Mapped Regions

Regions are pieces cut from a Euclidean (i.e. `flat') space. It is common to use regions that have been "stretched" by a mapping. Sometimes this is done to change the way a region is parameterized,

In other cases the stretching may be to accomodate differently shapped regions,

or to embed a lower dimensional region in a higher dimensional space, like creating a surface in three space,

or a curve in the plane

Or a plane in 3-space

To describe a mapped region it is necessary to specify

1. The dimension of space which contains the region to be mapped (the base space),
2. What points are in the region being mapped,
3. The dimension of the space to which it is mapped (the target space),
4. How to compute the mapping.

• getTargetSpaceDimension -- retrieve the dimension of the space in which the mapped region is embedded.

• getTargetSpace -- retrieve the space in which the mapped region is embedded (a Manifold).

• getPoint -- map a point from the base space (must be inChart), onto the region.

By not assuming 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).

Examples of Mapped Regions

• The Circle in the plane,
• The Plane in Three Space,

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