Local surface characteristics are associated with points, and are contrasted with global characteristics which are associated with entire surfaces. Local characteristics of a point are usually easily determined from the mathematical description of the surface. The study of entire classes of surfaces, such as minimal and CMC surfaces, is based on the global enforcement of some local characteristic.
Intrinsic versus Extrinsic Properties
Local surface characteristics, or properties, can be classified as either intrinsic or extrinsic. intrinsic properties could be deduced by a 'flatlander' on the surface unable to percieve the direction perpendicular to the surface in the embedding space, whereas extrinsic ones require knowledge of that space.
Various measures of curvature are based on a point's principal curvatures. They are multiplied to yield the gaussian curvature, an intrinsic property, and added to yield the mean curvature, an extrinsic property. Surfaces with zero mean curvature everywhere are minimal surfaces, while surfaces with constant mean curvature everywhere are often referred to as CMC surfaces. Mean and gaussian curvature of selected level surfaces are compared using color to indicate curvature.
Singularites on surfaces are points, or sets of points, where the surface becomes degenerate. There are a number of types of surface singularities. Point singularities are illustrated by Kumer Surface. Line singularities are illustrated by the Roemer Surface.
Sometimes surface self-intersections are called singularities. Such features are global surface characteristics and are quite distinct from the degenerate singularites described above