Cusp (singularity)

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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. 
2. 
3. The Hessian matrix of second derivatives has zero determinant.

[edit] Example

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

.

This curve can be expressed parametrically by the equations

This curve has a cusp at the origin.

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

[edit] See also

• Acnode
• Crunode
• Cusp catastrophe

[edit] References

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