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# Singular Point

A singular point of an algebraic curve is a point where the curve has "nasty" behavior such as a cusp or a point of self-intersection (when the underlying field is taken as the reals). More formally, a point on a curve is singular if the and partial derivatives of are both zero at the point . (If the field is not the reals or complex numbers, then the partial derivative is computed formally using the usual rules of calculus.)

The following table gives some representative named curves that have various types of singular points at their origin.

 singularity curve equation acnode cusp cusp curve crunode cardioid quadruple point quadrifolium ramphoid cusp keratoid cusp tacnode capricornoid triple point trifolium

Consider the following two examples. For the curve

the cusp at (0, 0) is a singular point. For the curve

is a nonsingular point and this curve is nonsingular.

Singular points are sometimes known as singularities, and vice versa.

Algebraic Curve, Cusp, Irregular Singularity, Ordinary Point, Regular Singular Point, Singularity

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## References

Arfken, G. "Singularities" and "Singular Points." §7.1 and 8.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 396-400 and 451-454, 1985.

Singular Point

## Cite this as:

Weisstein, Eric W. "Singular Point." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SingularPoint.html