Gary%20Miller

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Size Competitive Meshing without Large Angles
Gary L. Miller Carnegie Mellon Computer
Science Joint work with Todd Phillips and Don
Sheehy
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The Problem
Input A Planar Straight Line Graph
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The Problem
Input A Planar Straight Line Graph
Output A Conforming Triangulation
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The Problem
Input A Planar Straight Line Graph
Output A Conforming Triangulation
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The Problem
Input A Planar Straight Line Graph
Output A Conforming Triangulation
Quality
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What is a quality triangle?
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What is a quality triangle?
No Large Angles
No Small Angles
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What is a quality triangle?
No Large Angles
No Small Angles
Implies triangles have bounded aspect ratio.
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What is a quality triangle?
No Large Angles
No Small Angles
Implies triangles have bounded aspect ratio.
Implies triangles have bounded largest angles.
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What is a quality triangle?
No Large Angles
No Small Angles
Implies triangles have bounded aspect ratio.
Implies triangles have bounded largest angles.
Can be efficiently computed by Delaunay Refinement
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What is a quality triangle?
No Large Angles
No Small Angles
Sufficient for many applications.
Implies triangles have bounded aspect ratio.
Implies triangles have bounded largest angles.
Can be efficiently computed by Delaunay Refinement
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What is a quality triangle?
No Large Angles
No Small Angles
Sufficient for many applications.
Implies triangles have bounded aspect ratio.
Can be asymptotically smaller then Delaunay
Refinement triangulations.
Implies triangles have bounded largest angles.
Can be efficiently computed by Delaunay Refinement
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What is a quality triangle?
No Large Angles
No Small Angles
Sufficient for many applications.
Implies triangles have bounded aspect ratio.
Can be asymptotically smaller then Delaunay
Refinement triangulations.
Implies triangles have bounded largest angles.
More difficult to analyze.
Can be efficiently computed by Delaunay Refinement
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In Defense of Quality
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In Defense of Quality
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In Defense of Quality
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In Defense of Quality
What went wrong?
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In Defense of Quality
What went wrong?
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Interpolation Problem
No Large Angles
Large Angles
Large angles give large H1 errors that FEMs try
to minimize.
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Paying for the spread
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Paying for the spread
Spread L/s
L
s
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Paying for the spread
Optimal No-Large-Angle Triangulation
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Paying for the spread
What if we dont allow small angles?
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Paying for the spread
What if we dont allow small angles?
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Paying for the spread
What if we dont allow small angles?
O(L/s) triangles!
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Paying for the spread
What if we dont allow small angles?
O(L/s) triangles!
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Delaunay Refinement
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Delaunay Refinement
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Delaunay Refinement

• Theorem Delaunay Refinement on point sets
  terminates and returns a triangulation with
• all angles at least 30-e degrees
• O(n log L/s) triangles

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Patersons Example

• Requires O(n2) points.

O(n) points
O(n) lines
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Patersons Example

• Requires ?(n2) points.

O(n) points
O(n) lines
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Patersons Example

• Requires ?(n2) points.

O(n) points
O(n) lines
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Patersons Example

• Requires ?(n2) points.

O(n) points
O(n) lines
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Patersons Example

• Requires ?(n2) points.

O(n) points
O(n) lines
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Patersons Example

• Requires ?(n2) points.

O(n) points
O(n) lines
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Past Results

• O(n) triangles with 90o largest angles for
  polygons with holes. Bern, Mitchell, Ruppert,
  95
• ?(n2) lower bound for arbitrary PLSGs.
  Paterson
• O(n2) triangles with 132o angles on PSLGs. Tan,
  96

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Past Results
Delaunay Refinement Methods
No-Large-Angle Methods
Cons
Pros
Cons
Pros
Smaller Meshes Not well-graded Complicated to
Implement
Good Theory Optimal Runtime Graded Mesh Simple to
Implement Esthetically Nice
Huge Meshes O(L/s) Require Hacks to handle small
input angles. Size depends on smallest angle.
Smaller Meshes Worst-Case Optimal

