Cusp (singularity)

From Wikipedia, the free encyclopedia

For other uses, see Cusp.

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:

$f(x,y)=0\,$
${\partial f\over \partial x}={\partial f\over \partial y}=0$
The Hessian matrix of second derivatives has zero determinant. That is:

$\begin{bmatrix} \frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x\,\partial y} \\ \\ \frac{\partial^2 f}{\partial x\,\partial y} & \frac{\partial^2 f}{\partial y^2} \end{bmatrix} =0.$

[edit] Examples

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

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

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.

Cardioid
Cusps are frequently found in optics as a form of caustic.
They are also found in the projections of the profile of a surface.
Cusp curves caused by gravitational lensing may allow future Astronomers to infer the density and location of Dark Matter around galaxies.[1]
In economics, cusps in the demand curve occur at price points.