Coordinate system

Coordinate systems provide the essential framework through which mathematical, physical and geometric problems may be expressed numerically. By assigning ordered sets of values to points or geometric objects, these systems make it possible to translate abstract spatial relationships into analytical forms. Their development underpins vast areas of mathematics, including analytic geometry, differential geometry and the study of manifolds, while also being central to physics and engineering.
A coordinate system defines a rule by which one or more numbers uniquely determine the position of a point in a given space. These numbers, called coordinates, are typically real numbers, although more abstract numerical structures may be used in advanced contexts. The ordering of the coordinates is significant, and each coordinate often corresponds to a particular axis or direction. The construction of such systems allows geometric problems to be solved algebraically and, conversely, algebraic problems to be visualised geometrically.

Fundamental Concepts and Purpose

A coordinate system consists of a reference framework and a rule for assigning values to points. The reference framework may include a chosen origin, one or more axes, a set of basis vectors, or even a family of curves or surfaces. The purpose of coordinates is to standardise the location of points and geometric objects so that spatial properties may be analysed quantitatively.
Coordinates are used not only for points but also for representing lines, planes, curves and surfaces. Systems for different geometric elements may be equivalent in analytical structure, giving rise to dualistic relationships, particularly in projective geometry. Such equivalence ensures that results derived for one type of object are directly transferable to another according to the principle of duality.
Coordinate systems also permit the comparison of different representations of space through transformations. These transformations express coordinates in one system using the coordinates of another and are vital in mechanics, geometry, relativity and physics.

Common Coordinate Systems

Several standard coordinate systems are widely used due to their practical and theoretical significance.

Number Line

The simplest system is the real number line. An origin is selected on a straight line and each point is expressed by its signed distance from this origin. Every point has a unique coordinate, and every real number corresponds to a unique point. This system underpins much of elementary mathematics.

Cartesian Coordinate System

The Cartesian system is fundamental to analytic geometry. In two dimensions, two perpendicular axes are chosen, and the coordinate of a point is given by its signed distances from these axes. In three dimensions, three mutually orthogonal planes define three coordinate values. This concept can be generalised to n dimensions, yielding an n-tuple of coordinates. Depending on the orientation of the axes, the three-dimensional system may follow the right-hand or left-hand rule. Cartesian coordinates allow algebraic equations to represent geometric objects such as lines, circles, planes and spheres.

Polar Coordinate System

Polar coordinates define points in the plane using a radial distance r and an angular measure θ. A fixed point (the pole) and a reference ray (the polar axis) are chosen. A point is determined by the distance r from the pole and the angle θ formed by the polar axis and the line through the pole and the point. Multiple coordinate pairs can represent the same point, and the pole itself corresponds to r = 0 for any angle.

Cylindrical and Spherical Coordinate Systems

Cylindrical coordinates extend the polar system into three dimensions by adding a height coordinate z. Thus, a point is expressed as (r, θ, z). Spherical coordinates further generalise this approach by using radial distance, polar angle (θ) and azimuthal angle (φ). These systems are particularly useful in physics, where many problems exhibit cylindrical or spherical symmetry, such as those involving gravitational or electric fields.

Homogeneous Coordinates

Homogeneous coordinates are widely used in projective geometry. A point in the plane may be represented by a triple (x : y : z), where the ratios x/z and y/z give its Cartesian coordinates. This representation simplifies the handling of transformations such as projections, which are crucial in graphics, algebraic geometry and modelling perspective.

Other Coordinate Systems

Various specialised coordinate systems are used for particular mathematical or physical applications:

• Curvilinear coordinates, in which coordinate lines or surfaces may be curves rather than straight lines.
• Orthogonal coordinates, a class of curvilinear systems with mutually perpendicular coordinate surfaces.
• Log-polar coordinates, using the logarithm of the distance from the origin and an angular measure.
• Plücker coordinates, used for representing lines in three-dimensional projective space.
• Generalised coordinates, employed in Lagrangian mechanics to describe motion.
• Canonical coordinates, fundamental in Hamiltonian mechanics.
• Barycentric coordinates, used for triangles and ternary plots.
• Trilinear coordinates, also employed for triangle geometry.

Intrinsic descriptions of curves, such as the Whewell equation and the Cesàro equation, provide coordinate-free ways of specifying curvature and arc length but are nonetheless linked conceptually to coordinate methods.

Coordinate Curves, Surfaces and Axes

In any coordinate system, holding all but one coordinate constant and allowing that coordinate to vary traces a coordinate curve. If the curve is straight, it becomes a coordinate line. When the coordinate curves are not straight, the system is termed curvilinear. In Cartesian coordinates, all coordinate curves are lines and the axes are the lines obtained by setting all but one coordinate to zero.
In three dimensions, holding one coordinate constant while the others vary generates a coordinate surface. In spherical coordinates, fixing the radial distance defines a sphere, while in cylindrical coordinates fixing z yields a plane parallel to the base plane. Intersections of coordinate surfaces yield coordinate curves. In higher dimensions, fixing a single coordinate defines a coordinate hypersurface.

Transformations Between Coordinate Systems

The relationship between different coordinate systems is described through coordinate transformations. These transformations provide formulas to convert from one set of coordinates to another. For instance, the transformation from polar to Cartesian coordinates is given by:

• x = r cosθ
• y = r sinθ

More generally, any bijection of a space onto itself induces two corresponding coordinate transformations, depending on whether coordinates are transformed along with the points or held fixed while the points move. Such transformations underpin translation, rotation, scaling and general linear transformations.

Coordinate Maps and Manifolds

In advanced geometry and topology, coordinate systems are formalised through coordinate maps or charts, which assign coordinates to points in an open subset of a space in a one-to-one and continuous manner. A collection of charts forming a covering of a space is known as an atlas, and a space equipped with an atlas is termed a manifold.
The compatibility of overlapping charts determines the additional structure of the manifold. If the change of coordinates is differentiable, the manifold is differentiable; if smooth to all orders, it becomes a smooth manifold. These structures are fundamental in modern differential geometry and theoretical physics.