Point (geometry)
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A spatial point is an entity with a location in space but no extent (volume, area or length). In geometry, a point therefore captures the notion of location; no further information is captured. Points are used in the basic language of geometry, physics, vector graphics (both 2d and 3d), and many other fields. In mathematics generally, particularly in topology, any form of space is considered as made up of points as basic elements.
Points in Euclidean geometry
A point in Euclidean geometry has no size, orientation, or any other feature except position. Euclid's axioms or postulates assert in some cases that points exist: for example, they assert that if two lines on a plane are not parallel, there is exactly one point that lies on both of them. Euclid sometimes implicitly assumed facts that did not follow from the axioms (for example about the ordering of points on lines, and occasionally about the existence of points distinct from a finite list of points). Therefore the traditional axiomatization of point was not entirely complete and definitive.
Points in Cartesian geometry
Intuitively one can understand a location in the Cartesian 3D space. This location could be described with three real number coordinates: for instance
- P = (2, 6, 9).
The conceptual difference between these notions is significant, though: a point indicates a location, while a vector indicates a direction and length. If a distinguished point (the origin) is given, one can describe a location by giving the direction and distance from the origin to that point.
Points in topology
In topology, a point is simply an element of the underlying set of a topological space. Similar usage holds for similar structures such as uniform spaces, metric spaces, and so on.
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