coordinate

A coordinate is a value that locates a point relative to a reference frame. In an n-dimensional space, a point is defined by an ordered n-tuple of coordinates (x1, x2, ..., xn). The coordinate system consists of a set of axes or reference lines; it provides the rules for measuring and comparing coordinates.

Common coordinate systems include Cartesian, polar, cylindrical, spherical, and geographic coordinates. In a Cartesian system, perpendicular axes define positions; 2D points are written as (x, y) and 3D points as (x, y, z). Polar coordinates describe a point by (r, θ), with r ≥ 0 and θ the angle from a reference direction. Cylindrical coordinates add a height: (r, θ, z). Spherical coordinates use (ρ, φ, θ), where φ is the angle from the positive z-axis and θ the azimuth. Geographic coordinates specify location on the Earth via latitude and longitude (and optionally altitude).

Transformations between coordinate systems use specific formulas. For example, Cartesian to polar: r = sqrt(x^2 + y^2), θ = atan2(y, x); polar to Cartesian: x = r cos θ, y = r sin θ. In three dimensions, analogous relations connect Cartesian, cylindrical, and spherical systems.

In mathematics, coordinates enable description and analysis of geometric objects through equations relating coordinates, a field known as coordinate geometry. They are also used to represent vectors by their components in a chosen basis, yielding coordinate vectors. Coordinates underpin many applications across physics, computer graphics, geographic information systems, and engineering.