Unsolved Problems in Visibility Joseph O’Rourke Smith College

# Unsolved Problems in Visibility Joseph O’Rourke Smith College

## Unsolved Problems in Visibility Joseph O’Rourke Smith College

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##### Presentation Transcript

1. Unsolved Problems in VisibilityJoseph O’RourkeSmith College • Art Gallery Theorems • Illuminating Disjoint Triangles • Illuminating Convex Bodies • Mirror Polygons • Trapping Rays with Mirrors

2. Art Gallery Theorems • 360º-Guards: • Klee’s Question • Chvátal’s Theorem • Fisk’s Proof • 180º-Guards: • Tóth’s Theorem • 180º-Vertex Guards: • Urrutia’s Example

3. Klee’s Question • How many guards, • In fixed positions, • each with 360º visibility • are necessary • and sometimes sufficient • to visually cover • a polygon of n vertices

5. Chvátal’s Theorem [n/3] guards suffice (and are sometimes necessary) to visually cover a polygon of n vertices

6. Chvátal’s Comb Polygon

7. Fisk’s Proof • Triangulate polygon with diagonals • 3-color graph • Monochromatic guards cover polygon • Some color is used no more than [n/3] times

8. Polygon Triangulation

9. 3-coloring

10. 180º-Guards Csaba Tóth proved that [n/3] 180º-guards suffice.

11. π-floodlights

12. 180º-Vertex Guards

13. Urrutia’s 5/8’s Example

14. Outline • Art Gallery Theorems • Illuminating Disjoint Triangles • Illuminating Convex Bodies • Mirror Polygons • Trapping Rays with Mirrors

15. Illuminating Disjoint Triangles How might lights suffice to illuminate the boundary of n disjoint triangles? Boundary point is illuminated if there is a clear line of sight to a light source.

16. n=3

17. Current Status • n lights are sometimes necessary • [(5/4)n] lights suffice. • Conjecture (Urrutia): n+c lights suffice (for some constant c).

18. Outline • Art Gallery Theorems • Illuminating Disjoint Triangles • Illuminating Convex Bodies • Mirror Polygons • Trapping Rays with Mirrors

19. Illuminating Convex Bodies Boundary point illuminated* if light ray penetrates to interior of object. Status: • 2D: Settled • 3D: Open

20. Parallelogram: 22 = 4 lights

21. Parallelopiped: 23 = 8 lights

22. Open Problem Do 7 lights suffice to illuminate* the entire boundary for all other convex bodies (e.g., polyhedra) in 3D? (Hadwiger [1960])

23. Outline • Art Gallery Theorems • Illuminating Disjoint Triangles • Illuminating Convex Bodies • Mirror Polygons • Trapping Rays with Mirrors

24. Mirror Polygon: Illuminable?

25. Mirror Polygons Victor Klee (1973): Is every mirror polygon illuminable from each of its points? G. Tokarsky (1995): No: For some polygons, a light at a certain point will leave another point dark.

26. Room not illuminable from x

27. Tokarsky Polygon

28. Vertex Model?

29. Round Vertex Model

30. Conjectures Under round-vertex model, all mirror polygons are illuminable from each point. Under the vertex-kill model, the set of dark points has measure zero.

31. Open Question Are all mirror polygons illuminable from some point?

32. Outline • Art Gallery Theorems • Illuminating Disjoint Triangles • Illuminating Convex Bodies • Mirror Polygons • Trapping Rays with Mirrors

33. Trapping Light Rays with Mirrors • Arbitrary Mirrors • Circular Mirrors • Segment Mirrors ------------------------- • Narrowing Light Rays

34. Light from x is trapped!

35. Enchanted Forest of Mirror Trees