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Overlay Stitch Meshing. A competitive algorithm for no-large-angle triangulation. Don Sheehy Joint work with Gary Miller and Todd Phillips To appear at ICALP 2007. The Problem. Input: A Planar Straight Line Graph. The Problem. Input: A Planar Straight Line Graph.

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Overlay Stitch Meshing

Don Sheehy

Overlay Stitch Meshing


A competitive algorithm for no-large-angle triangulation

Don Sheehy

Joint work with Gary Miller and Todd Phillips

To appear at ICALP 2007

Don Sheehy

Overlay Stitch Meshing


The Problem

Input: A Planar Straight Line Graph

Don Sheehy

Overlay Stitch Meshing


The Problem

Input: A Planar Straight Line Graph

Output: A Conforming Triangulation

Don Sheehy

Overlay Stitch Meshing


The Problem

Input: A Planar Straight Line Graph

Output: A Conforming Triangulation

Don Sheehy

Overlay Stitch Meshing


Why would you want to do that?

Don Sheehy

Overlay Stitch Meshing


Why would you want to do that?

Don Sheehy

Overlay Stitch Meshing


Why would you want to do that?

Don Sheehy

Overlay Stitch Meshing


Why would you want to do that?

Don Sheehy

Overlay Stitch Meshing


Why would you want to do that?

Don Sheehy

Overlay Stitch Meshing


Why would you want to do that?

Don Sheehy

Overlay Stitch Meshing


Don Sheehy

Overlay Stitch Meshing


Don Sheehy

Overlay Stitch Meshing


Don Sheehy

Overlay Stitch Meshing


Don Sheehy

Overlay Stitch Meshing


Don Sheehy

Overlay Stitch Meshing


Don Sheehy

Overlay Stitch Meshing


What went wrong?

Don Sheehy

Overlay Stitch Meshing


?

?

What if you don’t know the function?

Don Sheehy

Overlay Stitch Meshing


2 Definitions of Quality

1. No Large Angles [Babuska, Aziz 1976]

Don Sheehy

Overlay Stitch Meshing


2 Definitions of Quality

1. No Large Angles [Babuska, Aziz 1976]

2. No Small Angles

Don Sheehy

Overlay Stitch Meshing


No Small Angles

Don Sheehy

Overlay Stitch Meshing


No Small Angles

You may have heard of these before.

Delaunay Refinement

Sparse Voronoi Refinement

Quadtree

Don Sheehy

Overlay Stitch Meshing


Paying for the spread

Don Sheehy

Overlay Stitch Meshing


Paying for the spread

Spread = L/s

L

s

Don Sheehy

Overlay Stitch Meshing


Paying for the spread

Optimal No-Large-Angle Triangulation

Don Sheehy

Overlay Stitch Meshing


Paying for the spread

What if we don’t allow small angles?

Don Sheehy

Overlay Stitch Meshing


Paying for the spread

What if we don’t allow small angles?

Don Sheehy

Overlay Stitch Meshing


Paying for the spread

What if we don’t allow small angles?

O(L/s) triangles!

Don Sheehy

Overlay Stitch Meshing


Paying for the spread

What if we don’t allow small angles?

Fact: For inputs with NO edges, no-small-angle meshing algorithms produce output with O(n log L/s) size and angles between 30o and 120o

O(L/s) triangles!

Don Sheehy

Overlay Stitch Meshing


What to do?

Don Sheehy

Overlay Stitch Meshing


What to do?

Small input angles can force even smaller ouput angles. [Shewchuk ’02]

Don Sheehy

Overlay Stitch Meshing


No Large Angles

Don Sheehy

Overlay Stitch Meshing


Polygons with Holes

[Bern, Mitchell, Ruppert 95] –

- All triangles are nonobtuse.

- Output has O(n) triangles.

Don Sheehy

Overlay Stitch Meshing


Polygons with Holes

[Bern, Mitchell, Ruppert 95] –

- All triangles are nonobtuse.

- Output has O(n) triangles.

Does not work for arbitrary PSLGs

Don Sheehy

Overlay Stitch Meshing


Paterson’s Example

Requires (n2) points.

O(n) points

O(n) lines

Don Sheehy

Overlay Stitch Meshing


Paterson’s Example

Requires (n2) points.

O(n) points

O(n) lines

Don Sheehy

Overlay Stitch Meshing


Paterson’s Example

Requires (n2) points.

