art gallery problems
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Art Gallery Problems
Prof. Silvia Fernández
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Star-shaped Polygons
A polygon is called star-shaped if there is a point P in its interior such that the segment joining P to any boundary point is completely contained in the polygon.
PG
F
E
D
C
B
A
H
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Krasnoseľskiĭ’s Theorem
Consider a painting gallery whose walls are completely hung with pictures.
If for each three paintings of the gallery there is a point from which all three can be seen,
then there exists a point from which all paintings can be seen.
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Krasnoseľskiĭ’s Theorem
Let S be a simple polygon such that
for every three points A, B, and C of S there exists a point M such that all three segments MA, MB, and MC are completely contained in S.
Then S is star-shaped.
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Krasnoseľskiĭ’s Theorem
Proof. Let P1, P2, … , Pnbe the vertices of the polygon S in counter-clockwise order.
P1
P2
P3
P4P5
P6
P7
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Krasnoseľskiĭ’s Theorem
Proof. Let P1, P2, … , Pnbe the vertices of the polygon S in counter-clockwise order.
For each side PiPi+1consider the half plane to its left.
P1
P2
P3
P4P5
P6
P7
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Krasnoseľskiĭ’s Theorem
For each side PiPi+1consider the half plane to its left.
P1P2
P1
P2
P3
P4P5
P6
P7
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Krasnoseľskiĭ’s Theorem
For each side PiPi+1consider the half plane to its left.
P2P3
P1
P2
P3
P4P5
P6
P7
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Krasnoseľskiĭ’s Theorem
For each side PiPi+1consider the half plane to its left.
P3P4
P1
P2
P3
P4P5
P6
P7
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Krasnoseľskiĭ’s Theorem
For each side PiPi+1consider the half plane to its left.
P4P5
P1
P2
P3
P4P5
P6
P7
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Krasnoseľskiĭ’s Theorem
For each side PiPi+1consider the half plane to its left.
P5P6
P1
P2
P3
P4P5
P6
P7
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Krasnoseľskiĭ’s Theorem
For each side PiPi+1consider the half plane to its left.
P6P7
P1
P2
P3
P4P5
P6
P7
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Krasnoseľskiĭ’s Theorem
For each side PiPi+1consider the half plane to its left.
P7P1
P1
P2
P3
P4P5
P6
P7
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Krasnoseľskiĭ’s TheoremThese n half planes:
are convexsatisfy Helly’s condition
P1P2 ∩ P4P5 ∩ P6P7
P1
P2
P3
P4P5
P6
P7
P1
P2
P3
P4P5
P6
P7
P1
P2
P3
P4P5
P6
P7
P1
P2
P3
P4P5
P6
P7
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Krasnoseľskiĭ’s TheoremThus the n half planes have a point P in common. Assume that P is not in S then
Let Q be the point in S closest to P.Q belongs to a side of S, say PkPk+1.But then P doesn’t belong to the
half-plane to the left of PkPk+1, getting a contradiction.
Therefore P is in S and our proof is complete.
Q
P1
P2
P3
P4P5
P6
P7P
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Guards in Art Galleries
In 1973, Victor Klee (U. of Washington) posed the following problem:
How many guards are required to guard an art gallery?
In other words, he was interested in the number of points (guards) needed to guard a simple plane polygon (art gallery or museum).
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Guards in Art Galleries
In 1975, Vašek Chvátal (Rutgers University) proved the following theorem:
Theorem. At most ⌊(n/3)⌋ guards are necessary to guard an art gallery with n walls, (represented as a simple polygon with n sides).
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Guards in Art GalleriesProof. A “book proof” by Steve Fisk, 1978 (Bowdoin College).
Consider any polygon P.
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Guards in Art Galleries
P can be triangulated in several ways.
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Guards in Art GalleriesFor any triangulation, the vertices of P can be colored with red, blue, and green in such a way that each triangle has a vertex ofeach color.
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Guards in Art Galleries
Since there are n vertices, there is one color that is used for at most ⌊(n/3)⌋ vertices.
212Green
243Blue
433Red
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Guards in Art Galleries
Placing the guards on vertices with that color concludes our proof.
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Applications
Placement of radio antennasArchitectureUrban planning Mobile roboticsUltrasonographySensors
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ReferencesChvátal, V. "A Combinatorial Theorem in Plane Geometry." J. Combin. Th. 18, 39-41, 1975.
Fisk, S. "A Short Proof of Chvátal's Watchman Theorem." J. Combin. Th. Ser. B 24, 374, 1978.
O'Rourke, J. Art Gallery Theorems and Algorithms. New York: Oxford University Press, 1987.
Urrutia, J., Art Gallery and Illumination Problems, in Handbook on Computational Geometry, J. Sac, and J. Urrutia, (eds.), Elsevier Science Publishers, Amsterdam, 2000, p. 973-1027.
DIMACS Research and Education Institute. "Art Gallery Problems."