A curve on a surface along which the surface exhibits cusp-type singularities in transverse cross-sections: locally the map from parameters to ambient space has a cusp in the normal direction while the singular set itself is a one-dimensional curve.

Principle

A generic corank-1 singularity of a map from R^2 to R^3 that appears along a curve: the differential drops rank transversely and the transverse profile is equivalent to a plane cusp (e.g., y^2 = x^3) while the singular locus is smooth.

Demonstration

Locally modelled by f(u,v) = (u, v^2, v^3): along the u-axis the surface has a cuspidal cross-section in the (v^2,v^3)-plane, producing a visible edge curve of cusp singularities.

Misapplication

Calling any sharp-looking line on a surface a cuspidal edge without checking the local singularity type; confusing a cuspidal edge with a crease or a sharp ridge where the transverse profile is not a true cusp.

Consequence

Geometry and analysis near the edge change: normals can fail to extend smoothly across the edge, curvature invariants are singular or one-sided, and standard manifold-based PDE or flow arguments may require modification to handle cusp behavior.

Reversal

A regular edge or smooth boundary curve: a curve on a surface where local parametrizations remain of full rank (no cusp in the transverse profile) and the surface is a smooth 2-manifold near the curve.

Boundary

Refers specifically to cusp-type transverse singularities along a one-dimensional locus; excludes isolated cusp points, pure creases (tangent discontinuity without cusp profile), and intersections arising purely from self-overlap.

Semantic Tension

Tension with crease/ridge vocabulary: cuspidal edges are singularities detected by differential rank and normal behavior, whereas creases are defined by tangent-plane discontinuity and ridges by curvature concentration—these phenomena can coincide but are conceptually distinct.

Synthesis

A cuspidal edge is a smooth curve along a surface where each transverse slice exhibits a cusp singularity: analytically a corank-1 degeneracy producing cusp-shaped profiles and geometrically a line of singular points that modifies normal and curvature behavior.