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A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two distinct lines in a projective plane intersect at exactly one point. Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic.
Later, the plane was described in a so-called Cartesian coordinate system, a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length.
In geometry, a Cartesian coordinate system (UK: / k ɑːr ˈ t iː zj ə n /, US: / k ɑːr ˈ t iː ʒ ə n /) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, called coordinate lines ...
Alternatively, a line can be described as the intersection of two planes. Let L be a line contained in distinct planes a and b with homogeneous coefficients (a 0 : a 1 : a 2 : a 3) and (b 0 : b 1 : b 2 : b 3), respectively. (The first plane equation is =, for example.)
A line m and a plane q in three-dimensional space, the line not lying in that plane, are parallel if and only if they do not intersect. Equivalently, they are parallel if and only if the distance from a point P on line m to the nearest point in plane q is independent of the location of P on line m.
The most basic example is the flat Euclidean plane, an idealization of a flat surface in physical space such as a sheet of paper or a chalkboard. On the Euclidean plane, any two points can be joined by a unique straight line along which the distance can be measured.
In analytic geometry, the intersection of a line and a plane in three-dimensional space can be the empty set, a point, or a line. It is the entire line if that line is embedded in the plane, and is the empty set if the line is parallel to the plane but outside it. Otherwise, the line cuts through the plane at a single point.
The flats in a plane (two-dimensional space) are points, lines, and the plane itself; the flats in three-dimensional space are points, lines, planes, and the space itself. The definition of flat excludes non-straight curves and non-planar surfaces, which are subspaces having different notions of distance: arc length and geodesic length ...