CUSP (SINGULARITY)

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

Contents

Example

See also

References

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.

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

★ Geometric Differentiation, , Ian, Porteous, Cambridge University Press, 1994, ISBN 0-521-39063-X