Unsolved Problems in Visibility
Joseph O’Rourke
Smith College
 Art
Gallery Theorems
 Illuminating Disjoint Triangles
 Illuminating Convex Bodies
 Mirror Polygons
 Trapping Rays with Mirrors
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
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
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
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
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Quad’s, Pentagons, Hexagons
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Chvátal’s Theorem
[n/3] guards suffice (and are sometimes
necessary) to visually cover a polygon of n
vertices
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Chvátal’s Comb Polygon
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Fisk’s Proof
1.
2.
3.
4.
Triangulate polygon with diagonals
3-color graph
Monochromatic guards cover polygon
Some color is used no more than [n/3]
times
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Polygon Triangulation
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
3-coloring
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
180º-Guards
Csaba Tóth proved that [n/3] 180º-guards
suffice.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
π-floodlights
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
180º-Vertex Guards
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Urrutia’s 5/8’s Example
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Outline
 Art
Gallery Theorems
 Illuminating Disjoint Triangles
 Illuminating Convex Bodies
 Mirror Polygons
 Trapping Rays with Mirrors
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
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.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
n=3
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Current Status
n
lights are sometimes necessary
 [(5/4)n] lights suffice.
 Conjecture
(Urrutia): n+c lights suffice (for
some constant c).
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Outline
 Art
Gallery Theorems
 Illuminating Disjoint Triangles
 Illuminating Convex Bodies
 Mirror Polygons
 Trapping Rays with Mirrors
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Illuminating Convex Bodies
Boundary point illuminated* if light ray
penetrates to interior of object.
Status:
 2D: Settled
 3D: Open
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Parallelogram: 22 = 4 lights
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Parallelopiped: 23 = 8 lights
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Open Problem
Do 7 lights suffice to illuminate* the entire
boundary for all other convex bodies (e.g.,
polyhedra) in 3D?
(Hadwiger [1960])
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Outline
 Art
Gallery Theorems
 Illuminating Disjoint Triangles
 Illuminating Convex Bodies
 Mirror Polygons
 Trapping Rays with Mirrors
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Mirror Polygon: Illuminable?
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
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.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Room not illuminable from x
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Tokarsky Polygon
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Vertex Model?
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Round Vertex Model
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
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.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Open Question
Are all mirror polygons illuminable from
some point?
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Outline
 Art
Gallery Theorems
 Illuminating Disjoint Triangles
 Illuminating Convex Bodies
 Mirror Polygons
 Trapping Rays with Mirrors
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Trapping Light Rays with Mirrors
 Arbitrary
Mirrors
 Circular Mirrors
 Segment Mirrors
------------------------ Narrowing Light Rays
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Light from x is trapped!
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Enchanted Forest of Mirror Trees
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Angular Spreading
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Ray approaching limit
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
10 Rays; 3 Segments
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
1000 mirrors vs. 1 ray
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
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
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
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?
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Narrowing Light Rays
Rays are narrowed to ε if the angle between
any pair or rays that escape to infinity is
less than ε > 0.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
20º → 10 º
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
10º → 5 º
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Necklace of Mirrors: 7 Disks
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Necklace of Mirrors: 13 Disks
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Narrowing Theorems
Given any ε > 0, the light emitted by a point
source can be narrowed by a finite number
of disjoint segment mirrors, or circle
mirrors.
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.
Favorite Open Problems





Art Gallery Theorems
 Do [(5/8)n] 180º vertex guards suffice?
Illuminating Disjoint Triangles
 Do n+c lights suffice?
Illuminating Convex Bodies
 Do 8 lights suffice in 3D?
Mirror Polygons
 Is every polygon illuminable from some point?
Trapping Rays with Mirrors
 Can segment mirrors trap all rays from one light
source?
Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.