(xy) is linear in x and y, and
2. L(xy) does not divide the cubic terms in the Taylor series of f(xy).

Ordinary cusps are very important geometrical objects. It can be shown that caustics in the plane generically comprise smooth points and ordinary cusp points. By generic we mean that an open and dense set of all caustics comprise smooth points and ordinary cusp points. Caustics are, informally, points of exceptional brightness caused by the reflection of light from some object. In the teacup picture light is bouncing off the side of the teacup and interacting in a non-parallel fashion with itself. This results in a caustic. The bottom of the teacup represents a two-dimensional cross section of this caustic.

The ordinary cusp is also important in wavefronts. A wavefront can be shown to generically comprise smooth points and ordinary cusp points. By generic we mean that an open and dense set of all wavefronts comprise smooth points and ordinary cusp points.
• A rhamphoid cusp (coming from the Greek meaning beak-like) is given by x2y5 = 0, i.e. the zero-level-set of a type A4-singularity. These cusps are non-generic as caustics and wavefronts. The rhamphoid cusp and the ordinary cusp are non-diffeomorphic.

For a type A4-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 A4).

To see where these extra divisibility conditions come from, assume that f has a degenerate quadratic part L2 and that L divides the cubic terms. It follows that the third order taylor series of f is given by L2 ± LQ where Q is quadratic in x and y. We can complete the square to show that L2 ± LQ = (L ± ½Q)2 – ¼Q4. We can now make a diffeomorphic change of variable (in this case we simply substitute polynomials with linearly independent linear parts) so that (L ± ½Q)2 − ¼Q4 → x12 + P1 where P1 is quartic (order four) in x1 and y1. The divisibility condition for type A≥4 is that x1 divides P1. If x1 does not divide P1 then we have type exactly A3 (the zero-level-set here is a tacnode). If x1 divides P1 we complete the square on x12 + P1 and change coordinates so that we have x22 + P2 where P2 is quintic (order five) in x2 and y2. If x2 does not divide P2 then we have exactly type A4, i.e. the zero-level-set will be a rhamphoid cusp.