O(n) points

O(n) lines

Don Sheehy

Overlay Stitch Meshing


Paterson’s Example

Requires (n2) points.

O(n) points

O(n) lines

Don Sheehy

Overlay Stitch Meshing


Paterson’s Example

Requires (n2) points.

O(n) points

O(n) lines

Don Sheehy

Overlay Stitch Meshing


Paterson’s Example

Requires (n2) points.

O(n) points

O(n) lines

Don Sheehy

Overlay Stitch Meshing


Propagating Paths

Don Sheehy

Overlay Stitch Meshing


Propagating Paths

Don Sheehy

Overlay Stitch Meshing


Propagating Paths

Don Sheehy

Overlay Stitch Meshing


Propagating Paths

Don Sheehy

Overlay Stitch Meshing


Propagating Paths

Don Sheehy

Overlay Stitch Meshing


Propagating Paths

First introduced by Mitchell [93]

Don Sheehy

Overlay Stitch Meshing


Propagating Paths

First introduced by Mitchell [93]

Later Improved by Tan [96]

Don Sheehy

Overlay Stitch Meshing


Propagating Paths

First introduced by Mitchell [93]

Later Improved by Tan [96]

Worst Case Optimal Size O(n2)

Angle bounds: 132o

Don Sheehy

Overlay Stitch Meshing


The OSM Algorithm(Overlay Stitch Meshing)

Don Sheehy

Overlay Stitch Meshing


The OSM Algorithm(Overlay Stitch Meshing)

  • Results:

  • Angles bounded by 170o

  • O(log L/s) competitive size

Don Sheehy

Overlay Stitch Meshing


The OSM Algorithm(Overlay Stitch Meshing)

Don Sheehy

Overlay Stitch Meshing


The OSM Algorithm(Overlay Stitch Meshing)

Don Sheehy

Overlay Stitch Meshing


The OSM Algorithm(Overlay Stitch Meshing)

Don Sheehy

Overlay Stitch Meshing


The OSM Algorithm(Overlay Stitch Meshing)

Don Sheehy

Overlay Stitch Meshing


The OSM Algorithm(Overlay Stitch Meshing)

Don Sheehy

Overlay Stitch Meshing


The OSM Algorithm(Overlay Stitch Meshing)

Don Sheehy

Overlay Stitch Meshing


The OSM Algorithm(Overlay Stitch Meshing)

Don Sheehy

Overlay Stitch Meshing


The OSM Algorithm(Overlay Stitch Meshing)

Don Sheehy

Overlay Stitch Meshing


The OSM Algorithm(Overlay Stitch Meshing)

Don Sheehy

Overlay Stitch Meshing


The OSM Algorithm(Overlay Stitch Meshing)

Don Sheehy

Overlay Stitch Meshing


The OSM Algorithm(Overlay Stitch Meshing)

Don Sheehy

Overlay Stitch Meshing


The OSM Algorithm(Overlay Stitch Meshing)

Don Sheehy

Overlay Stitch Meshing


The OSM Algorithm(Overlay Stitch Meshing)

  • An Overlay Edge is kept if

    1. It does not intersect the input,

    OR

Don Sheehy

Overlay Stitch Meshing


The OSM Algorithm(Overlay Stitch Meshing)

  • An Overlay Edge is kept if

    1. It does not intersect the input,

    OR

    2. It forms a good intersection with the input.

Don Sheehy

Overlay Stitch Meshing


The OSM Algorithm(Overlay Stitch Meshing)

  • An Overlay Edge is kept if

    1. It does not intersect the input,

    OR

    2. It forms a good intersection with the input.

at least 30o

Don Sheehy

Overlay Stitch Meshing


Angle Guarantees

Don Sheehy

Overlay Stitch Meshing


Angle Guarantees

Don Sheehy

Overlay Stitch Meshing


Angle Guarantees

We want to show that angles at stitch vertices are bounded by 170o

Don Sheehy

Overlay Stitch Meshing


Angle Guarantees

An Overlay Triangle

Don Sheehy

Overlay Stitch Meshing


Angle Guarantees

An Overlay Triangle

An Overlay Edge

Don Sheehy

Overlay Stitch Meshing


Angle Guarantees

Gap Ball

An Overlay Triangle

An Overlay Edge

Don Sheehy

Overlay Stitch Meshing


Angle Guarantees

A Good Intersection

Don Sheehy

Overlay Stitch Meshing


Angle Guarantees

A Bad Intersection

Don Sheehy

Overlay Stitch Meshing


Angle Guarantees

A Bad Intersection

Don Sheehy

Overlay Stitch Meshing


Angle Guarantees

A Bad Intersection

Don Sheehy

Overlay Stitch Meshing


Angle Guarantees

How Bad can it be?

