Some formulas for numbers of line segments and lines in a rectangular grid

TL;DR

This paper derives formulas for counting line segments and lines through grid points in multi-dimensional rectangular grids, generalizing known two-dimensional results and providing recursive formulas for specific cases.

Contribution

It introduces new formulas for counting line segments and lines in multi-dimensional grids, extending classical two-dimensional results and offering recursive formulas for special cases.

Findings

Formulas for line segments connecting q+1 points in n_1 x ... x n_k grids

Formulas for lines passing through at least q points

Recursive formulas for the case k=2, n_1=n_2

Abstract

We present a formula for the number of line segments connecting q+1 points of an n_1 x...x n_k rectangular grid. As corollaries, we obtain formulas for the number of lines through at least q points and, respectively, through exactly q points of the grid. The well-known case k=2 is so generalized. We also present recursive formulas for these numbers assuming k=2, n_1=n_2. The well-known case q=2 is so generalized.