## Singular Point (Algebraic Curve)

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.)

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.