A Bad Intersection

Don Sheehy

Overlay Stitch Meshing


Angle Guarantees

How Bad can it be?

Don Sheehy

Overlay Stitch Meshing


Angle Guarantees

How Bad can it be?

Don Sheehy

Overlay Stitch Meshing


Angle Guarantees

How Bad can it be?

About 10o

Don Sheehy

Overlay Stitch Meshing


Angle Guarantees

How Bad can it be?

About 10o

About 170o

Don Sheehy

Overlay Stitch Meshing


Angle Guarantees

How Bad can it be?

Theorem: If any input edge makes a good intersection with an overlay edge then any other intersection on that edge is not too bad.

About 10o

About 170o

Don Sheehy

Overlay Stitch Meshing


Size Bounds

Don Sheehy

Overlay Stitch Meshing


Size Bounds

How big is the resulting triangulation?

Don Sheehy

Overlay Stitch Meshing


Size Bounds

How big is the resulting triangulation?

Goal: log(L/s)-competitive with optimal

Don Sheehy

Overlay Stitch Meshing


Local Feature Size

lfs(x) = distance to second nearest input vertex.

Note: lfs is defined on the whole plane before we do any meshing.

x

lfs(x)

Don Sheehy

Overlay Stitch Meshing


Ruppert’s Idea

  • r = Q(lfs(c))

  • area(D) = Q(lfs(c)2)

  • # of triangles = (ss 1/lfs(x,y)2 dxdy)

r

c

Don Sheehy

Overlay Stitch Meshing


Ruppert’s Idea

  • r = Q(lfs(c))

  • area(D) = Q(lfs(c)2)

  • # of triangles = (ss 1/lfs(x,y)2 dxdy)

r

c

dx

This represents a (dxdy/lfs(x,y)2) fraction of the triangle.

dy

Don Sheehy

Overlay Stitch Meshing


Ruppert’s Idea

  • r = Q(lfs(c))

  • area(D) = Q(lfs(c)2)

  • # of triangles = (ss lfs(x,y)-2 dxdy)

r

c

Caveat: Only Works for meshes

with bounded smallest angle.

Don Sheehy

Overlay Stitch Meshing


Extending Ruppert’s Idea

# of triangles = (ss lfs(x,y)-2 dxdy)

Don Sheehy

Overlay Stitch Meshing


Extending Ruppert’s Idea

# of triangles = (ss lfs(x,y)-2 dxdy)

An input edge e intersecting

the overlay mesh

Don Sheehy

Overlay Stitch Meshing


Extending Ruppert’s Idea

# of triangles = (ss lfs(x,y)-2 dxdy)

# of triangles along e = (sz2 e lfs(z)-1 dz)

An input edge e intersecting

the overlay mesh

Don Sheehy

Overlay Stitch Meshing


Extending Ruppert’s Idea

# of triangles = (ss lfs(x,y)-2 dxdy)

# of triangles along e = (sz2 e lfs(z)-1 dz)

Now we can compute the size of our output mesh. We just need to compute these integrals.

Don Sheehy

Overlay Stitch Meshing


Output Size

Overlay Mesh Size = (ss lfs(x,y)-2 dxdy)

Stitch Vertices Added = (sz2 E lfs(z)-1 dz)

Don Sheehy

Overlay Stitch Meshing


Output Size

Overlay Mesh Size = (ss lfs(x,y)-2 dxdy)

O(n log L/s)

Stitch Vertices Added = (sz2 E lfs(z)-1 dz)

O(log L/s)£ |OPT|

Don Sheehy

Overlay Stitch Meshing


Output Size

Overlay Mesh Size = (ss lfs(x,y)-2 dxdy)

O(n log L/s)

Stitch Vertices Added = (sz2 E lfs(z)-1 dz)

O(log L/s)£ |OPT|

We’ll Prove this now

Don Sheehy

Overlay Stitch Meshing


What can we say about the optimal no-large-angle triangulation?

