Finding Large Sticks and Potatoes in Polygons.

1. 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

2. Motivation • Natural Optimization Problems • Shape Approximation • Visibility Culling for Computer Graphics

3. Biggest Potato

4. Peeling Potato inside Smooth Closed Curves

5. Biggest French Fry

6. Longest Stick

7. Related and Prior Work

8. Convex Polygons on Point Sets

9. Related Work: Longest Stick

10. Our Results (On Peeling)

11. Approximate Largest Stick • Divide and Conquer Algorithm • Uses balanced cuts (Chazelle Cuts) d c a c b e b d

12. Approximate Largest Stick • Compute weak visibility region from anchor edge (diagonal) e. • (p) has combinatorial type (u,v) • Optimize for each of the O(n) elementary intervals. 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. Theorem: One can compute a ½-approximation for longest stick in a simple polygon in O(nlogn) time. Open Problem: Can we get O(1)-approx in O(n) time?

13. 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

14. Pixels and the visibility problem.

15. Pixels and the visibility problem.

16. Approximate Largest Stick

17. Approximate Largest Convex-gon • Suffices to look for a large triangle to get a O(1)-approximation. • 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).

18. Approximate Largest Convex-gon • Suffices to look for a large triangle to get a O(1)-approximation. • 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).

19. Approximate Largest Triangular potato

20. Approximate FAT Largest triangular potato

21. Approximate Fat Triangles : Results

22. A Sampling approach

23. Largest Area Triangle using Sampling

24. Largest Area Triangle by Sampling: A difficulty

25. Peeling an ellipse

26. Max Area ellipse inside sampled curves

27. Linearized convex hull + Normal cond. + Inside Test

28. An Example output

29. An Example output

30. Questions? Future Work • 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?