This document discusses open and closed guard edges in simple polygons. It defines key terms like open guard edges, closed guard edges, and star-shaped polygons. It presents two lemmas: 1) A point p is visible from an open/closed edge depending on the common vertices of paths from p to the edge's endpoints. 2) Given open guard edges ab and cd, their associated paths are disjoint and straight lines. The document proves that a non-starshaped polygon has at most 1 open guard edge and at most 3 closed guard edges. It also shows the intersection of paths associated with closed guard edges is in the polygon's kernel.

1. Open Guard Edges and Edge
Guards in Simple Polygons
,with Csaba D. Tóth and Godfried T. Toussaint, Proceedings of 23rd
Canadian Conference on Computational Geometry, 449-454, 2011.
http://www.eecs.tufts.edu/~awinslow/
Presenter : Oscar, openguards@olife.org
2013/06/10

2. • Guard edges
• Assign. 2 - art gallery problem

3. • Terms explanation
• Upper bound of open/closed guard edges
of Non-starshaped simple polygon.
• Lemma
Outline

4. Terms explanation

5. Closed guard edges

6. Open guard edges

7. Geodesic path(p,q)
• Geodesic path(p,q) →
path(p,q)
• Shortest directed path from p to q that lies
entirely in P.
• path(p,q) is straight line p and q see each⇔
other.

8. weakly visible
• A point is weakly visible to a set of points
• this point is visible from some point in
that set.

9. A starshaped n-gon P with kernel(P)

10. Star-shaped polygon
https://en.wikipedia.org/wiki/Star-shaped_polygon

11. Non-starshaped simple polygon

12. Lemma 1(1) open edge
• Let p be a point inside a simple polygon P.
1. Point p is visible from an open edge uv ⇔
p is the only common vertex of path(p, u)
and path(p, v)

13. Point p is visible from an open edge uv p is⇔
the only common vertex of path(p, u) and
path(p, v)

14. Lemma 1(2) closed edge
• Let p be a point inside a simple polygon P.
2.p is visible from a closed edge uv {p, u,⇔
v} are only three possible common
vertices of path(p, u) and path(p, v).

15. p is visible from a closed edge uv ⇔ {p, u, v}
are only three possible common vertices of
path(p, u) and path(p, v).

16. OPEN GUARD EDGES

17. A simple polygon with open
guard edges
• ≥ 2 starshaped.⇔
• ≤ 1 Non-starshaped⇔

18. Lemma 2
Given ab, cd are open guard edges.
⇒path(b, c) and path(a, d) are disjoint.
⇒path(a, c) and path(b, d) are straight line.

19. • quadrilateral Q {← ab, path(b,c), cd, and path(a,d)}
• Assume interior vertex q path(b,c) path(a,d)∈ ∩
• a or b is not visible from the open edge cd (Lemma 1)
• cd is not a guard edge (contradiction)
• Conclude
path(b,c) and path(a,d) are disjoint
Q is a simple polygon

20. • Assume an interior vertex of path(a,c) in path(b,c)
• c is not visible from ab
• ab is not a guard edge (contradiction)
• Conclude
 path(a,c) has no interior vertices.
 path(b,d) has no interior vertices as well.

21. Lemma 3
Given ab, cd are open guard edges.
⇒The intersection point x = ac ∩ bd is in
the kernel of P

22. • To show that an arbitrary point p is visible from x.
• ac and bd are diagonals of P (lemma 2)
• assume p (abx) (cdx)∈ △ ∪ △
• px lies in the same triangle
• p can be seen by x.

23. • Assume p is outside of both triangles.
• ∵ ab and cd are open guard edges
• p sees q and o (relative interiors)
• quadrilateral Q = (o, p, q, x) is simple and inside P
• diagonal px lies inside Q
• ∴p can be seen by x.

24. Theorem 4
Non-starshaped simple polygon open⇔
guard edge ≤ 1

25. Theorem 4
• Non-starshaped simple polygon open guard edge ≤ 1⇒
• Open guard edge ≥
2
• has kernel(P)
(Lemma 3)
• A starshaped polygon
• Contradiction

26. CLOSED GUARD EDGES

27. Theorem 5
non-starshaped simple polygon
⇔
closed guard edges ≤ 3

28. Lemma 6
• Given g1,g3 are closed guard edges.
the path(b, c) and path(a, d) are disjoint⇒
path(a, c) and path(b, d) in⇒ □{a, b, c, d}.

29. • path(a,c) and path(b,d) lie in Q
• Any interior vertex of path(a,c) and path(b,d) is in Q
• Assume path(a,c) and path(b,c) have a common interior vertex
• c is not visible from ab
• ab is not a guard edge (contradiction)
• Conclude
 all interior vertices of path(a,c) and path(b,d) are in {a,b,c,d}

30. Corollary 7
• If □{a, b, c, d} is convex path(a,c) and⇒
path(b,d) are straight line.

31. Corollary 7
convex({a, b, c, d})= (abc)△
⇒
path(a,c) = (a,d,c) and path(b,d) = bd.

32. Lemma 8
• The intersection point x = path(a, c) ∩
path(b, d) is in the kernel of P .
g2

33. • To show that an arbitrary point p is visible from x.
• △ (abx) , (cdx) are diagonals in P△ (Corollary 7)
• Assume p (abx) (cdx)∈ △ ∪ △
• px lies in the same triangle
• p can be seen by x.

34. • Assume p is outside of both triangles.
• w.l.o.g. assume, p is on the right side of the
directed path(a,c) and path(b,d).
• p and the guard edge g4 are on opposite sides
• If path(p,x) = px, then p is visible from x (done)

35. • path(p,x) is not a straight line
• w.l.o.g. assume, path(p,x) makes a right turn at its last
interior vertex q
• = path(p,d) also makes a right turn at q
• ∵ p is visible from the guard edge cd
• ∴ q = c (Lemma 1b)
• ∵any paths from g4 to p make a right turn at c
• ∴ p is not visible from g4 (Lemma 1b) (contradiction)
• ∵ from g4 to p is straight line
• ∴ path(p,x) is a straight line

36. Theorem 5
non-starshaped simple polygon
⇔
closed guard edges ≤ 3

37. TO FIND KERNEL(P)

38. Left and right kernels