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Dominance for Containment Problems

Waseem Akram, Sanjeev Saxena

TL;DR

This work considers the containment problem where input objects are homothetic triangles and the query objects considered are line segments, circles, and trapezoids with bases parallel to either axis and shows that this problem can be solved using the 3-d query dominance problem.

Abstract

In a containment problem, the goal is to preprocess a set of geometric objects so that, given a geometric query object, we can report all the objects containing the query object. We consider the containment problem where input objects are homothetic triangles and the query objects considered are line segments, circles, and trapezoids with bases parallel to either axis. We show that this problem can be solved using the 3-d query dominance problem. The solutions presented can also be extended for higher dimensions.

Dominance for Containment Problems

TL;DR

This work considers the containment problem where input objects are homothetic triangles and the query objects considered are line segments, circles, and trapezoids with bases parallel to either axis and shows that this problem can be solved using the 3-d query dominance problem.

Abstract

In a containment problem, the goal is to preprocess a set of geometric objects so that, given a geometric query object, we can report all the objects containing the query object. We consider the containment problem where input objects are homothetic triangles and the query objects considered are line segments, circles, and trapezoids with bases parallel to either axis. We show that this problem can be solved using the 3-d query dominance problem. The solutions presented can also be extended for higher dimensions.
Paper Structure (14 sections, 5 figures)

This paper contains 14 sections, 5 figures.

Table of Contents

  1. Introduction
  2. The Interval Overlapping Problem
  3. Containment in Isosceles Right-angled Triangles
  4. Line Segments
  5. Orthogonal Segments
  6. Segments with Positive Slope
  7. Segments with Negative Slope
  8. Axes-parallel Rectangle
  9. Ellipse
  10. Trapezoids
  11. Arbitrary Right-angled Triangles
  12. Arbitrary Homothetic Triangles
  13. Extension for Higher Dimensions
  14. Conclusion

Figures (5)

  • Figure 1: Triangle $T_i$ containing horizontal segment $PQ$.
  • Figure 2: Triangle $T_i$ containing segment $PQ$ with a positive slope.
  • Figure 3: Triangle $T_i$ containing segment $PQ$ with negative slopes. In $(a)$, slope of $PQ$ is smaller than that of hypotenuse's while in $(b)$$PQ$'s slope is larger than that of hypotenuse.
  • Figure 4: Triangle $T_i$ containing circle $C$.
  • Figure 5: Triangle $T_i$ containing trapezoid $PQRS$ with bases parallel to the $x$-axis.