Hidden Mobile Guards in Simple Polygons

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Presentation transcript:

Hidden Mobile Guards in Simple Polygons Sarah Cannon Diane Souvaine Andrew Winslow

Art gallery problems

Art gallery problems

Art gallery problems

Art gallery problems

Art gallery problems

Art gallery problems

Art gallery problems

Art gallery problems

Goal: all locations seen by some guard. Art gallery problems Goal: all locations seen by some guard.

Hidden guards

Hidden guards

Hidden guards

Hidden guards

Hidden guards Claim: This polygon does not admit a hidden guard set. Proof: Any single guard is insufficient, any two are not hidden.

Guard varieties Open Closed Edge Diagonal Mobile

Polygon classes Simple Starshaped Monotone Orthogonal

Results for hidden guards Do all polygons in class admit a hidden guard set?

Results for hidden guards Do all polygons in class admit a hidden guard set?

Claim: this monotone polygon does not admit a hidden open edge guard set.

Each ear needs an edge guard inside it. Proof: Each ear needs an edge guard inside it. Any ear edge guard pair can’t be augmented into a hidden guard set.

Claim: this simple polygon does not admit a hidden open mobile guard set.

Proof by 27 subcases Subcases 3a-o Cases 1-3 Subcases 3p-u Subcases 1a-c, 2a-c Subcases 3a-o Subcases 3p-u

Claim: every monotone or starshaped polygon admits a hidden open mobile guard set. Approach: use the set of edges of a geodesic path.

Results for hidden guards Open problem!