Plane arrangements are a useful tool for surface and volume modelling. However, their main drawback is poor scalability. We introduce two key novelties that enable the construction of plane arrangements for complex objects and entire scenes: an ordering scheme for the plane insertion and the direct use of input points during arrangement construction. Both ingredients reduce the number of unwanted splits, resulting in improved scalability of the construction mechanism by up to two orders of magnitude compared to existing algorithms. We further introduce a remeshing and simplification technique that allows us to extract low-polygon surface meshes and lightweight convex decompositions of volumes from the arrangement. We show that our approach leads to state-of-the-art results for the aforementioned tasks by comparing it to learning-based and traditional approaches on various different datasets.