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.
|ramphoid cusp||keratoid cusp|
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.