Given a residually connected incidence geometry $Γ$ that satisfies two conditions, denoted $(B_1)$ and $(B_2)$, we construct a new geometry $H(Γ)$ with properties similar to those of $Γ$. This new geometry $H(Γ)$ is inspired by a construction of Percsy, Percsy and Leemans [1]. We show how $H(Γ)$ relates to the classical halving operation on polytopes, allowing us to generalize the halving operation to a broader class of geometries, that we call non-degenerate leaf hypertopes. Finally, we apply this generalization to cubic toroids in order to generate new examples of regular hypertopes.