Pinch point (mathematics)

In mathematics, particularly in the fields of differential geometry and singularity theory, a pinch point (sometimes called a cusp point or a Whitney umbrella point) refers to a type of singular point on a surface. A surface containing a pinch point exhibits a specific kind of non-smooth behavior at that location.

Specifically, a pinch point is characterized by the surface locally resembling the equation z = xy in a suitably chosen coordinate system. Geometrically, near the pinch point, the surface appears to fold back on itself. Imagine holding a piece of fabric taut and then pinching it sharply in the middle; the point of the pinch is analogous to a pinch point on a mathematical surface.

The presence of pinch points can significantly alter the properties of a surface and influences its overall topology and geometry. These points are crucial in understanding the qualitative behavior of surfaces, especially those defined by implicit equations or parametric representations.

Pinch points are also important in the study of projections of surfaces from higher-dimensional spaces into lower-dimensional spaces. For instance, the projection of a smooth surface in three-dimensional space onto a plane may exhibit pinch points, even if the original surface is smooth. Understanding these singular points is essential for visualizing and analyzing the original higher-dimensional object.

The concept of a pinch point is related to other types of singularities such as folds and cusps. All these singularities are classified within the framework of singularity theory, which provides a systematic way to study and understand the behavior of mappings and surfaces near such points.