Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Unsolved...

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


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


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


Quad’s, Pentagons, HexagonsQuad’s, Pentagons, Hexagons


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


ChvChváátal’s Comb Polygontal’s Comb Polygon


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


Polygon TriangulationPolygon Triangulation


3-coloring3-coloring


180180ºº-Guards-Guards

Csaba TCsaba Tóóth proved that [n/3] 180th proved that [n/3] 180ºº-guards -guards suffice.suffice.


ππ-floodlights-floodlights


180180ºº-Vertex Guards-Vertex Guards


Urrutia’s 5/8’s ExampleUrrutia’s 5/8’s Example


OutlineOutline



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.


n=3n=3


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).


OutlineOutline



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


Parallelogram: 2Parallelogram: 222 = 4 lights = 4 lights


Parallelopiped: 2Parallelopiped: 233 = 8 lights = 8 lights


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])


OutlineOutline



Mirror Polygon: Illuminable?Mirror Polygon: Illuminable?


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.


Room not illuminable from xRoom not illuminable from x


Tokarsky PolygonTokarsky Polygon


Vertex Model?Vertex Model?


Round Vertex ModelRound Vertex Model


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..


Open QuestionOpen Question

Are all mirror polygons illuminable from Are all mirror polygons illuminable from somesome point? point?


OutlineOutline



Trapping Light Rays with Trapping Light Rays with MirrorsMirrors

Arbitrary MirrorsArbitrary Mirrors Circular MirrorsCircular Mirrors Segment MirrorsSegment Mirrors

-------------------------------------------------- Narrowing Light RaysNarrowing Light Rays


Light from Light from xx is trapped! is trapped!


Enchanted Forest of Mirror TreesEnchanted Forest of Mirror Trees


Angular SpreadingAngular Spreading


Ray approaching limitRay approaching limit


10 Rays; 3 Segments10 Rays; 3 Segments


1000 mirrors vs. 1000 mirrors vs. 1 ray1 ray


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


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?


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.


2020ºº →→ 10 10 ºº


1010ºº →→ 5 5 ºº


Necklace of Mirrors: 7 DisksNecklace of Mirrors: 7 Disks


Necklace of Mirrors: 13 DisksNecklace of Mirrors: 13 Disks


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.


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

Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Unsolved...

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