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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 $K$ 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.)

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.

See also Algebraic Curve, Cusp

© 1996-9 Eric W. Weisstein