A piecewise-linear surface is a surface represented by a locally finite collection of planar polygons joined face-to-face along common edges and vertices. The resulting polygonal complex is a two-dimensional manifold, possibly with boundary. The faces incident to each vertex therefore form a disk-like or half-disk-like neighborhood.
Each face is planar, so the tangent plane is constant in the interior of a face and can change across an edge. When every face is a triangle, a triangle mesh gives the standard representation. A polygon mesh that fails the local manifold conditions is nonmanifold and does not represent a surface in this sense (Botsch et al. 2010).
Piecewise-linear surfaces are commonly used to approximate smooth surfaces in computational geometry and computer graphics.
The graph of a piecewise linear function of two variables on a polygonal domain is a piecewise-linear surface. The graph of a one-variable piecewise linear function is its one-dimensional analog, a polygonal curve.
