Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (−1.5, −2.5) in blue, and the origin n-dimensional Euclidean space for any dimension (x − a) + (y − b) = r where a and b are the coordinates of the center y satisfy the equation 2</sup> + ''y''<sup>2</sup> = 4"}},"i":0}}]}'/>x + y = 4; the area, the perimeter and the tangent line at any point can be computed from this equation by using integrals and derivatives, in a way that can be applied to any curve.
Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing.
+ or x↦ax+b) taking a specific point's coordinate in one system to its coordinate in the other system. Choosing a coordinate system for each of two different lines establishes an affine map from one line to the other taking each point on one line to the point on the other line with the same coordinate.Two dimensions
editIn mathematics, physics, and engineering, the first axis is usually defined or depicted as horizontal and oriented to the right, and the second axis is vertical and oriented upwards. (However, in some computer graphics contexts, the ordinate axis may be oriented downwards.) The origin is often labeled O, and the two coordinates are often denoted by the letters X and Y, or x and y. The axes may then be referred to as the X-axis and Y-axis. The choices of letters come from the original convention, which is to use the latter part of the alphabet to indicate unknown values. The first part of the alphabet was used to designate known values.
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 (x, y), then its distances from the X-axis and from the Y-axis are | · | denotes the absolute value of a number.
Three dimensions
editA Cartesian coordinate system for a three-dimensional space consists of an ordered triplet of lines (the axes) that go through a common point (the origin), and are pair-wise perpendicular; an orientation for each axis; and a single unit of length for all three axes. As in the two-dimensional case, each axis becomes a number line. For any point P of space, one considers a plane through P perpendicular to each coordinate axis, and interprets the point where that plane cuts the axis as a number. The Cartesian coordinates of P are those three numbers, in the chosen order. The reverse construction determines the point P given its three coordinates.
Alternatively, each coordinate of a point P can be taken as the distance from P to the plane defined by the other two axes, with the sign determined by the orientation of the corresponding axis.
Each pair of axes defines a coordinate plane. These planes divide space into eight octants. The octants are:
(t, u + v, π/2). Thus, the origin has coordinates (0, 1, 0), and (x, y, z). The z-axis is vertical and the x-axis is highlighted in green. Thus, the red plane shows the points with y = −1. The three surfaces intersect at the point P (shown as a black sphere) with the Cartesian coordinates ℝ is the set of all real numbers. In the same way, the points in any Euclidean space of dimension n be identified with the tuples (lists) of n real numbers; that is, with the Cartesian product
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