  $\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}}
\newcommand{\SetX}{\mathsf{X}} \newcommand{\eps}{\varepsilon}
\newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}}
\newcommand{\IntRange}[1]{[ #1 ]} \newcommand{\Space}{\ovebarline{\mathsf{m}}}
\newcommand{\pth}[2][\!]{#1\left({#2}\right)} \newcommand{\Arr}{{\cal A}}$
  Let $H$ be a set of $n$ planes in three dimensions, and let $r \leq n$ be a
parameter. We give a simple alternative proof of the existence of a
$(1/r)$-cutting of the first $n/r$ levels of $\Arr(H)$, which consists of
$O(r)$ semi-unbounded vertical triangular prisms. The same construction yields
an approximation of the $(n/r)$-level by a terrain consisting of $O(r/\eps^3)$
triangular faces, which lies entirely between the levels $(1\pm\eps)n/r$. The
proof does not use sampling, and exploits techniques based on planar separators
and various structural properties of levels in three-dimensional arrangements
and of planar maps. The proof is constructive, and leads to a simple randomized
algorithm, with expected near-linear running time. An application of this
technique allows us to mimic Matousek's construction of cuttings in the plane,
to obtain a similar construction of "layered" $(1/r)$-cutting of the entire
arrangement $\Arr(H)$, of optimal size $O(r^3)$. Another application is a
simplified optimal approximate range counting algorithm in three dimensions,
competing with that of Afshani and Chan.
