Cusp (singularity)

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:

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.

Example

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.

File:Cusp in teacup.gif

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.

See also

• Acnode
• Crunode
• Cusp catastrophe

References

• Porteous, Ian (1994). Geometric Differentiation. Cambridge University Press. ISBN 0-521-39063-X.