integercoordinate

An integercoordinate is a point in Euclidean space whose coordinates are all integers. In two dimensions, this means points of the form (x, y) with x and y integers; in three dimensions, (x, y, z) with x, y, z integers, and similarly in higher dimensions. The collection of all integercoordinates in R^n is denoted Z^n and forms a regular lattice with unit spacing along each axis.

This lattice structure makes integercoordinates fundamental in discrete geometry and number theory. Lattice points are closed under addition and subtraction, and their arrangement underpins problems involving area, tilings, and visibility on a grid. Distances between integercoordinates are computed by the Euclidean distance formula, but the resulting distances are not guaranteed to be integers. The geometry of lattice points often interacts with concepts such as convexity, lattice polygons, and counting methods like Pick’s theorem in two dimensions.

Integercoordinates have broad practical and theoretical uses. In computer graphics and digital imaging, grid-based coordinates align with pixel and voxel representations. They are central to grid-based pathfinding, robotics, and geographic information systems where space is discretized. In mathematics, integercoordinates appear in Diophantine equations, lattice point counting, and optimization problems such as integer programming and lattice-based cryptography. The study of integercoordinates thus bridges geometry, combinatorics, and number theory, providing a simple yet rich model of discrete space.