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<p>Two triangles in the plane are called <i>homothetic</i> if one can be obtained from the other using scaling and translation operations. In this paper, we investigate an enclosure problem involving homothetic triangles. Specifically, the problem is to preprocess a given set of <i>n</i> homothetic triangles so that the triangles enclosing (or containing) a query object can be found quickly. The query objects we consider include points, line segments, trapezoids, and ellipses. We propose an efficient solution to the problem using the 3-D dominance searching. Our results can also be extended to higher dimensions. Moreover, we also present a negative result that suggests that the enclosure problem involving (arbitrarily) axis-parallel right-angled triangles may not be solvable using 4-D dominance searching.</p>

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Dominance for enclosure problems

  • 期刊论文

  • 发表日期：

    2025年11月8日

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  • Waseem Akram,
  • Sanjeev Saxena

摘要

Two triangles in the plane are called homothetic if one can be obtained from the other using scaling and translation operations. In this paper, we investigate an enclosure problem involving homothetic triangles. Specifically, the problem is to preprocess a given set of n homothetic triangles so that the triangles enclosing (or containing) a query object can be found quickly. The query objects we consider include points, line segments, trapezoids, and ellipses. We propose an efficient solution to the problem using the 3-D dominance searching. Our results can also be extended to higher dimensions. Moreover, we also present a negative result that suggests that the enclosure problem involving (arbitrarily) axis-parallel right-angled triangles may not be solvable using 4-D dominance searching.