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Visibility Via Reflection

Vaezi, Arash | 2021

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 54703 (19)
  4. University: Sharif University of Technology
  5. Department: Computer Engineering
  6. Advisor(s): Ghodsi, Mohammad
  7. Abstract:
  8. This document deals with the following results in details:Given a simple polygon P with $n$ vertices, the visibility polygon (VP) of a point q, or a segment pq inside P can be computed in linear time. It is known that the visibility polygon of a point inside a polygon in the presence of a mirror can be computed in linear time. We propose a linear time algorithm to extend the VP of a viewer (point or segment) by converting some edges of P into mirrors (reflecting-edges with specular type of reflection), such that a given non-visible segment uw can also be seen from the viewer.Various definitions for the visibility of a segment, such as weak, strong, or complete visibility, are considered. Our algorithm finds every edge that, when converted to a mirror, makes uw visible to our viewer. We find out exactly which interval of uw becomes visible, by every edge middling as a mirror, all in linear time. In other words, we present an algorithm that for every edge e of P reveals precisely which part of uw is mirror-visible through e.As mentioned, we study visibility with specular and diffuse reflections. The number of times a ray can be reflected can be taken as a parameter. We prove that finding edges to add exactly k units of area to the visibility polygon of a query point q in most cases is NP-complete, and its minimization is NP-hard. These cases contain single and multiple, either with secular or diffuse reflections. Considering the diffuse type of reflection when the multiple reflection is allowed, even more than two reflections, the problem remains open. Also, the problem of finding edges to add at least k units of area with the minimum number of diffuse reflecting-edges is NP-hard too. Extending visibility polygons with specular reflection considering multiple reflections remains open. Furthermore, we illustrate that if P is a funnel or a weak visibility polygon, then the problem of extending visibility polygons via diffuse reflecting-edges becomes more straightforward and can be solved in polynomial time.In the document, we also study a variant of the Art Gallery problem in which the ``walls" can be replaced by reflecting-edges, which allows the guard to see further and thereby see a larger portion of the gallery.Clearly, if we let an edge of the polygon allow reflections, then the visibility region should be changed accordingly.Given a simple polygon P, in the Art Gallery problem, the goal is to find the minimum number of guards needed to cover the entire P, where a guard is a point and can see another point q when pq does not cross the edges of P.This document studies a variant of the Art Gallery problem in which the boundaries of P are replaced by single specular-reflection edges, which allows the view rays to reflect once per collision with an edge. This property allows the guards to see through the reflections and thereby see a larger portion of the polygon. For this problem, the position of the guards in P can be determined with our proposed O(1)-approximation algorithm.We also prove that if P (possibly with holes) can be guarded by α guards without reflections, then P can be guarded by at most ⌈α/(1-⌊r/4⌋ ) ⌉guards when r diffuse reflections are permitted.Additionally, we show that for vertex guards, the art gallery problem considering $r$ reflections, for both the diffuse and specular reflection are solvable in O(n^(4^(r+1)+2))time, giving an approximation ratio of O(log⁡n)
  9. Keywords:
  10. Mirror ; Specular Reflection ; Diffuse Reflection ; Single Reflection ; Multiple Reflection ; Art Gallery Problem ; Visibility Algorithm ; NP-Hard Problems

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  • مقدمه
    • مسئله‌ی حفاظت موزه‌ی هنر
    • مسائل مورد بحث در پایان‌نامه
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    • کلیت الگوریتم
      • الگوریتم 2.1.1
    • محاسبه‌ی رئوس LBV و RBV
    • الگوریتم 2.3
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      • الگوریتم 2.4.1
      • تحلیل الگوریتم 2.4.1
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      • تمام نقاط ناظر به طور کامل پاره‌خط هدف را ببینند.
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      • چندضلعی کاهش
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