The geometric layer

This documentation is still incomplete!

Until now, we have seen grids as purely combinatorial constructs. Adding geometric functionality in a separate layer has the advantage to make a combinatorial grid usable in a broader context, because there may be many different geometric embeddings for a grid. Also, there are a lot of algorithms that do not require any geometry at all.

Some of the major points of variation of the mathematical aspects of geometric embeddings are

• embedding in higher-dimensional space (2D grid in 3D space)
• embedding in curved geometry (e.g. on a sphere)
• non-linear elements
• time-dependent embeddings

Furthermore, geometric embeddings can differ in computational aspects:

• arithmetic: exact or floating-point?
• storage or calculation of entities?
• exact or approximate calculation (e. g. cell volume for nonlinear embeddings)

A linear and a non-linear geometry for the same grid

What kind of functionality is available in a class implementing a geometric embedding for a grid? This question cannot be answered in general. The most basic geometry concept is that of Vertex Grid Geometry, which just allows access to vertex coordinates. A more advanced concept is Volume Grid Geometry, which defines geometric counterparts for all combinatorial grid elements, as shown in the table.

types
typedef coord_type GeomCoord	point in space
typedef segment_type GeomSegment	1D segment corr. to Edge
typedef polygon_type GeomPolygon	2D polygon corr. to Face
typedef polyhedron_type GeomPolyhedron	3D polyhedron corr. to Cell (in 3D Grids)
functionals from combinatorics to geometry
coord_type coord(Vertex) segment_type segment(Edge) polygon_type polygon(Face) polyhedron_type polyhedron(Cell)	mappings of combinatorial entities to geometric entities

There may be a lot more functionality available, depending on the actual geometry. For example, if the combinatorial and the geometric dimension are equal, one may define outward normal in the centers of facets.