Cusp (singularity)

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A cusp on the curve x3–y2=0
A cusp on the curve x3–y2=0

In singularity theory a cusp is a singular point of a curve. Spinode is an alternative name, but this is less commonly used today.

For a curve defined as the zero set of a function of two variables f(x,y) = 0, the cusps on the curve will have the following properties:

  1. f(x,y)=0\,
  2. {\partial f\over \partial x}={\partial f\over \partial y}=0
  3. The Hessian matrix of second derivatives has zero determinant.

A classic example of a curve that exhibits a cusp is the curve defined by

x^3-y^2=0\,.

This curve can be expressed parametrically by the equations

x=t^2, y=t^3\,.

This curve has a cusp at the origin.

A cusp occurring in the reflection of light in the bottom of a teacup.
A cusp occurring in the reflection of light in the bottom of a teacup.

Cusps are frequently found in optics as a form of caustic. They are also found in the projections of the profile of a surface.


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