A pointed singularity on a curve or boundary where smoothness fails because two tangent directions coalesce into a single tangent with higher contact; the curve is not regular there and curvature typically blows up.

Principle

A cusp occurs where a parametrization has vanishing derivative to higher order (e.g., a map t↦(t^2,t^3) at t=0) so that branches meet with tangents coinciding and the usual smooth manifold structure breaks down.

Demonstration

Example: the semicubical parabola defined by y^2=x^3 has a cusp at the origin; the curve has a single tangent direction but is not differentiable as a regular embedded smooth curve at that point.

Misapplication

Calling any sharp corner or vertex a cusp; corners have distinct one-sided tangents (angle discontinuity), whereas a cusp involves coincident tangents and higher-order contact between branches.

Consequence

Cusps change local geometric and analytic properties: curvature and arclength parametrizations behave singularly, local intersection multiplicities increase, and classification in singularity theory (A2-type) applies.

Reversal

A node or transverse crossing is the opposite: branches intersect with distinct tangents and the singularity is of simpler intersection type rather than cuspidal higher-contact type.

Boundary

Applies primarily to plane algebraic or analytic curves and to boundary regularity questions; excludes ordinary corners, crossings, tacnodes of higher multiplicity that are a different singularity class.

Semantic Tension

Tension exists between cusp and corner terminology in geometry and PDE boundary regularity: both look 'pointed' but have different analytic signatures (higher-order vanishing versus angle discontinuity).

Synthesis

A cusp is a higher-contact pointed singularity where derivative orders vanish so branches coalesce tangentially; it is a nonregular point with strong geometric and analytic consequences distinct from corners or crossings.