localmap

Local map is a term used in mathematics and computer science to describe a function or representation that describes structure only in a vicinity of a point or region, rather than globally. A local map is defined on a neighborhood of a point in its domain and is used to study local properties; a map may not be defined or may behave differently away from that neighborhood.

In topology and differential geometry, the most standard local notion is that of a local homeomorphism or local diffeomorphism: for each point, there exists a neighborhood on which the map restricts to a homeomorphism or diffeomorphism onto its image. In the theory of manifolds, local charts are homeomorphisms from a neighborhood of a point to Euclidean space; an atlas patches these local maps to describe the global structure.

Exponential maps at a point are important examples of local maps: they send a neighborhood of 0 in the tangent space TpM to the manifold M, often being a local diffeomorphism near 0.

In geographic information systems and navigation, a local map may refer to a map view focused on a specific area, such as a city or neighborhood, as opposed to a global map of an entire country or planet. Such local maps support zooming, panning, and detail appropriate to the region.

The use of local maps emphasizes local validity and enables analysis that does not assume global consistency, but may require combining several local maps to cover a larger space or dataset.