Manifolds

Not all geometrical objects can be represented as mapped regions with non-singular mappings. Mapped regions may have singular maps, but we need to provide a way to describe geometrical regions using only non-singular maps. The prototypical example of a region which can't 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 are a collection of mapped regions, (called charts, and the collection is called an Atlas). To be what most mathematicians would call a Manifold, there must be some overlap between the regions after they have been mapped. There must also be a way of mapping between regions which overlap. The analogy is to an atlas of the world, which has a number of charts (maps), each of which describes a small piece of the world. A road, like Route 129, may span many maps, and when you drive your car off one map the road atlas must provide a pointer to the next map which shows the road.

Also, in order for you to get your bearings on the new map, the new map has to show the area around the point you arrive at from the old map (i.e. the maps must overlap).

The member functions which Manifolds provide (in addition to those provided by mapped regions) are

• getNumberOfChartsInAtlas -- query the number of charts.
• mapBetweenCharts -- map a point from one chart to another (the point must be in both charts).
• getChartsThatOverlap -- return a list of the charts that overlap a chart.

The member functions for the sphere manifold are given below.

Examples of General Manifolds

• The Surface of the Sphere,

[Research home page]

[ IBM home page | Order | Privacy | ContactIBM | Legal ]