Definition:Singular Point/Real

Definition

Let $C$ be a locus.

Definition 1

A point $P \in C$ is called a singular point if and only if $P$ does not have a unique tangent to $C$ which is itself differentiable.

Categories of Singular Points

Acnode

An acnode is a singular point of the locus of an equation describing a curve which is not actually on that curve.

Double Point

Let $C$ be a locus.

A point $P \in C$ is called a double point if and only if $C$ intersects itself at $P$ such that there are $2$ tangents to $C$ at $P$.

Cusp

A cusp is a singular point on a curve at which there are two different tangents which coincide.

Thus a cusp is a special case of a double point in which the tangents are coincident.

Crunode

A crunode is a double point $P$ of the locus of an equation describing a curve which intersects itself in such a way that there are $2$ distinct tangents at $P$.

Also see

• Equivalence of Definitions of Singular Point

• Results about singular points can be found here.