Create Presentation
Download Presentation

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

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Unsolved Problems in VisibilityJoseph O’RourkeSmith**College • Art Gallery Theorems • Illuminating Disjoint Triangles • Illuminating Convex Bodies • Mirror Polygons • Trapping Rays with Mirrors**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**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**Chvátal’s Theorem**[n/3] guards suffice (and are sometimes necessary) to visually cover a polygon of n vertices**Fisk’s Proof**• Triangulate polygon with diagonals • 3-color graph • Monochromatic guards cover polygon • Some color is used no more than [n/3] times**180º-Guards**Csaba Tóth proved that [n/3] 180º-guards suffice.**Outline**• Art Gallery Theorems • Illuminating Disjoint Triangles • Illuminating Convex Bodies • Mirror Polygons • Trapping Rays with Mirrors**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.**Current Status**• n lights are sometimes necessary • [(5/4)n] lights suffice. • Conjecture (Urrutia): n+c lights suffice (for some constant c).**Outline**• Art Gallery Theorems • Illuminating Disjoint Triangles • Illuminating Convex Bodies • Mirror Polygons • Trapping Rays with Mirrors**Illuminating Convex Bodies**Boundary point illuminated* if light ray penetrates to interior of object. Status: • 2D: Settled • 3D: Open**Open Problem**Do 7 lights suffice to illuminate* the entire boundary for all other convex bodies (e.g., polyhedra) in 3D? (Hadwiger [1960])**Outline**• Art Gallery Theorems • Illuminating Disjoint Triangles • Illuminating Convex Bodies • Mirror Polygons • Trapping Rays with Mirrors**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.**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.**Open Question**Are all mirror polygons illuminable from some point?**Outline**• Art Gallery Theorems • Illuminating Disjoint Triangles • Illuminating Convex Bodies • Mirror Polygons • Trapping Rays with Mirrors**Trapping Light Rays with Mirrors**• Arbitrary Mirrors • Circular Mirrors • Segment Mirrors ------------------------- • Narrowing Light Rays**Conjectures**No collection of disjoint segment mirrors can trap all the light from one source. No collection of disjoint circle mirrors can trap all the light from one source**Conjectures (continued)**A collection of disjoint segment mirrors may trap only X nonperiodic rays from one source. X = • countable number of • finite number of • zero?