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Maintaining the visibility map of spheres while moving the viewpoint on a circle at infinity

MPS-Authors
http://pubman.mpdl.mpg.de/cone/persons/resource/persons44909

Lenhof,  Hans-Peter
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons45509

Smid,  Michiel
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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92-102.pdf
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Citation

Lenhof, H.-P., & Smid, M.(1992). Maintaining the visibility map of spheres while moving the viewpoint on a circle at infinity (MPI-I-92-102). Saarbrücken: Max-Planck-Institut für Informatik.


Cite as: http://hdl.handle.net/11858/00-001M-0000-0014-B059-2
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.