A cuspidal singularity (cusp) is a point where a single branch of a curve has a self-tangency producing a pointed, nontransverse local geometry; algebraically the quadratic part is a square and higher-order terms produce a single tangent direction with multiplicity.

Principle

Cusps occur when the lowest-degree homogeneous part of the local defining equation has a repeated linear factor so that the branch does not split into distinct tangents; the contact order between coordinate parametrizations is higher and normalization yields a single branch.

Demonstration

A standard plane model is y^2 = x^3 at the origin: the gradient vanishes there and the curve has a single tangent direction but a sharper 'point' than a smooth inflection, so the origin is a cusp.

Misapplication

Labeling any sharp-looking point as a cusp without checking algebraic multiplicity or splitting of the quadratic part, or confusing an inflection point of a smooth curve with a cuspidal singularity.

Consequence

Cusps are more severe than nodes: they are not stable under small perturbations in the same way, they change the normalization differently than nodes, and they contribute differently to invariants such as δ and the genus formula.

Reversal

An ordinary node where two distinct tangents exist and the branches cross transversely, or a smooth inflection point where curvature changes but the point remains regular.

Boundary

Specifically a unibranch singularity with tangent of multiplicity greater than one; excludes multi-branch degeneracies, higher-order cusp types (ramphoid cusps), and nonreduced phenomena. Characteristic of the ground field can alter the classification.

Semantic Tension

Tension arises with the notion of an inflection point or a higher-order contact that is nonetheless regular; a cusp is singular and unibranch, whereas an inflection point is a smooth phenomenon with vanishing higher derivatives but regular local ring.

Synthesis

A cusp is a unibranch singular point where the local equation's lowest-degree part is a repeated linear factor, producing a single tangent of higher multiplicity and a pointed, nontransverse local geometry (classically modeled by y^2=x^3).