TY - JOUR
T1 - On the number of views of translates of a cube and related problems
AU - Aronov, Boris
AU - Schiffenbauer, Robert
AU - Sharir, Micha
N1 - Funding Information:
Keywords: Visibility; Aspect graphs; Orthographic views; Perspective views; Fat objects; Polyhedral terrains; Arrangements; Envelopes; Combinatorial geometry ✩ Work on this paper by Boris Aronov and Micha Sharir has been supported by a joint grant from the US–Israeli Binational Science Foundation. Work by Boris Aronov has also been supported by NSF Grants CCR-99-72568 and ITR CCR-00-81964. Work by Micha Sharir has also been supported by a grant from the Israel Science Fund (for a Center of Excellence in Geometric Computing), by NSF Grants CCR-97-32101 and CCR-00-98246, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. * Corresponding author. E-mail addresses: [email protected] (B. Aronov), [email protected] (R. Schiffenbauer), [email protected] (M. Sharir).
PY - 2004/2
Y1 - 2004/2
N2 - It is known that a general polyhedral scene of complexity n has at most O(n6) combinatorially different orthographic views and at most O(n9) combinatorially different perspective views, and that these bounds are tight in the worst case. In this paper we show that, for the special case of scenes consisting of a collection of n translates of a cube, these bounds improve to O(n4+ε) and O(n6+ε), for any ε>0, respectively. In addition, we present constructions inducing Ω(n4) combinatorially different orthographic views and Ω(n6) combinatorially different perspective views, thus showing that these bounds are nearly tight in the worst case. Finally, we show how to extend the upper and lower bounds to several classes of related scenes.
AB - It is known that a general polyhedral scene of complexity n has at most O(n6) combinatorially different orthographic views and at most O(n9) combinatorially different perspective views, and that these bounds are tight in the worst case. In this paper we show that, for the special case of scenes consisting of a collection of n translates of a cube, these bounds improve to O(n4+ε) and O(n6+ε), for any ε>0, respectively. In addition, we present constructions inducing Ω(n4) combinatorially different orthographic views and Ω(n6) combinatorially different perspective views, thus showing that these bounds are nearly tight in the worst case. Finally, we show how to extend the upper and lower bounds to several classes of related scenes.
KW - Arrangements
KW - Aspect graphs
KW - Combinatorial geometry
KW - Envelopes
KW - Fat objects
KW - Orthographic views
KW - Perspective views
KW - Polyhedral terrains
KW - Visibility
UR - http://www.scopus.com/inward/record.url?scp=84867994588&partnerID=8YFLogxK
U2 - 10.1016/j.comgeo.2003.10.001
DO - 10.1016/j.comgeo.2003.10.001
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AN - SCOPUS:84867994588
SN - 0925-7721
VL - 27
SP - 179
EP - 192
JO - Computational Geometry: Theory and Applications
JF - Computational Geometry: Theory and Applications
IS - 2
ER -