On the number of views of translates of a cube and related problems

Authors:
Boris Aronov;Robert Schiffenbauer;Micha Sharir
Affiliations:
Department of Computer and Information Science, Polytechnic University, Brooklyn, NY;Department of Computer and Information Science, Polytechnic University, Brooklyn, NY;School of Computer Science, TelAviv University, TelAviv 69978, Israel and Courant Institute of Mathematical Sciences, New York University, New York, NY
Venue:
Computational Geometry: Theory and Applications
Year:
2004

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Abstract

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.