gridthough

Gridthrough is a term used in digital geometry and grid-based algorithms to describe a characteristic path or traversal that moves through a lattice in a continuous, grid-respecting manner. In practice, gridthrough paths are those that align with the underlying square grid, progressing from cell to adjacent cell in a way that preserves a coherent through-line across the grid.

Origin and usage: The word is a portmanteau of grid and through and has appeared in scholarly articles and technical glossaries since the early 2010s to describe grid-aware discretizations of curves and routes. It is not a standard mathematical notion with a single formal definition, but a family of related ideas used across subfields such as digital topology, grid routing, and image processing.

Formal aspects: In two-dimensional square grids, a gridthrough path typically is defined by two conditions: (1) the path intersects the interior of a contiguous sequence of unit squares without jumping across nonadjacent cells, and (2) the sequence of cell intersections induces a connected trace that can be represented on the grid as a polyline with steps along grid lines. Variants may require monotone projections or minimal turn counts.

Applications: Gridthrough concepts are used in discrete curve extraction, rasterization, mesh generation on grids, and routing in network-on-chip or mesh networks. They provide a framework for converting continuous curves into grid-based representations with predictable connectivity, and for designing routing schemes that minimize backtracking.

See also: Digital topology, lattice path, grid graph, grid routing, NoC routing.