Coordinate system

A coordinate system assigns numerical tuples to geometric points relative to a chosen origin, axes, or basis. Different systems can represent the same geometry with different numbers.

Coordinate: A number that specifies one component of a point’s position in a coordinate system. Each coordinate supplies one entry in the numerical description of a point.
Change of coordinates: A rule that converts coordinate descriptions between two systems on the same space. It translates a geometric description into a different choice of axes or parameters.
Cartesian coordinate system: A coordinate system using perpendicular axes and signed distances from their intersection. Its straight axes make rectangular geometry and many algebraic equations direct.
Coordinate-free formulation: A mathematical description stated without choosing particular coordinate values or axes. It expresses the same geometry while avoiding dependence on a chosen coordinate system.
Origin (mathematics): The reference point assigned zero coordinates in a coordinate system. Choosing the origin fixes the reference from which positions are measured.
Coordinate transformation: A mapping that expresses points or geometric quantities in a new coordinate description. Transformations make explicit how numerical descriptions change when coordinates change.
Polar coordinate system: A two-dimensional system locating points by distance from an origin and direction angle. It simplifies problems with rotational symmetry around a point.
Reference frame: A system of spatial and temporal references used to describe positions and events. Coordinate systems encode positions within frames, but a frame also specifies physical reference conditions.
Coordinate axis: A directed reference line used to measure one coordinate component. Axes establish the directions along which coordinates are measured.
Jacobian matrix: A matrix of first derivatives describing how a differentiable mapping changes locally. Its entries govern local effects of changing coordinates, including area and volume scaling.

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