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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.
