We study the relative homology group of an affine hyperplane arrangement and its Poincaré dual, the cohomology at finite distance of the complement. We give an Orlik--Solomon-type description of the latter, and identify it with the vector space of logarithmic forms having vanishing residues at infinity. To this end, we introduce a partial version of wonderful compactifications, which could be relevant in other contexts where blow-ups only occur at infinity. Finally, we show that the cohomology at finite distance coincides with the vector space of canonical forms in the sense of positive geometry.