a triangulated surface

Take a smooth surface — the curved hood of a car, a human face, a terrain — and approximate it by a quilt of flat triangles whose corners sit on the surface. The result is a triangulated surface: a two-dimensional simplicial complex in which the triangles tile the shape so that, around every vertex, the triangles fan out into a single disk and along every interior edge exactly two triangles meet. This is the workhorse object of computer graphics and geometry processing, where it is usually called a triangle mesh.

Precisely, a triangulated surface is a simplicial complex K together with the combinatorial-manifold condition: the link of every vertex is a single cycle of edges (or a single path, at a boundary vertex), so that |K| is a topological surface, possibly with boundary. The data are three lists — vertex positions in R^3, the edges, and the triangular faces — plus the incidence relations among them; the classic half-edge or quad-edge data structures store this so you can walk from a triangle to its neighbors in constant time. Curvature and other smooth quantities are not given for free: they must be re-derived discretely, for example concentrating Gaussian curvature at vertices as an angle defect, or attaching a discrete mean curvature to edges.

Triangulated surfaces matter because almost everything we scan, simulate, or render lives as a mesh. The combinatorics control the topology — Euler's formula V - E + F = 2 - 2g pins down the genus g — while the vertex positions carry the geometry, and the two are independent: refining the triangulation changes V, E, F together but not the genus. A subtle and important honesty point is that mesh quantities depend on the triangulation, not only on the underlying smooth surface: a poorly shaped mesh (slivers, very obtuse triangles) can make discrete curvatures and the cotangent Laplacian behave badly even when the surface it approximates is perfectly smooth.