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).

But one can also describe points in 1, 2 or more than 3 dimensions. The description of a point is quite similar to the description of a spatial vector, which also can exist in space with dimensions from one to many.

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.

One could argue that in this world it makes no sense to say that a point is in a one or two dimensional space, because we experience space in 3 dimensions, where one or two dimensions exists within this space, thus forcing 1d and 2d points to actually be 3d points. This way one could say that the only real spatial points are 3d points. And one could also argue that by giving more than 3 coordinates one starts to describe features which are not related to space (how would you describe the fourth dimension in spatial terms?) This is really a question about what we mean by space.

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