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Finding Large Sticks and Potatoes in Polygons. Matya Katz and Arik Sityon Ben-Gurion University. Olaf Hall-Holt St. Olaf College. Joseph S.B. Mitchell Stony Brook University. Piyush Kumar Florida State University. Motivation. Natural Optimization Problems - PowerPoint PPT Presentation
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Finding Large Sticks and Potatoes in Polygons.
Olaf Hall-HoltSt. Olaf College
Matya Katz and Arik Sityon Ben-Gurion University
Joseph S.B. MitchellStony Brook University
Piyush KumarFlorida State University
1. Natural Optimization Problems
2. Shape Approximation
3. Visibility Culling for Computer Graphics
Motivation
Biggest Potato
Peeling Potato inside Smooth Closed Curves
Biggest French Fry
Longest Stick
Related and Prior Work
Convex Polygons on Point Sets
Related Work: Longest Stick
Our Results (On Peeling)
1. Divide and Conquer Algorithm
2. Uses balanced cuts (Chazelle Cuts)
Approximate Largest Stick
e
a
b c
db
cd
Approximate Largest Stick
1. Compute weak visibility region from anchor edge
(diagonal) e.
2. (p) has combinatorial type (u,v)
3. Optimize for each of the O(n) elementary intervals.
Theorem:
One can compute a ½-approximation for longest stick in a simple polygon in O(nlogn) time.
Algorithm:
At each level of the recursive decomposition of P, compute longest anchored sticks from each diagonal cut: O(n) per level.
Longest Anchored stick is at least ½ the length of the longest stick.
Open Problem:Can we get O(1)-approx in O(n) time?
Approximate Largest Stick: Improved Approx.
Algorithm:
Bootstrap from the O(1)-approx, discretize search space more finely, reduce to a visibility problem, and apply efficient data structures
Pixels and the visibility problem.
Pixels and the visibility problem.
Approximate Largest Stick
Approximate Largest Convex-gon
1. Suffices to look for a large triangle to get a O(1)-
approximation.
2. For any convex body B, there is an inscribed triangle T*
of area at least c.area(B). There exists a O(1)
approximation to T* anchored at a cut computable in
O(nlogn).
Approximate Largest Convex-gon
1. Suffices to look for a large triangle to get a O(1)-
approximation.
2. For any convex body B, there is an inscribed triangle T*
of area at least c.area(B). There exists a O(1)
approximation to T* anchored at a cut computable in
O(nlogn).
Approximate Largest Triangular potato
Approximate FAT Largest triangular potato
Approximate Fat Triangles : Results
A Sampling approach
Largest Area Triangle using Sampling
Largest Area Triangle by Sampling: A difficulty
Peeling an ellipse
Max Area ellipse inside sampled curves
Linearized convex hull + Normal cond. + Inside Test
An Example output
An Example output
• PTAS for largest triangle ?
• Find exact solutions/approximations for biggest potato ?
• Packing convex sets in shapes.
• Sub quadratic bounds for max area star shaped
polygons?
• Find k convex potatoes to max the area of the union?
Sum? Max area k-gon (Non-convex)?
• d-D?
Future WorkQuestions?