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

Examples

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

.

This curve can be expressed parametrically by the equations

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.

See also

• Acnode
• Crunode
• Cusp catastrophe

References

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

http://www.sciencedaily.com/releases/2009/04/090414160801.htm

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