A narrow, concentrated region on a surface where curvature attains large values along a curve, producing a pronounced ridge feature; smoothness degrades but the tangent plane may remain continuous unlike a crease.

Principle

Arises when curvature (principal curvatures or their derivatives) concentrates along a one-dimensional locus: the surface remains at least C^1 but second derivatives become large or singular in the limit, creating a visually sharp ridge.

Demonstration

The rounded but sharp-looking edge of a bent metal sheet or the smoothed limit of a dihedral fold: smoothing a polyhedral edge yields a band where curvature concentrates and appears as a sharp ridge though tangent directions vary continuously.

Misapplication

Calling every visually sharp feature a sharp ridge when in fact the feature may be a true crease (tangent discontinuity) or a cuspidal edge (transverse cusp); or assuming curvature blow-up implies topological singularity.

Consequence

Affects geometric processing and mechanics: ridge lines guide feature detection, control streamline and geodesic flow, local bending energy concentrates there in elasticity, and mesh refinement must capture curvature gradients for accurate representation.

Reversal

A broad smooth undulation: curvature remains moderate and smoothly varying so that no narrow concentration or visually sharp ridge is present along the curve.

Boundary

Refers to concentrated curvature without necessarily implying a discontinuity of the tangent plane or multiple preimages; excludes creases (tangent jump), cuspidal edges (cusp transverse profile), and self-intersection seams.

Semantic Tension

Tension with crease and cuspidal-edge notions: a sharp ridge emphasizes scalar curvature concentration and continuous tangents, whereas a crease emphasizes a jump in first derivatives and a cuspidal edge emphasizes transverse singularity type.

Synthesis

A sharp ridge is a narrow locus on a surface where curvature is highly concentrated, producing a pronounced geometric feature that may limit to a crease but is distinguished by (typically) continuous tangent behavior and strong second-derivative localization.