Cartesian coordinate system

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 geometry, a Cartesian coordinate system (UK: /kɑːrˈtiːzjən/, US: /kɑːrˈtiːʒən/) 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, coordinate axes or just axes (plural of axis) of the system. The point where they meet is called the origin and has $(0, 0)$ as coordinates.

Similarly, the position of any point in three-dimensional space can be specified by three Cartesian coordinates, which are the signed distances from the point to three mutually perpendicular planes. More generally, $n$ Cartesian coordinates specify the point in an $n$ -dimensional Euclidean space for any dimension $n$ . These coordinates are the signed distances from the point to $n$ mutually perpendicular fixed hyperplanes.

Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is (x − a)² + (y − b)² = r² where a and b are the coordinates of the center (a, b) and r is the radius.

Cartesian coordinates are named for René Descartes, whose invention of them in the 17th century revolutionized mathematics by allowing the expression of problems of geometry in terms of algebra and calculus. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by equations involving the coordinates of points of the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates $x$ and $y$ satisfy the equation $x 2 + y 2 = 4$ ; 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.