A localized non-smooth point on a curve or surface where the tangent (or tangent plane) degenerates so that the local geometry has a pointed, sharp form; analytically, it is a singular point at which derivatives fail to define a regular tangent direction, for example the plane cusp parametrized by t ↦ (t^2,t^3).

Principle

A cusp arises when two branches of a curve meet with coincident tangent direction but differing contact order, producing a failure of differentiability characterized by vanishing linear terms and leading higher-order terms that create a pointed shape.

Demonstration

Plane example: the semicubical parabola given by y^2 = x^3 at the origin is a standard cusp — the tangent direction is undefined because both coordinate derivatives vanish to the first order. Surface example: the edge of a folded sheet may produce a one-dimensional cusp curve where normal directions collapse.

Misapplication

Calling any sharp corner or polygonal vertex a cusp. A polygonal corner has two distinct, well-defined tangents meeting at an angle; a cusp requires tangents to coincide or degenerate and is a higher-order analytic singularity.

Consequence

Cusps obstruct classical differential geometric notions (curvature may blow up or be undefined), affect local analytic classification (they require resolution or blow-up to desingularize), and influence topology and enumerative invariants of algebraic curves.

Reversal

A regular point where a unique nondegenerate tangent or tangent plane exists; contrasted also with a transverse intersection or a corner where distinct tangents exist instead of a degeneracy.

Boundary

Applies to analytic or smooth (C^k) curves and surfaces where tangent structure is meaningful; excludes generic polygonal corners, fractal roughness without tangent collapse, and removable coordinate singularities after reparametrization.

Semantic Tension

Cusp versus corner or node: a corner has distinct tangent directions; a node is an intersection of distinct branches with separate tangents; a cusp is a single branch with tangent degeneracy. The terms are often confused in informal geometry but have distinct analytic signatures.

Synthesis

A cusp singularity is the analytic phenomenon on a curve or surface where the linear tangent description fails and higher-order contact produces a pointed, non-smooth local geometry; recognizing it requires checking vanishing of first-order terms and classifying the next-order behavior.