TY - GEN

T1 - Efficient hidden surface removal for objects with small union size

AU - Katz, Matthew J.

AU - Overmars, Mark H.

AU - Sharir, Micha

N1 - Funding Information:
*Work by Mark Over-mars has been partially supported by the ESPRIT Basic Research Action No. 3075 (project ALCOM) and by the Dutch Organisation for Scientific Research. Work on this paper by Matthew Katz and Micha Sharir has been supported by a Grant from the G .I.F., the German-Israefi Foundation for Scientific Research and Development. Work by Micha Sharir has also been supported by Office of Naval Research Grant Nooo14-90-J-1284, by National Science Foundation Grant CCR-8%01484, and by grants from the U .S .-Israeli Binationaf Science Foundation, and the Fund for Basic Research administered by the Israeli Academy of Sciences.
Publisher Copyright:
© 1991 ACM.

PY - 1991/6/1

Y1 - 1991/6/1

N2 - Let 5 be a set of n non-intersecting objects in space for which we want to determine the portions visible from some viewing point. We assume that the objects are ordered by depth from the viewing point (e.g., they are all horizontal and are viewed from infinity from above). In this paper we give two algorithms that compute the visible portions in time O((U(n) + k)log2n), where U(n') is a super-additive bound on the maximal complexity of the union of (the projections on a viewing plane of) any n' objects from the family under consideration, and k is the complexity of the resulting visibility map. Both algorithms use O(U(n) log n) working storage. The algorithms are useful when the objects are "fat" in the sense that the union of the projection of any subset of them has small (i.e., subquadratic) complexity. We present three applications of these general techniques: (i) For disks (or balls in space) we have U(n) - O(n), thus the visibility map can be computed in time O((n + k) log2 n). (ii) For 'fat' triangles (where each internal angle is at least some fixed 6 degrees) we have U(n) = O(n log log n) and the algorithms run in time O((n log log n + k)log2 n). (iii) The methods also apply to computing the visibility map for a polyhedral terrain viewed from a fixed point, and yield O((nα(n) + k) log n) algorithms.

AB - Let 5 be a set of n non-intersecting objects in space for which we want to determine the portions visible from some viewing point. We assume that the objects are ordered by depth from the viewing point (e.g., they are all horizontal and are viewed from infinity from above). In this paper we give two algorithms that compute the visible portions in time O((U(n) + k)log2n), where U(n') is a super-additive bound on the maximal complexity of the union of (the projections on a viewing plane of) any n' objects from the family under consideration, and k is the complexity of the resulting visibility map. Both algorithms use O(U(n) log n) working storage. The algorithms are useful when the objects are "fat" in the sense that the union of the projection of any subset of them has small (i.e., subquadratic) complexity. We present three applications of these general techniques: (i) For disks (or balls in space) we have U(n) - O(n), thus the visibility map can be computed in time O((n + k) log2 n). (ii) For 'fat' triangles (where each internal angle is at least some fixed 6 degrees) we have U(n) = O(n log log n) and the algorithms run in time O((n log log n + k)log2 n). (iii) The methods also apply to computing the visibility map for a polyhedral terrain viewed from a fixed point, and yield O((nα(n) + k) log n) algorithms.

UR - http://www.scopus.com/inward/record.url?scp=0142101507&partnerID=8YFLogxK

U2 - 10.1145/109648.109652

DO - 10.1145/109648.109652

M3 - Conference contribution

AN - SCOPUS:0142101507

SN - 0897914260

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 31

EP - 40

BT - Proceedings of the Annual Symposium on Computational Geometry

PB - Association for Computing Machinery

T2 - 7th Annual Symposium on Computational Geometry, SCG 1991

Y2 - 10 June 1991 through 12 June 1991

ER -