acnode

An acnode is a type of singularity that can occur in algebraic curves. It is characterized by being an isolated point on the curve where the curve has no tangent. In simpler terms, imagine a smooth curve that suddenly stops and has a single, isolated point without any direction. This isolated point is the acnode.

Mathematically, an acnode can be identified by analyzing the local behavior of the algebraic curve near that point. If a point on the curve is an acnode, its local parametrization will exhibit certain properties. For example, if the curve is defined by an equation f(x, y) = 0, the point (x0, y0) is an acnode if it is a solution to f(x, y) = 0, and the Hessian matrix of f evaluated at (x0, y0) is non-zero and has two non-zero eigenvalues of opposite signs. Alternatively, a point is an acnode if it is an isolated real point of the curve in the real plane, while the complex curve in the complex plane at that point has a specific structure that relates to a node (a crossing point).

Acnodes are distinct from other types of singularities like nodes (where the curve crosses itself, forming two tangent lines) and cusps (where the curve has a sharp point with a single tangent line). The concept of acnodes is primarily studied in algebraic geometry.