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Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Unsolved Problems in VisibilityUnsolved Problems in VisibilityJoseph O’RourkeJoseph O’Rourke
Smith CollegeSmith College
Art Gallery TheoremsArt Gallery Theorems Illuminating Disjoint TrianglesIlluminating Disjoint Triangles Illuminating Convex BodiesIlluminating Convex Bodies Mirror PolygonsMirror Polygons Trapping Rays with MirrorsTrapping Rays with Mirrors
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Art Gallery TheoremsArt Gallery Theorems
360360ºº-Guards: -Guards: Klee’s QuestionKlee’s Question ChvChváátal’s Theoremtal’s Theorem Fisk’s ProofFisk’s Proof
180180ºº-Guards: -Guards: TTóóth’s Theoremth’s Theorem
180180ºº-Vertex Guards: -Vertex Guards: Urrutia’s ExampleUrrutia’s Example
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Klee’s QuestionKlee’s Question
How many guards,How many guards, In In fixedfixed positions, positions, each with each with 360360ºº visibility visibility are are necessarynecessary and sometimes and sometimes sufficientsufficient to visually to visually cover cover a polygon of a polygon of nn vertices vertices
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Quad’s, Pentagons, HexagonsQuad’s, Pentagons, Hexagons
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
ChvChváátal’s Theoremtal’s Theorem
[n/3] guards suffice (and are sometimes [n/3] guards suffice (and are sometimes necessary) to visually cover a polygon of n necessary) to visually cover a polygon of n verticesvertices
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
ChvChváátal’s Comb Polygontal’s Comb Polygon
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Fisk’s ProofFisk’s Proof
1.1. Triangulate polygon with diagonalsTriangulate polygon with diagonals
2.2. 3-color graph3-color graph
3.3. Monochromatic guards cover polygonMonochromatic guards cover polygon
4.4. Some color is used no more than [n/3] Some color is used no more than [n/3] timestimes
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Polygon TriangulationPolygon Triangulation
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
3-coloring3-coloring
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
180180ºº-Guards-Guards
Csaba TCsaba Tóóth proved that [n/3] 180th proved that [n/3] 180ºº-guards -guards suffice.suffice.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
ππ-floodlights-floodlights
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
180180ºº-Vertex Guards-Vertex Guards
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Urrutia’s 5/8’s ExampleUrrutia’s 5/8’s Example
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
OutlineOutline
Art Gallery TheoremsArt Gallery Theorems Illuminating Disjoint TrianglesIlluminating Disjoint Triangles Illuminating Convex BodiesIlluminating Convex Bodies Mirror PolygonsMirror Polygons Trapping Rays with MirrorsTrapping Rays with Mirrors
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Illuminating Disjoint TrianglesIlluminating Disjoint Triangles
How might lights suffice to illuminate the How might lights suffice to illuminate the boundary of n disjoint triangles?boundary of n disjoint triangles?
Boundary point is Boundary point is illuminatedilluminated if there is a if there is a clear line of sight to a light source.clear line of sight to a light source.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Current StatusCurrent Status
n lights are sometimes necessaryn lights are sometimes necessary [(5/4)n] lights suffice.[(5/4)n] lights suffice.
Conjecture (Urrutia): n+c lights suffice (for Conjecture (Urrutia): n+c lights suffice (for some constant c).some constant c).
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
OutlineOutline
Art Gallery TheoremsArt Gallery Theorems Illuminating Disjoint TrianglesIlluminating Disjoint Triangles Illuminating Convex BodiesIlluminating Convex Bodies Mirror PolygonsMirror Polygons Trapping Rays with MirrorsTrapping Rays with Mirrors
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Illuminating Convex BodiesIlluminating Convex Bodies
Boundary point Boundary point illuminated*illuminated* if light ray if light ray penetrates to interior of object.penetrates to interior of object.
Status:Status: 2D: Settled2D: Settled 3D: Open3D: Open
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Parallelogram: 2Parallelogram: 222 = 4 lights = 4 lights
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Parallelopiped: 2Parallelopiped: 233 = 8 lights = 8 lights
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Open ProblemOpen Problem
Do 7 lights suffice to illuminate* the entire Do 7 lights suffice to illuminate* the entire boundary for all other convex bodies (e.g., boundary for all other convex bodies (e.g., polyhedra) in 3D?polyhedra) in 3D?
