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A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane. The set of the ordered pairs of real numbers (the real coordinate plane), equipped with the dot product, is often called the Euclidean plane or standard Euclidean plane, since every Euclidean plane is isomorphic to it.
A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane. In a Cartesian plane, one can define canonical representatives of certain geometric figures, such as the unit circle (with radius equal to the length unit, and center at the origin), the unit square (whose diagonal has endpoints at (0, 0) and (1, 1)), the ...
In a Cartesian coordinate system with coordinates (x, y), a unit square is defined as a square consisting of the points where both x and y lie in a closed unit interval from 0 to 1. That is, a unit square is the Cartesian product I × I , where I denotes the closed unit interval.
Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple. In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates.
A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane. The set R 2 {\displaystyle \mathbb {R} ^{2}} of the ordered pairs of real numbers (the real coordinate plane ), equipped with the dot product , is often called the Euclidean plane or standard Euclidean plane , since every Euclidean plane is isomorphic to it.
Euclid never used numbers to measure length, angle, or area. The Euclidean plane equipped with a chosen Cartesian coordinate system is called a Cartesian plane; a non-Cartesian Euclidean plane equipped with a polar coordinate system would be called a polar plane. Three parallel planes. A plane is a ruled surface.
A point in the plane may be represented in homogeneous coordinates by a triple (x, y, z) where x/z and y/z are the Cartesian coordinates of the point. [10] This introduces an "extra" coordinate since only two are needed to specify a point on the plane, but this system is useful in that it represents any point on the projective plane without the ...
Cartesian coordinates identify points of the Euclidean plane with pairs of real numbers. In mathematics, the real coordinate space or real coordinate n-space, of dimension n, denoted R n or , is the set of all ordered n-tuples of real numbers, that is the set of all sequences of n real numbers, also known as coordinate vectors.