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 K is taken as the reals). More formally, a point (a,b) on a curve f(x,y)=0 is singular if the x and y partial derivatives of f are both zero at the point (a,b). (If the field K 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.

cuspcusp curvex^3-y^2=0
quadruple pointquadrifolium(x^2+y^2)^3=4a^2x^2y^2
ramphoid cuspkeratoid cuspx^4+x^2y^2-2x^2y-xy^2+y^2=0
triple pointtrifolium(x^2+y^2)^2=a(x^3-3xy^2)

Consider the following two examples. For the curve


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


(0,i) is a nonsingular point and this curve is nonsingular.

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

See also

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

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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.

Referenced on Wolfram|Alpha

Singular Point

Cite this as:

Weisstein, Eric W. "Singular Point." From MathWorld--A Wolfram Web Resource.

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