A mapping f of a triangulated surface into space is polyhedral provided it is linear on triangles. Intuitively, this means that it mapped as the union of finitely many planar triangles.

Often it is convenient to join adjacent triangles that are coplanar into a single facet. In this case, a polyhedral surface is the union of finitely many planar polygonal regions; here the faces are the maximal 2-dimensional planar subsets of the mapping, the edges are the maximal 1-dimensional linear subsets of the boundaries of the faces, and the vertices are the endpoints of the edges. Regions that are not simply connected may be generated in this way.

Since any plane polygonal region can be decomposed into triangles, there is no loss of generality in defining polyhedral surfaces in terms of triangles only, though it is often more convenient in practice to use arbitrary polygonal faces.

See also:

[More] Smooth surfaces

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