Sweeping shapes optimal crumb cleanup
Download
1 / 13

Sweeping Shapes: Optimal Crumb Cleanup - PowerPoint PPT Presentation


  • 146 Views
  • Uploaded on

Sweeping Shapes: Optimal Crumb Cleanup. Yonit Bousany, Mary Leah Karker, Joseph O’Rourke, Leona Sparaco. What does it mean to sweep a shape?.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Sweeping Shapes: Optimal Crumb Cleanup' - aitana


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Sweeping shapes optimal crumb cleanup

Sweeping Shapes: Optimal Crumb Cleanup

Yonit Bousany, Mary Leah Karker, Joseph O’Rourke, Leona Sparaco


What does it mean to sweep a shape
What does it mean to sweep a shape?


Restricting our attention to two-sweeps and triangles, the minimum sweeping cost is always achieved by enclosing the triangle in a minimum perimeter parallelogram.


One flush lemma
One-flush Lemma 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]


Conjecture
Conjecture minimum sweeping cost is always achieved by enclosing the triangle in a minimum perimeter parallelogram.

However.... Conjecture: Minimal cost sweeping can be achieved with two sweeps for any convex shape.


All enclosing parallelograms for acute triangles
All enclosing parallelograms for acute triangles. minimum sweeping cost is always achieved by enclosing the triangle in a minimum perimeter parallelogram.


All enclosing parallelograms for obtuse triangles
All enclosing parallelograms for obtuse triangles. minimum sweeping cost is always achieved by enclosing the triangle in a minimum perimeter parallelogram.


Theorem
Theorem minimum sweeping cost is always achieved by enclosing the triangle in a minimum perimeter parallelogram.

Normalize 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.


Proof of one subcase
Proof of One Subcase minimum sweeping cost is always achieved by enclosing the triangle in a minimum perimeter parallelogram.

hb < 1

hb (1-b) < (1-b)

hb- b hb < 1-b

b·hb = 1·h1

hb - h1 < 1 - b

hb + b < 1 + h1


The best way of sweeping a shape is not necessarily achieved with two sweeps
The best way of sweeping a shape is not necessarily achieved with two sweeps:

An example requiring three sweeps.


ad