sweeping shapes: optimal crumb cleanup
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Sweeping Shapes: Optimal Crumb Cleanup. Yonit Bousany, Mary Leah Karker, Joseph O’Rourke, Leona Sparaco. What does it mean to sweep a shape?. - PowerPoint PPT PresentationTRANSCRIPT
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Sweeping Shapes: Optimal Crumb Cleanup
Yonit Bousany, Mary Leah Karker, Joseph O’Rourke, Leona Sparaco
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What does it mean to sweep a shape?
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Restricting our attention to two-sweeps and triangles, the minimum sweeping cost is always achieved by enclosing the triangle in a minimum perimeter parallelogram.
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One-flush Lemma
The minimal perimeter enclosing parallelogram is always flush against at least one edge of the convex hull.
[Mitchell and Polishchuk 2006]
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Conjecture
However.... Conjecture: Minimal cost sweeping can be achieved with two sweeps for any convex shape.
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All enclosing parallelograms for acute triangles.
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All enclosing parallelograms for obtuse triangles.
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TheoremNormalize triangle so that the
longest edge=1. Let θ be the ab-apex.
• If θ ≥ 90, the min cost sweep is determined by the parallelogram 2-flush against a and b.
• If θ ≤ 90, the min cost sweep is determined by the rectangle 1-flush against the shortest side.
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Proof of One Subcase
hb < 1hb (1-b) < (1-b)hb- b hb < 1-bb·hb = 1·h1hb - h1 < 1 - b hb + b < 1 + h1
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The best way of sweeping a shape is not necessarily achieved with two sweeps:
An example requiring three sweeps.