The Plane
The Plane is a two dimensional mapped region, which maps
the real eucliden plane into three space. It has no boundaries.
In terms of the Base Classes:
- Base Space Dimension = 2.
- isitInChart((x,y))=TRUE.
- Target Space Dimension = 3.
- Target Space = Euclidean 3-space.
- getPoint((x,y))=O+x*b1/|b1|+y*b2/|b2|.
Mapped regions are derived from Manifolds, and the other member functions that
a Manifold requires will be provided. The Mapped Region will not be discrete,
and by default will not be Differentiable. For the Plane, the information
necessary for Differentiability is:
- TangentSpaceDimension=2.
- getTangentSpace(0) = b1.
- getTangentSpace(1) = b2.
- NormalSpaceDimension=1.
- getNormalSpace = b1xb2
- getMetric = identity matrix.
- getConnection = 0.
- getDistance = Euclidean Distance
- isGeodesicTangentConstant = TRUE.
- GeodesicTangent = (x1-x2)/|x1-x2|.
- projectOntoManifold(no direction given) = (x,y,z)->(((x,y,z)-O).b1,((x,y,z)-O).b2)
- projectOntoManifold(orthogonal direction (lx,ly,lz) given) =
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