<p>The classical problem of discrete structure recognition is revisited in this paper. We focus on pieces of naive lines and, more generally, naive arithmetic hyperplanes, and present a new approach to recognising these discrete structures based on the Stern–Brocot tree. The algorithm for pieces of lines in dimension 2 proposes an alternative method to the state of the art, retaining linear complexity and incrementality for the segments. While most of the concepts can be generalised to planes in dimension 3 and hyperplanes in higher dimensions, certain points in the management of the descent in the Stern–Brocot tree merit further study. The proposed algorithm calculates separating chords characterising the membership of planes to cones generated by the branch of the Stern–Brocot tree. This generalisation shows the close link between arithmetic hyperplanes and the generalised Stern–Brocot tree and opens up interesting prospects for recognising pieces of arithmetic hyperplanes. Finally, we propose a geometric interpretation of separating chords and an interpretation of plane probing algorithms in the Stern–Brocot tree, showing both the links and the differences with our approach.</p>