Definition:Double Point

Definition

Let $C$ be a locus.

A singular 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 $P$.

Categories of Double Points

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

• Results about double points can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): double point
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): multiple point ($k$-tuple point)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): double point
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): multiple point ($k$-tuple point)