1 / 30

Finding Large Sticks and Potatoes in Polygons.

This paper presents innovative algorithms for solving natural optimization problems related to shape approximation and visibility culling in computer graphics. Specifically, we focus on the computation of the largest stick and potato within smooth closed curves and convex polygons. Our methodology includes a divide-and-conquer algorithm with balanced cuts, achieving a O(n log n) time complexity for approximating the longest stick. We also explore theoretical bounds and potential future work on exact solutions and PTAS for various geometric configurations.

hazel
Télécharger la présentation

Finding Large Sticks and Potatoes in Polygons.

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. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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?

More Related