TY - GEN

T1 - New bounds for lower envelopes in three dimensions, with applications to visibility in terrains

AU - Halperin, Dan

AU - Sharir, Micha

PY - 1993

Y1 - 1993

N2 - We consider the problem of bounding the complexity of the lower envelope of n 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 of n such surface patches is O(n2 · 2c√log n), for some constant c 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 all rays in 3-space that lie above a given polyhedral terrain K with n 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(n3 · 2c√log n) for some constant c; in particular, there are at most that many rays that pass above the terrain and touch it in 4 edges. This bound, combined with the analysis of de Berg et al. [2], gives an upper bound (which is almost tight in the worst case) on the number of topologically-different orthographic views of such a terrain.

AB - We consider the problem of bounding the complexity of the lower envelope of n 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 of n such surface patches is O(n2 · 2c√log n), for some constant c 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 all rays in 3-space that lie above a given polyhedral terrain K with n 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(n3 · 2c√log n) for some constant c; in particular, there are at most that many rays that pass above the terrain and touch it in 4 edges. This bound, combined with the analysis of de Berg et al. [2], gives an upper bound (which is almost tight in the worst case) on the number of topologically-different orthographic views of such a terrain.

UR - http://www.scopus.com/inward/record.url?scp=0027803461&partnerID=8YFLogxK

U2 - 10.1145/160985.160989

DO - 10.1145/160985.160989

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AN - SCOPUS:0027803461

SN - 0897915828

SN - 9780897915823

T3 - Proceedings of the 9th Annual Symposium on Computational Geometry

SP - 11

EP - 18

BT - Proceedings of the 9th Annual Symposium on Computational Geometry

PB - Association for Computing Machinery (ACM)

T2 - Proceedings of the 9th Annual Symposium on Computational Geometry

Y2 - 19 May 1993 through 21 May 1993

ER -