Cusp (singularity)

A cusp, in the context of singularity theory, refers to a type of singularity in a curve or surface. It is a point where the curve or surface changes its character abruptly. More specifically, it's a point where the tangent to the curve or surface is not well-defined, or where the tangent lines change direction in a non-smooth way. Different types of cusps exist, depending on the specific mathematical description of the singularity.

The term "cusp" derives from its resemblance to the pointed end of a crescent moon.

Several mathematical approaches can describe and classify cusps:

• Geometric Approach: This involves examining the local geometry of the curve or surface near the cusp point. This might involve analyzing the curvature, tangent lines, and osculating circles in the vicinity of the singularity.

• Analytic Approach: This approach utilizes techniques from calculus and analysis, such as the study of derivatives and partial derivatives at the cusp point. The behavior of the function defining the curve or surface near the singularity is analyzed to characterize the cusp type.

• Topological Approach: This approach focuses on the properties of the curve or surface that are preserved under continuous deformations. The topological classification of cusps considers how the neighborhood of the cusp point changes.

While the concept of a cusp is readily understood geometrically, the rigorous mathematical classification and analysis can become quite intricate, depending on the dimensionality of the space and the specific type of singularity being studied. Different branches of mathematics, including algebraic geometry, differential geometry, and singularity theory itself, contribute to a comprehensive understanding of cusps. Furthermore, the concept finds application in various fields like physics, computer graphics, and catastrophe theory.

The study of cusps often involves analyzing the unfolding of the singularity, which describes how the cusp point changes when small perturbations are applied to the system.