Sweeping Shapes: Optimal Crumb Cleanup. Yonit Bousany, Mary Leah Karker, Joseph O’Rourke, Leona Sparaco. What does it mean to sweep a shape?.
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Yonit Bousany, Mary Leah Karker, Joseph O’Rourke, Leona Sparaco
Restricting our attention to two-sweeps and triangles, the minimum sweeping cost is always achieved by enclosing the triangle in a minimum perimeter parallelogram.
The minimal perimeter enclosing parallelogram is always flush against at least one edge of the convex hull.
[Mitchell and Polishchuk 2006]
However.... Conjecture: Minimal cost sweeping can be achieved with two sweeps for any convex shape.
Normalize triangle so that the longest edge=1. Let θ be the ab-apex.
hb < 1
hb (1-b) < (1-b)
hb- b hb < 1-b
b·hb = 1·h1
hb - h1 < 1 - b
hb + b < 1 + h1
An example requiring three sweeps.