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Local coordinates are measurement indices into a local coordinate system or a local coordinate space. A simple example is using house numbers to locate a house on a street; the street is a local coordinate system within a larger system composed of city townships, states, countries, etc.
Local systems exist for convenience. Continuing the geographic example, one could use latitude and longitude for all terrestrial locations, but unless one has a highly precise GPS device, this is impractical.
Local coordinate spaces are also useful for their ability to model independently transformable aspects of geometrical scene graphs. When modeling a car, for example, it is desirable to describe the center of each wheel with respect to the car's coordinate system, but then specify the shape of each wheel in separate local spaces centered about these points. This way, the information describing each wheel can be simply duplicated four times, and independent transformations (e.g., steering rotation) can be similarly effected. The disadvantage is computational cost: the rendering system must access the higher-level coordinate system of the car and combine it with the space of each wheel in order to draw everything in its proper place.
Local coordinates also afford digital designers a means around the finite limits of numerical representation. The tread marks on a tire, for example, can be described using millimetres by allowing the whole tire to occupy the entire range of numeric precision available. The larger aspects of the car, such as its frame, might be described in centimetres, and the terrain that the car travels on could be specified in meters.
In differential topology, local coordinates on a manifold are defined by means of an atlas of charts. The basic idea behind coordinate charts is that each small patch of a manifold can be endowed with a set of local coordinates. These are collected together into an atlas, and stitched together in such a way that they are self-consistent on the manifold.