surfaceswhere

Surfaceswhere is a concept used in differential geometry and computational geometry to describe the subset of a surface where a given predicate holds. Let S be a smooth surface embedded in three-dimensional space, and let P be a predicate that assigns true or false to points on S, possibly depending on the point’s position, its normal, or an associated scalar field on S. The surfaceswhere of P on S, denoted W(P,S), is the set of points x in S for which P(x) is true. If P is defined by a smooth function f on S, with P(x) equivalent to a relation such as f(x) = 0 or f(x) ≤ 0, then W(P,S) corresponds to a level set or a sublevel region on S under appropriate regularity conditions.

Examples illustrate the concept. P(x) = z(x) > h selects the portion of S lying above a height h. P(x) = K(x) > 0 selects regions of positive Gaussian curvature. P(x) tests whether the surface normal N(x) lies within a small angle of a fixed direction d, yielding the portion of S aligned with d. More generally, W can be defined by any predicate that combines coordinates, normals, and fields defined on S.

Computationally, surfaceswhere regions on meshes can be extracted by testing vertex or face values, and their boundaries by locating where P changes value. When P is defined by a smooth function, the boundary of W often forms curves on S. This concept generalizes level sets and contours to surface settings and is used in mesh segmentation, feature detection, and surface modeling. Note that surfaceswhere is a descriptive term introduced for exposition and is not a standard, widely adopted terminology. See also level set methods, contours, and surface segmentation.