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Abstract

We investigate 3D visibility problems for scenes that consist of $n$ non-intersecting spheres. The viewing point $v$ moves on a flightpath that is part of a ``circle at infinity'' given by a plane $P$ and a range of angles $\{\alpha(t)|t\in [0:1]\}\subset [0:2\pi]$. At ``time'' $t$, the lines of sight are parallel to the ray $r(t)$ in the plane $P$, which starts in the origin of $P$ and represents the angle $\alpha(t)$ (orthographic views of the scene). We describe algorithms that compute the visibility graph at the start of the flight, all time parameters $t$ at which the topology of the scene changes, and the corresponding topology changes. We present an algorithm with running time $O((n+k+p)\log n)$, where $n$ is the number of spheres in the scene; $p$ is the number of transparent topology changes (the number of different scene topologies visible along the flightpath, assuming that all spheres are transparent); and $k$ denotes the number of vertices (conflicts) which are in the (transparent) visibility graph at the start and do not disappear during the flight.