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?