Cusp (singularity) - Examples

Examples

• An ordinary cusp is given by x2 − y3 = 0, i.e. the zero-level-set of a type A₂-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 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(x, y)2, where L(x, y) is linear in x and y, and
2. L(x, y) does not divide the cubic terms in the Taylor series of f(x, y).

Ordinary cusps are very important geometrical objects. It can be shown that caustic 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 x2 – y5 = 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.