Don Sheehy

Overlay Stitch Meshing


Competitive Analysis

Warm-up

Don Sheehy

Overlay Stitch Meshing


Competitive Analysis

Warm-up

90o

Don Sheehy

Overlay Stitch Meshing


Competitive Analysis

Warm-up

?o

Don Sheehy

Overlay Stitch Meshing


Competitive Analysis

Warm-up

Don Sheehy

Overlay Stitch Meshing


Competitive Analysis

Warm-up

Don Sheehy

Overlay Stitch Meshing


Competitive Analysis

Warm-up

Don Sheehy

Overlay Stitch Meshing


Competitive Analysis

Warm-up

Don Sheehy

Overlay Stitch Meshing


Competitive Analysis

Warm-up

Don Sheehy

Overlay Stitch Meshing


Competitive Analysis

Warm-up

Suppose we have a triangulation with all angles at least 170o.

Don Sheehy

Overlay Stitch Meshing


Competitive Analysis

Warm-up

Suppose we have a triangulation with all angles at least 170o.

the optimal

Don Sheehy

Overlay Stitch Meshing


Competitive Analysis

Warm-up

Suppose we have a triangulation with all angles at least 170o.

the optimal

Every input edge is covered by empty lenses.

Don Sheehy

Overlay Stitch Meshing


Competitive Analysis

This is what we have to integrate over.

Suppose we have a triangulation with all angles at least 170o.

the optimal

Every input edge is covered by empty lenses.

Don Sheehy

Overlay Stitch Meshing


Competitive Analysis

This is what we have to integrate over.

We show that in each lens, we put O(log (L/s)) points on the edge.

Don Sheehy

Overlay Stitch Meshing


Competitive Analysis

This is what we have to integrate over.

e’

We show that in each lens, we put O(log (L/s)) points on the edge.

In other words:

se’ 1/lfs(z) dz = O(log (L/s))

Don Sheehy

Overlay Stitch Meshing


Competitive Analysis

sz2 e’ 1/lfs(z) dz = O(log (L/s))

e’

Don Sheehy

Overlay Stitch Meshing


Competitive Analysis

sz2 e’ 1/lfs(z) dz = O(log (L/s))

e’

Parametrize e’ as [0, l] where l = length(e’)

Don Sheehy

Overlay Stitch Meshing


Competitive Analysis

sz2 e’ 1/lfs(z) dz = O(log (L/s))

e’

Parametrize e’ as [0, l]

1st trick: lfs ¸ s everywhere

Don Sheehy

Overlay Stitch Meshing


Competitive Analysis

sz2 e’ 1/lfs(z) dz = O(log (L/s))

t

e’

Parametrize e’ as [0, l]

1st trick: lfs ¸ s everywhere

2nd trick: lfs ¸ ct for t2 [0,l/2]

Don Sheehy

Overlay Stitch Meshing


Competitive Analysis

sz2 e’ 1/lfs(z) dz = O(log (L/s))

t

e’

sz2 e’ 1/lfs(z) dz · 2s0s 1/s + 2ssl/2 1/x dx

= O(1) + O(log l/s)

= O(log L/s)

Parametrize e’ as [0, l]

1st trick: lfs ¸ s everywhere

2nd trick: lfs ¸ ct for t2 [0,l/2]

Don Sheehy

Overlay Stitch Meshing


Conclusions

Don Sheehy

Overlay Stitch Meshing


Conclusion

  • A new algorithm for no-large-angle triangulation.

  • The output has bounded degree triangles.

  • The first log-competitive analysis for such an algorithm.

Don Sheehy

Overlay Stitch Meshing


Conclusion

  • A new algorithm for no-large-angle triangulation.

  • The output has bounded degree triangles.

  • The first log-competitive analysis for such an algorithm.

  • We used some of the high school calculus you thought you forgot.

Don Sheehy

Overlay Stitch Meshing


Where to go from here?

  • 3D?

Don Sheehy

Overlay Stitch Meshing


Where to go from here?

  • 3D?

  • Better angle bounds?

Don Sheehy

Overlay Stitch Meshing


Where to go from here?

  • 3D?

  • Better angle bounds?

  • Leverage lower bound technology for constant competitive algorithm.

Don Sheehy

Overlay Stitch Meshing


Thank you.

Don Sheehy

Overlay Stitch Meshing


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