(Hadwiger [1960])(Hadwiger [1960])
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
OutlineOutline
Art Gallery TheoremsArt Gallery Theorems Illuminating Disjoint TrianglesIlluminating Disjoint Triangles Illuminating Convex BodiesIlluminating Convex Bodies Mirror PolygonsMirror Polygons Trapping Rays with MirrorsTrapping Rays with Mirrors
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Mirror Polygon: Illuminable?Mirror Polygon: Illuminable?
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Mirror PolygonsMirror Polygons
Victor Klee (1973): Is every mirror polygon Victor Klee (1973): Is every mirror polygon illuminable from illuminable from eacheach of its points? of its points?
G. Tokarsky (1995): No: For some polygons, G. Tokarsky (1995): No: For some polygons, a light at a certain point will leave another a light at a certain point will leave another point dark.point dark.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Room not illuminable from xRoom not illuminable from x
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Tokarsky PolygonTokarsky Polygon
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Vertex Model?Vertex Model?
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Round Vertex ModelRound Vertex Model
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
ConjecturesConjectures
Under Under round-vertexround-vertex model, all mirror model, all mirror polygons are illuminable from polygons are illuminable from eacheach point. point.
Under the Under the vertex-killvertex-kill model, the set of dark model, the set of dark points has points has measure zeromeasure zero..
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Open QuestionOpen Question
Are all mirror polygons illuminable from Are all mirror polygons illuminable from somesome point? point?
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
OutlineOutline
Art Gallery TheoremsArt Gallery Theorems Illuminating Disjoint TrianglesIlluminating Disjoint Triangles Illuminating Convex BodiesIlluminating Convex Bodies Mirror PolygonsMirror Polygons Trapping Rays with MirrorsTrapping Rays with Mirrors
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Trapping Light Rays with Trapping Light Rays with MirrorsMirrors
Arbitrary MirrorsArbitrary Mirrors Circular MirrorsCircular Mirrors Segment MirrorsSegment Mirrors
-------------------------------------------------- Narrowing Light RaysNarrowing Light Rays
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Light from Light from xx is trapped! is trapped!
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Enchanted Forest of Mirror TreesEnchanted Forest of Mirror Trees
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Angular SpreadingAngular Spreading
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Ray approaching limitRay approaching limit
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
10 Rays; 3 Segments10 Rays; 3 Segments
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
1000 mirrors vs. 1000 mirrors vs. 1 ray1 ray
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
ConjecturesConjectures
No collection of disjoint No collection of disjoint segment mirrorssegment mirrors can can trap all the light from one source.trap all the light from one source.
No collection of disjoint No collection of disjoint circle mirrorscircle mirrors can can trap all the light from one sourcetrap all the light from one source
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Conjectures (continued)Conjectures (continued)
A collection of disjoint segment mirrors may A collection of disjoint segment mirrors may trap only trap only XX nonperiodic rays from one nonperiodic rays from one source.source.
XX = = countable number ofcountable number of finite number offinite number of zero?zero?
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Narrowing Light RaysNarrowing Light Rays
Rays are Rays are narrowed to narrowed to εε if the angle between if the angle between any pair or rays that escape to infinity is any pair or rays that escape to infinity is less than less than εε > 0. > 0.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
2020ºº →→ 10 10 ºº
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
1010ºº →→ 5 5 ºº
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Necklace of Mirrors: 7 DisksNecklace of Mirrors: 7 Disks
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Necklace of Mirrors: 13 DisksNecklace of Mirrors: 13 Disks
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Narrowing TheoremsNarrowing Theorems
Given any Given any εε > 0, the light emitted by a point > 0, the light emitted by a point source can be narrowed by a finite number source can be narrowed by a finite number of disjoint of disjoint segmentsegment mirrors, or mirrors, or circlecircle mirrors.mirrors.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Art Gallery TheoremsArt Gallery Theorems Do [(5/8)n] 180Do [(5/8)n] 180ºº vertex guards suffice? vertex guards suffice?
Illuminating Disjoint TrianglesIlluminating Disjoint Triangles Do n+c lights suffice?Do n+c lights suffice?
Illuminating Convex BodiesIlluminating Convex Bodies Do 8 lights suffice in 3D?Do 8 lights suffice in 3D?
Mirror PolygonsMirror Polygons Is every polygon illuminable from some point?Is every polygon illuminable from some point?
Trapping Rays with MirrorsTrapping Rays with Mirrors Can segment mirrors trap all rays from one light Can segment mirrors trap all rays from one light
source?source?
Favorite Open ProblemsFavorite Open Problems
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