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In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the ...
In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted or . It is a geometric space in which two real numbers are required to determine the position of each point . It is an affine space , which includes in particular the concept of parallel lines .
A plane segment or planar region (or simply "plane", in lay use) is a planar surface region; it is analogous to a line segment. A bivector is an oriented plane segment, analogous to directed line segments. [a] A face is a plane segment bounding a solid object. [1] A slab is a region bounded by two parallel planes.
Mathematical spaces are often defined or represented using numbers rather than geometric axioms. One of the most fundamental two-dimensional spaces is the real coordinate space , denoted R 2 , {\displaystyle \mathbb {R} ^{2},} consisting of pairs of real-number coordinates.
Euclidean space was introduced by ancient Greeks as an abstraction of our physical space. Their great innovation, appearing in Euclid's Elements was to build and prove all geometry by starting from a few very basic properties, which are abstracted from the physical world, and cannot be mathematically proved because of the lack of more basic tools.
Then the coordinate planes can be referred to as the xy-plane, yz-plane, and xz-plane. In mathematics, physics, and engineering contexts, the first two axes are often defined or depicted as horizontal, with the third axis pointing up. In that case the third coordinate may be called height or altitude.
In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space. [1] If the space is two-dimensional, then a half-space is called a half-plane (open or closed). [2] [3] A half-space in a one-dimensional space is called a half-line [4] or ray.
In geometry, a hyperplane of an n-dimensional space V is a subspace of dimension n − 1, or equivalently, of codimension 1 in V.The space V may be a Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly since the definition of subspace differs in these settings; in all cases however, any hyperplane can ...