Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length.

History and Etymology

The system is named after the French mathematician and philosopher René Descartes (1596–1650), who published the idea in his 1637 work La Géométrie. It was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. The development of the Cartesian coordinate system played a fundamental role in the development of analytic geometry by providing a bridge between Euclidean geometry and algebra. The adjective Cartesian refers to the Latin form of Descartes' name, Cartesius.

Description

Each reference line is called a coordinate axis or just axis (plural axes) of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes.

Two-Dimensional System

In the two-dimensional plane, the Cartesian coordinate system is defined by an ordered pair of perpendicular lines (axes), a single unit of length for both axes, and an orientation for each axis. The point where the axes meet is taken as the origin for both, thus turning each axis into a number line. Any point P in the plane is identified by its x-coordinate (or abscissa), which is the signed distance from the y-axis to the point along a line parallel to the x-axis, and its y-coordinate (or ordinate), which is the signed distance from the x-axis to the point along a line parallel to the y-axis. The coordinates are written as an ordered pair (x, y). The choice of letters is arbitrary, but x and y are the most common. The axes divide the plane into four infinite regions, called quadrants, each bounded by two half-axes. The quadrants are typically numbered I, II, III, and IV, starting from the upper right (where both x and y are positive) and proceeding counterclockwise.

Three-Dimensional System

A three-dimensional Cartesian coordinate system is formed by a point called the origin and a basis consisting of three mutually perpendicular vectors. These vectors define the three coordinate axes: the x-axis, y-axis, and z-axis. They are usually oriented so that they follow the right-hand rule, forming a right-handed system. The coordinates of a point P are the signed distances from P to the three planes defined by the axes (the yz-plane, xz-plane, and xy-plane). They are written as an ordered triple (x, y, z). The xy-plane, xz-plane, and yz-plane divide space into eight octants, analogous to the quadrants of a plane. The orientation and labeling of the axes is a matter of convention, with common choices including the right-handed system where the z-axis is the vertical axis.

Higher Dimensions

A Cartesian coordinate system for an n-dimensional space is defined analogously. It consists of n mutually perpendicular coordinate axes, which meet at a common origin O. Each axis is a copy of the real number line. The coordinates of a point P are the n-tuple (x₁, x₂, ..., xₙ) of signed distances from the hyperplanes perpendicular to the respective axes. This system establishes a one-to-one correspondence between points in the n-dimensional Euclidean space and n-tuples of real numbers (the Cartesian product of n copies of ℝ). This identification allows the tools of algebra and calculus to be applied to geometric problems.

Applications

The Cartesian coordinate system is the foundation of analytic geometry and provides a geometric interpretation for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, and multivariable calculus. It is essential in physics and engineering for modeling physical space and describing the positions, velocities, and trajectories of objects. In computer graphics and geographic information systems (GIS), Cartesian coordinates (often in two or three dimensions) are fundamental for representing shapes, images, and maps. The system's simplicity and power lie in its ability to represent geometric figures by algebraic equations and inequalities, enabling the solving of geometric problems through algebraic manipulation.