structural:scholar_edition, stylistic:style_transfer, structural:summarization, linguistic:translation
Singular point

A singular point of an implicit surface (in R 3 {\displaystyle \mathbb {R} ^{3}}) is a point of the surface where the implicit equation holds and the three partial derivatives of its defining function are all zero. Therefore, the singular points are the solutions of a system of four equations in three indeterminates. As most such systems have no solution, many surfaces do not have any singular point. A surface with no singular point is called regular or non-singular.

The study of surfaces near their singular points and the classification of the singular points is singularity theory. A singular point is isolated if there is no other singular point in a neighborhood of it. Otherwise, the singular points may form a curve. This is in particular the case for self-crossing surfaces.