Index:
A Region is a piece of some Euclidean Space. Some examples of regions are
A Region must provide
It is sometimes convenient to use regions which have been "stretched" by a mapping. Sometimes this is done to change the way that a region is parameterized. In other cases the stretching may be used to embed a region in a higher dimensional space, for example a surface in three space, a curve in the plane, or a plane in 3-space.
Some examples of Mapped Regions are
A Mapped region must provide the same member functions as a Region, but in addition must provide
A Region is a Mapped Region with the target space a Eulcidean N-space of the same dimension as the base space, and with the identity mapping.
Note that we do not assume that the target space is R^n.
This allows objects like a curve on a surface in R^3 to be
expressed.
The curve might define points in R^3 (the target
space of the surface), but information is lost, and the
resulting points on the curve may not lie exactly on the
surface. If instead the curve defines points in R^2 (the
base space of the surface), then the points the curve
produceshave to lie on the surface. This is done simply
by defining the target space of the curve (mapped region)
to be the surface itself, not R^3. The getPoint
member function would then return a point in the base space
of the surface (R^2).
For a detailed description of Mapped Regions, click here.
Not all geometrical objects can be represented as mapped regions with non-singular mappings. Mapped regions may have singular maps, but there also must be a way to describe geometrical regions using only non-singular maps. The prototypical example of a region which cannot be represented as a "non-singular" mapped region is the surface of the sphere { x in R^3 | ||x||=1}. It requires at least two "mapped regions" to represent the surface of the sphere.
Manifolds consist of a set of Mapped Regions with the same base space and target space. When Mapped Regions are associated with a Manifold they are called charts, and the list of charts is called an atlas. The analogy is to navigational maps, where charts of sections of the coastline (for example) are bound together into an atlas which covers a particular region. Note: Manifolds do not have to be implemented as a set of Mapped Regions.
Manifolds must provide member functions for determining how many charts are in the atlas, member functions corresponding to the member functions of each chart, and member functions for moving between charts. These are
A Mapped Region is a Manifold with one chart.
For a detailed description of Manifolds, click here.
We have now encountered the main features of General Manifolds, and two special types of Manifolds: Regions and Mapped Regions. These are special in that much of the information that is present in the General Manifold takes on known values. For example, for a Region the embedding space is always the same as the base space, and the single chart is the identity mapping.
There are other special types of Manifolds which add properties or information to the General Manifold. NAO supports three of these additional properties, and allows Manifolds with any combination of the three. They are
There are three member functions which determine which of these properties is present:
Many Manifolds, like Euclidean n-space, the sphere, and tori, are either infinite in extent, or are periodic. There are also many "geometric" objects, like polygonal and polyhedral regions, and disks(balls), which have boundaries. NAO Manifolds support this through the "Open/Not-Open" attribute.
A boundary is a lower dimensional Manifold which divides the interior of the Manifold from the exterior (the base space of the boundary is of one dimension less, and the target space is the same). Manifolds may include thier boundaries, or not. (i.e. there is the open unit interval, (0,1) and unit intervals with one or more of the endpoints [0,1)). NAO Manifolds which are not-Open have a list of Boundaries, and provide member functions to determine how many boundaries are present, and to extract the boundaries. For a detailed description of Manifolds with boundaries, click here.
Points in a Euclidean space are written as a linear combination of basis vectors. These are almost always the coordinate directions. We have used this notation here to describe points in R^3 as (x,y,z). For flat spaces this works well, but for mapped regions there are two coordinate systems available, one in the base space (which becomes a basis for the tangent space), and one in the target space. Working in the target space is straightforward, but often not convenient. The information which a Differentiable Manifold provides is necessary to write differential equations on manifolds in terms of derivatives in the base space.
For a detailed description of Differentiable Manifolds, click here.
A Manifold is a way of describing a set of points in the target space. A grid, or mesh, represents a continuous Manifold with a set of points, or a set of polygonal regions. We call these meshes Discrete Manifolds. They are essentially cellular complexes, which are the most abstract type of mesh. They provide an interface to either regular grids, or finite element grids, and can be any dimension.
For a detailed description of Discrete Manifolds, click here.
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