The art gallery problem is formulated in geometry as the minimum number of guards that need to be placed in an n-vertex simple polygon such that all points of the interior are visible. Visibility is defined such that two points u and v are mutually visible if the line segment joining them lies inside the polygon
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The Art Gallery Problem
1. Mathematical Problem Solving
Guards
Olive Byukusenge,Sabah Mirghani Yagoub Mohammed, Akor Stanley
Yusuf Brima, Patrick Ntwari Shema
Group Seven(7)
October 9, 2020
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3. Problem formulation
Consider a plot of farm owned by Brian, what is the minimum
number of stationary guards needed to protect the farm?
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4. Problem formulation
Given a n-vertex simple non-regular polygon, what is the
minimum number of guards to see every point of the interior
of the polygon?
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7. Theorem
Theorem
Every polygon can be triangulated, it is possible to split a complex
polygon to smaller triangles [1] [2].
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8. Theorem
Number of line segments equal to n − 3 and where n is the number of
sides of the polygon.
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9. Proofs
Proof.
Base Case:
Considering the simplest polygon (a triangle) with sides n = 3, the
triangulation theorem works perfectly.
Next we try to triangulate any n-sided polygon and observe that they can
indeed be triangulated.
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10. Theorem
Theorem
For any triangulated polygon, we can assign three colours to the vertices,
such that all the vertices of that polygon are expressed in three colours.
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11. Proof
Proof.
Base Case:
Considering the very trivial case of a simple polygon [that is a triangle]
where n = 3, we actually see that for each vertex we can assign a
different colour.
The 3-colorable theorem can then be generalized for larger polygons
n = 4, 5, ........., k and this holds true as shown in the figure below.
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16. Conclusion
The minimum number of guards required by Brian to secure the entire
perimeter of his farm is 7 as indicated by the red colours of the 24-sided
polygons in figure 6.
The number of guards required to cover any n-gon shaped field is
proportional to the number of sides of the polygon, the solution is found
by taking the floor of n
3 where n is the number of sides of the n-gon,
however this value is not often the minimum number of guards
required.
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18. References
[1] N. Chesnokov, The art gallery problem: An overview and extension
to chromatic coloring and mobile guards, 2018.
[2] C. D. T´oth, G. T. Toussaint, and A. Winslow, “Open guard edges
and edge guards in simple polygons,” in Spanish Meeting on
Computational Geometry, Springer, 2011, pp. 54–64.
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