Cusp

A cusp is a point on a curve where the curve fails to be smooth, producing a sharp or pointed end. In the plane, a cusp often occurs at a singular point where the curve has a unique tangent line, but the curve is not differentiable there because the velocity of a parametric representation vanishes. As a result, the two branches of the curve meet with a common tangent but do not cross.

A classic example is the curve given by y^2 = x^3, which has a cusp at the origin. A standard parametric form is x = t^2, y = t^3, and as t approaches 0 the curve approaches a single point with tangent along the x-axis. In algebraic geometry, a cusp is a type of singularity with multiplicity two, distinct from a node, where two branches cross with different tangents.

Cusp points are used to distinguish between different local behaviors of curves. They are contrasted with smooth points, where the curve has a well-defined tangent through every nearby point, and with nodes, where two branches intersect.

Beyond mathematics, the term cusp has other uses. In dentistry, a cusp is the pointed projection on a tooth crown, such as the cusps of molars and canines, which play a role in chewing and occlusion. In advanced mathematics, the concept of a cusp also appears in catastrophe theory and related ideas as a simple type of singularity describing how a system can change abruptly.