coordinateframe

A coordinate frame, or reference frame, is a system used to assign numerical coordinates to points in space. It consists of an origin point and a set of axes that define direction and scale. The orientation of the axes gives the frame its handedness, commonly right-handed in 3D, and the units determine the scale of coordinates. In a Cartesian frame, points are described by orthogonal unit vectors along three axes, resulting in coordinates (x, y, z). More general frames may use non-orthogonal or curvilinear bases, such as cylindrical or spherical coordinate systems, or arbitrary affine bases defined by a basis matrix.

Coordinates of a point are expressed relative to the chosen frame. Changing from one frame to another requires a transformation that typically combines rotation and translation. In 3D, a rigid transformation uses a rotation matrix R and a translation vector t, mapping coordinates x in one frame to x' in another via x' = R x + t. Homogeneous coordinates and 4x4 transformation matrices are widely used in fields such as computer graphics and robotics to represent both rotation and translation compactly and to enable chaining of multiple frame changes.

Frames can be global (world) frames or local (object, sensor, or body) frames. A collection of frames related by transformations forms a frame graph, with the relative pose between frames described by transformation matrices. Applications span robotics, computer graphics, navigation, geodesy, and physics, where the choice of frame influences interpretation and measurement of positions and orientations.