Cusps

Cusps are pointed features that occur in various contexts, typically referring to a sharp point or a singular point on a shape. The term is used in geometry, anatomy, and even astrology to describe a boundary or projection that stands out from its surroundings.

In mathematics, a cusp is a type of singular point on a curve where the curve has a sharp point and a well-defined tangent does not exist. A classic example is the plane curve given by y^2 = x^3, which has a cusp at the origin. Another way to see it is through a parametric form x = t^2, y = t^3, where t = 0 yields the cusp. A cusp is different from a node, where a curve intersects itself with two distinct tangents. Cusps represent a single, unbranched point where the smoothness of the curve fails in a specific, localized manner.

In algebraic geometry, cusps are commonly discussed as simple singularities of plane curves. They reflect a particular kind of degeneracy where the two branches of the curve meet with a common tangent, producing a sharp tip rather than a crossing.

In anatomy and dentistry, a cusp refers to a pointed projection on a tooth, such as cusp tips on molars and premolars. These projections play a role in how teeth grasp and chew and vary in number and arrangement across species and individuals.

In astrology, the term cusp denotes the boundary between two zodiac signs, and people born near these dividing lines are sometimes described as having traits of both adjacent signs.

Overall, cusps denote pointed or singular features that mark a transition or an extremal point in a given context.