Leapers: generalising the knight

Generalise the knight to an $(m, n)$-leaper, which jumps $m$ squares one way and $n$ the other. The knight is the $(1, 2)$-leaper. Its reachable squares and colour behaviour depend on $m$ and $n$: if $m + n$ is odd the leaper changes colour every jump like the knight, and if $m + n$ is even it stays on one colour, limiting where it can go.

Which leapers can reach every square is a question of parity and common factors: a leaper whose colour is fixed, or whose steps share a common divisor, cannot cover the whole board. Designing a leaper and mapping its reach ties together vectors, parity, and reachability.