# Polyhedron with faces that are not flat

Short version: Which term generalizes "polyhedron" to include shapes whose faces are not necessarily flat?

Long version:

The finite volume method is not very restrictive when it comes to the shape of the grid cells. OpenFOAM (a library that implements FVM), for example, requires that a cell be "contiguous, convex and closed" and defined as "a list of faces". A face is defined by a "list of points such that each two neighbouring points are connected by an edge"; the face center "needs to be inside the face", and not "all points of the face need to be coplanar".

First I thought of describing the class of allowed shapes for a cell as "polyhedra". However, a polyhedron must have flat faces, which is not a requirement for the cell. I wonder if "warped polyhedra" would better describe the shape or if there is an appropriate geometrical term for that class.

• I guess you could call it a topological polyhedron.
– user856
Mar 8, 2018 at 18:23
• The problem is not with the shape itself, it's with the description. Think of a similar example in plane. If I have two points, it's easy to describe the straight segment that connects them. If you connect them in a more complicated way, you have more parameters, and significantly more computations to perform. Mar 8, 2018 at 18:28
• @Rahul, wouldn't a 3D topological polyhedron still have flat faces? Mar 8, 2018 at 18:34
• What about "curved polyhedron with straight edges"? Mar 8, 2018 at 18:35
• I don't know if Wikipedia is the best reference, but an important polytope was originally conceived this way, the associahedron. Wikipedia describes Stasheff's original realization as a "curvilinear polytope." A web search for this term yields a similar question, because I left a similar comment there. Oh, internet... Mar 8, 2018 at 18:56

Such faces (rank 2 faces which don't lie in a 2-dimensional plane, or more generally rank-$j$ faces which don't lie in a $j$-dimensional subspace) are commonly called "skew faces". Often they are conceived as "skeletal", so they consist solely of a set of vertices and the edges between them, not necessarily lying in a single plane. It sounds like you want the faces to also include a surface bounded by the straight line-segment edges. Then you need to decide whether or not you want to allow the surfaces to intersect or overlap.