Sparse Arrangements and the Number of Views of Polyhedral Scenes

Mark de Berg, Dan Halperin, Mark Overmars, Marc van Kreveld
1997 International journal of computational geometry and applications  
In this paper we study several instances of the problem of determining the maximum number of topologically distinct two-dimensional images that three-dimensional scenes can induce. To bound this number, we investigate arrangements of curves and of surfaces that have a certain sparseness property. Given a collection of n algebraic surface patches of constant maximum degree in 3-space with the property that any vertical line stabs at most k of them, we show that the maximum combinatorial
more » ... y of the entire arrangement that they induce is (n 2 k). We extend this result to collections of hypersurfaces in 4-space and to collections of (d ? 1)-simplices in d-space, for any xed d. We show that this type of arrangements (sparse arrangements) is relevant to the study of the maximum number of topologically di erent views of a polyhedral terrain. Given a polyhedral terrain with n edges and vertices, we derive an upper bound O(n 5 2 c p log n ) on the maximum number of views of the terrain from in nity, where c is some positive constant. Moreover, we show that this bound is almost tight in the worst case, by introducing a lower bound construction inducing (n 5 (n)) distinct views. We also analyze the case of perspective views, point to the potential role of sparse arrangements in obtaining a sharp bound for this case, and present a lower bound construction inducing (n 8 (n)) distinct views.
doi:10.1142/s0218195997000120 fatcat:65v2uco2z5clvpjmglydky2hde