Deducing a mystery piece

A hidden piece has an unknown move-set. You are shown the squares it can reach from a given square and must deduce its rule. This is reasoning from data to a model: collect evidence, propose a rule, then test the rule against new evidence.

Describe a move-set as a set of vectors, the allowed steps $(\Delta \text{file}, \Delta \text{rank})$. A rook is the vectors along the axes of any length; a knight is the eight vectors whose components are $1$ and $2$ in some order and sign. Once you write the reached squares as offsets from the start, the pattern of offsets is the rule, and you can predict the next legal squares.