Abstract
We consider the problem of bounding the complexity of the lower envelope ofn surface patches in 3-space, all algebraic of constant maximum degree, and bounded by algebraic arcs of constant maximum degree, with the additional property that the interiors of any triple of these surfaces intersect in at most two points. We show that the number of vertices on the lower envelope ofn such surface patches is\(O(n^2 \cdot 2^{c\sqrt {\log n} } )\), for some constantc depending on the shape and degree of the surface patches. We apply this result to obtain an upper bound on the combinatorial complexity of the “lower envelope” of the space of allrays in 3-space that lie above a given polyhedral terrainK withn edges. This envelope consists of all rays that touch the terrain (but otherwise lie above it). We show that the combinatorial complexity of this ray-envelope is\(O(n^3 \cdot 2^{c\sqrt {\log n} } )\) for some constantc; in particular, there are at most that many rays that pass above the terrain and touch it in four edges. This bound, combined with the analysis of de Berget al. [4], gives an upper bound (which is almost tight in the worst case) on the number of topologically different orthographic views of such a terrain.
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Work on this paper by the first author has been supported by a Rothschild Postdoctoral Fellowship. Work on this paper by the second author has been supported by NSF Grant CCR-91-22103, and by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F., the German-Israeli Foundation for Scientific Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences.
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Halperin, D., Sharir, M. New bounds for lower envelopes in three dimensions, with applications to visibility in terrains. Discrete Comput Geom 12, 313–326 (1994). https://doi.org/10.1007/BF02574383
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DOI: https://doi.org/10.1007/BF02574383