Ray shooting and stone throwing with near-linear storage

  • Authors:

  • Micha Sharir;Hayim Shaul

  • Affiliations:

  • School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel and Courant Institute of Mathematical Sciences, New York University, New York, NY;Courant Institute of Mathematical Sciences, New York University, New York, NY

  • Venue:
  • Computational Geometry: Theory and Applications
  • Year:
  • 2005

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Abstract

The paper presents two algorithms involving shooting in three dimensions. We first present an algorithm for performing ray shooting amid several special classes of n triangles in three dimensions, including sets of fat triangles, and sets of triangles stabbed by a common line. In all these special cases, our technique requires near-linear pre-processing and storage, and answers a query in O(n2/3+ε) time. This improves the best known result of O(n3/4+ε) query time (with near-linear storage) for general triangles. The second algorithm handles stone-throwing amid arbitrary triangles in 3-space, where the curves along which we shoot are vertical parabolic arcs that are trajectories of stones thrown under gravity. We present an algorithm that answers stone-throwing queries in O(n3/4+ε) time, using near linear storage and preprocessing. As far as we know, this is the first nontrivial solution of this problem. Several extensions of both algorithms are also presented.