an algorithm for triangulating multiple 3d polygons m zou, t ju, n carr eurographics symposium on...

An algorithm for triangulating multiple 3D polygonsM Zou, T Ju, N CarrEurographics Symposium on Geometry Processing2013

Who? Kundan Krishna1

From?

When?

1Department of Computer Science & Engineering, IIT KanpurE-mail: kkrishna@iitk.ac.in

5 September, 2014

Introduction Previous Work New work Is it really good? References

Contents

Introduction

Previous Work

New work

Is it really good?

References

IIT Kanpur 0

Introduction Previous Work New work Is it really good? References

Introduction

What is triangulation?

IIT Kanpur 1

Introduction Previous Work New work Is it really good? References

Introduction

What is triangulation?

Generate a surface whose boundaries are given.

IIT Kanpur 1

Introduction Previous Work New work Is it really good? References

Introduction

What is triangulation?

Generate a surface whose boundaries are given.

Boundary specified in the form of points of one or manypolygons.

IIT Kanpur 1

Introduction Previous Work New work Is it really good? References

Introduction

What is triangulation?

Generate a surface whose boundaries are given.

Boundary specified in the form of points of one or manypolygons.

Topologically equivalent to a sphere with holes.

IIT Kanpur 1

Introduction Previous Work New work Is it really good? References

Introduction

What is triangulation?

Generate a surface whose boundaries are given.

Boundary specified in the form of points of one or manypolygons.

Topologically equivalent to a sphere with holes.

IIT Kanpur 1

Introduction Previous Work New work Is it really good? References

Introduction

What is triangulation?

Generate a surface whose boundaries are given.

Boundary specified in the form of points of one or manypolygons.

Topologically equivalent to a sphere with holes.

IIT Kanpur 1

Introduction Previous Work New work Is it really good? References

New work

IIT Kanpur 2

Introduction Previous Work New work Is it really good? References

New work

The algorithm aims to optimize a certain metric.

IIT Kanpur 2

Introduction Previous Work New work Is it really good? References

New work

The algorithm aims to optimize a certain metric.

The metric can be based on individual triangles or pairsof them.

IIT Kanpur 2

Introduction Previous Work New work Is it really good? References

New work

The algorithm aims to optimize a certain metric.

The metric can be based on individual triangles or pairsof them.

Eg. Individual metric : Minimize sum of areas oftriangle.

IIT Kanpur 2

Introduction Previous Work New work Is it really good? References

New work

The algorithm aims to optimize a certain metric.

The metric can be based on individual triangles or pairsof them.

Eg. Individual metric : Minimize sum of areas oftriangle.

Eg. Shared metric : Minimize dihedral angle betweentriangles.

IIT Kanpur 2

Introduction Previous Work New work Is it really good? References

New work

The algorithm aims to optimize a certain metric.

The metric can be based on individual triangles or pairsof them.

Eg. Individual metric : Minimize sum of areas oftriangle.

Eg. Shared metric : Minimize dihedral angle betweentriangles.First algorithm to give a provably optimal algorithmfor minimizing such metrics, for general polygons.

IIT Kanpur 2

Introduction Previous Work New work Is it really good? References

New work

The algorithm aims to optimize a certain metric.

The metric can be based on individual triangles or pairsof them.

Eg. Individual metric : Minimize sum of areas oftriangle.

Eg. Shared metric : Minimize dihedral angle betweentriangles.First algorithm to give a provably optimal algorithmfor minimizing such metrics, for general polygons.

IIT Kanpur 2

Introduction Previous Work New work Is it really good? References

One polygon

IIT Kanpur 3

Introduction Previous Work New work

One polygon

One polygonVery well researched problem.Optimal algorithms known.

Is it really good? References

IIT Kanpur 3

Introduction Previous Work New work

One polygon

One polygonVery well researched problem.Optimal algorithms known.Delaunay triangulation

Is it really good? References

IIT Kanpur 3

Introduction Previous Work New work

One polygon

One polygonVery well researched problem.Optimal algorithms known.Delaunay triangulation

Is it really good? References

IIT Kanpur 3

Introduction Previous Work New work

One polygon

One polygonVery well researched problem.Optimal algorithms known.Delaunay triangulation

Is it really good? References

IIT Kanpur 3

Introduction Previous Work New work Is it really good? References

More than one polygons

Algorithm known only for parallel polygons.

