koordinatrum

Koordinatrum is a term used in theoretical geometry and computer science to denote a unifying framework for describing coordinates and transformations across multiple spaces. In its general form, a koordinatrum consists of a coordinate set, a reference space, and a mapping that assigns to each object a coordinate vector relative to a chosen frame. It emphasizes the idea that coordinates are not intrinsic properties of objects but manifestations relative to a frame of reference and a transform group.

Etymology and terminology: The word koordinatrum is formed from the root related to coordinates, with a Latin-like suffix to denote a structured system. In literature, coordinatrum is sometimes used interchangeably with terms like coordinate frame or coordinate space, though its usage stresses the interoperability of different frames via well-defined transformations.

Formal structure: A koordinatrum comprises three components: a set S of objects, a set C of coordinate descriptors, and a projection map coord: S -> C. Additionally, a group G of transformations acts on C, implementing changes of frame. An invariant feature under G is called a coordinatral invariant. In practice, koordinatrum provides rules for composing transforms and for deriving relational coordinates between frames.

Applications: The concept is used conceptually in robotics, computer graphics, and geographic information systems to compare data described in different frames. It supports operations such as frame alignment, coordinate interpolation, and multi-frame aggregation.

History and status: Koordinatrum is a relatively new or theoretical construct rather than a standardized mathematical object. It appears in speculative discussions and pedagogical expositions about coordinate systems and frame transformations, rather than as a widely adopted formalism.