koordinaatioluku

Koordinaatioluku is a numeric value that encodes the coordinates of a point in a coordinate system into a single scalar. In mathematics and computer science, it denotes the result of a bijective mapping from a coordinate tuple to one number, enabling compact storage, indexing, and hashing of spatial data.

For discrete coordinates in d dimensions, several standard constructions exist. One common method in two dimensions is the Cantor pairing function: pi(x,y) = (x+y)(x+y+1)/2 + y, defined for nonnegative integers and extendable to all integers by shifting. This gives a unique koordinaatioluku for each point in the grid. Another widely used approach is the Morton code (Z-order), which interleaves the binary digits of the coordinates to produce a single code; it is especially popular in spatial indexing and image processing. Higher dimensions can be handled by analogous generalizations of these schemes.

In practice, real-valued coordinates are typically not encoded directly. Instead, coordinates are first quantized to a grid with a chosen resolution, and then the resulting integer coordinates are encoded using a Cantor-type or Morton-type koordinaatioluku.

Applications include database and GIS indexing, computer graphics, spatial hashing, and efficient spatial joins. The primary properties are locality and computability: the code is deterministic, and nearby points often map to nearby codes under certain encodings, though this is not guaranteed for all schemes.

Limitations include the need to fix a domain and resolution, as well as potential inefficiency for very large or high-dimensional datasets. Decode procedures exist to recover coordinates from the code, given the same encoding. See also Cantor pairing function, Morton order, and space-filling curves.