IIT Kanpur 4

Introduction Previous Work New work Is it really good? References

More than one polygons

Algorithm known only for parallel polygons.

IIT Kanpur 4

Introduction Previous Work New work Is it really good? References

More than one polygons

Algorithm known only for parallel polygons.

IIT Kanpur 4

Introduction Previous Work New work Is it really good? References

New work

IIT Kanpur 5

Introduction Previous Work New work Is it really good? References

New work

Proposed algorithm which works for arbitrary number ofpolygons which are arbitrarily positioned in space.

IIT Kanpur 5

Introduction Previous Work New work Is it really good? References

New work

Proposed algorithm which works for arbitrary number ofpolygons which are arbitrarily positioned in space.

The algorithm uses divide and conquer strategy tosuccesively triangulate smaller partitions of the set ofpolygon points.

IIT Kanpur 5

Introduction Previous Work New work Is it really good? References

New work

Proposed algorithm which works for arbitrary number ofpolygons which are arbitrarily positioned in space.

The algorithm uses divide and conquer strategy tosuccesively triangulate smaller partitions of the set ofpolygon points.

Major Achievement

It avoids non-manifold edges, which are edges belongingto more than 2 triangles in the triangulation.

IIT Kanpur 5

Introduction Previous Work New work Is it really good? References

New work

Proposed algorithm which works for arbitrary number ofpolygons which are arbitrarily positioned in space.

The algorithm uses divide and conquer strategy tosuccesively triangulate smaller partitions of the set ofpolygon points.

Major Achievement

It avoids non-manifold edges, which are edges belongingto more than 2 triangles in the triangulation.

IIT Kanpur 5

Introduction Previous Work New work Is it really good? References

New work

Proposed algorithm which works for arbitrary number ofpolygons which are arbitrarily positioned in space.

The algorithm uses divide and conquer strategy tosuccesively triangulate smaller partitions of the set ofpolygon points.

Major Achievement

It avoids non-manifold edges, which are edges belongingto more than 2 triangles in the triangulation.

IIT Kanpur 5

Introduction Previous Work New work Is it really good? References

Shortcomings

IIT Kanpur 6

Introduction Previous Work New work Is it really good? References

Shortcomings

The algorithm for optimization is a naive one which isexponential in the number of polygons.

IIT Kanpur 6

Introduction Previous Work New work Is it really good? References

Shortcomings

The algorithm for optimization is a naive one which isexponential in the number of polygons.

Not scalable.

IIT Kanpur 6

Introduction Previous Work New work Is it really good? References

Shortcomings

The algorithm for optimization is a naive one which isexponential in the number of polygons.

Not scalable.The paper proposes to reduce the running time by

considering a subset of triangles as possible candidatesin trianglulation. Turns out that Delaunay triangles cando the job, with only 1% of triangles lying outside theset of Delaunay triangles entered.

Reduces the running time, but no formal proof has beengiven.

IIT Kanpur 6

Introduction Previous Work New work Is it really good? References

Shortcomings

The algorithm for optimization is a naive one which isexponential in the number of polygons.

Not scalable.The paper proposes to reduce the running time by

considering a subset of triangles as possible candidatesin trianglulation. Turns out that Delaunay triangles cando the job, with only 1% of triangles lying outside theset of Delaunay triangles entered.

Reduces the running time, but no formal proof has beengiven.

IIT Kanpur 6

Introduction Previous Work New work Is it really good? References

References

An algorithm for triangulating multiple 3D polygons byM Zou, T Ju, N Carr Eurographics Symposium onGeometry Processing 2013.

Some images taken from Google Image Search, andsome fromhttp://www.cse.wustl.edu/ zoum/projects/TriMultPoly.

IIT Kanpur 7

Thanks to Sharbat for the LATEXtemplate