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Manifolds with Boundaries

As a way of understanding what a boundary is, consider regions which are intervals on the real line. These can be represented easily in terms of regions. However, there is one thing that intervals have that regions do not (no, Dorothy, not a testimonial). Intervals have end points. If intervals always contained thier endpoints (or never contained endpoints), this notion could be supported by Regions. However, intervals can be open, closed, or one of two flavors of half-open:

This notion carries over to higher dimensions. For example, the region below

has a boundary consisting of two pieces, which are half open intervals on the curve separating points inside the region from those outside the region.

This idea is provided in Region, MappedRegions and Manifolds via the isitOpen member Function:

An Open Manifold is one which includes no points on its boundary. Technically this means that every point on the Manifold has some neighborhood surrounding it, which only contains points which are "inside". A Manifold which is not open has a list of lower dimensional Manifolds which make up its boundary. The number of Boundaries, and the individual boundaries can be accessed using the member functions

The following figure shows how getBoundary works.

For this example, getNumberOfBoundaries() returns 2. Notice that the boundaries which are returned may themselves have boundaries. In some situations it may be preferably to consider this as four boundaries, all open. The extra two boundaries would be the points. This is left to the implementation.

Note that while "boundary" has a meaning in terms of topology, there is no restriction on NAO Manifolds. The usual meaning requires that the boundary of an m-manifold be an m-1-manifold. We impose no such restriction. The Boundaries returned need not be smooth, need not be of one lower dimension, and could just as well be special sets of points where a different boundary condition is applied (that is, it need not even be the boundary). This is left to the implementation, though we would recommend following common usage.

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