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Past Results
OUR
Delaunay Refinement Methods
No-Large-Angle Methods
Cons
Pros
Cons
Pros
Smaller Meshes Not well-graded Complicated to
Implement
Good Theory Optimal Runtime Graded Mesh Simple to
Implement Esthetically Nice
Huge Meshes O(L/s) Require Hacks to handle small
input angles. Size depends on smallest angle.
Smaller Meshes Worst-Case Optimal
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Past Results
OUR
Delaunay Refinement Methods
No-Large-Angle Methods
Cons
Pros
Cons
Pros
Only Worst-Case Bounds Not well-graded Complicated
to Implement
Good Theory Optimal Runtime Graded Mesh Simple to
Implement Esthetically Nice
Huge Meshes O(L/s) Require Hacks to handle small
input angles. Size depends on smallest angle.
Smaller Meshes Worst-Case Optimal
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Past Results
OUR
Delaunay Refinement Methods
No-Large-Angle Methods
Cons
Pros
Cons
Pros
Only Worst-Case Bounds Not well-graded Complicated
to Implement
Good Theory Optimal Runtime Graded Mesh Simple to
Implement Esthetically Nice
Huge Meshes O(L/s) Require Hacks to handle small
input angles. Size depends on smallest angle.
Smaller Meshes Worst-Case Optimal size
Log L/s -competitive
Graded on Average
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Past Results
OUR
Delaunay Refinement Methods
No-Large-Angle Methods
Cons
Pros
Cons
Pros
Only Worst-Case Bounds Not well-graded Complicated
to Implement
Good Theory Optimal Runtime Graded Mesh Simple to
Implement Esthetically Nice
Huge Meshes O(L/s) Require Hacks to handle small
input angles. Size depends on smallest angle.
Smaller Meshes Worst-Case Optimal size
Log L/s -competitive
Graded on Average
Our Angle bounds are not as good, 170o versus
140o
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Local Feature Size
lfs(x) distance to second nearest vertex.
Note lfs is defined on the whole plane.
x
lfs(x)
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The OSM Algorithm(Overlay Stitch Meshing)
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The OSM Algorithm(Overlay Stitch Meshing)
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The OSM Algorithm(Overlay Stitch Meshing)
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The OSM Algorithm(Overlay Stitch Meshing)
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The OSM Algorithm(Overlay Stitch Meshing)
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The OSM Algorithm(Overlay Stitch Meshing)
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The OSM Algorithm(Overlay Stitch Meshing)
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The OSM Algorithm(Overlay Stitch Meshing)
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The OSM Algorithm(Overlay Stitch Meshing)
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The OSM Algorithm(Overlay Stitch Meshing)
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The OSM Algorithm(Overlay Stitch Meshing)
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The OSM Algorithm(Overlay Stitch Meshing)
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The OSM Algorithm(Overlay Stitch Meshing)
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The OSM Algorithm(Overlay Stitch Meshing)

• An Overlay Edge is kept if
• 1. It does not intersect the input,
• OR

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The OSM Algorithm(Overlay Stitch Meshing)

• An Overlay Edge is kept if
• 1. It does not intersect the input,
• OR
• 2. It forms any good intersection with the input.

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The OSM Algorithm(Overlay Stitch Meshing)

• An Overlay Edge is kept if
• 1. It does not intersect the input,
• OR
• 2. It forms any good intersection with the input.

at least 30o
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Angle Guarantees
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Angle Guarantees
We want to prove that the stitch vertices that we
keep do not form bad angles.
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Angle Guarantees
An Overlay Triangle
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Angle Guarantees
An Overlay Triangle
An Overlay Edge
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Angle Guarantees
Gap Ball
An Overlay Triangle
An Overlay Edge
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Angle Guarantees
A Good Intersection
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Angle Guarantees
A Bad Intersection
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Angle Guarantees
A Bad Intersection
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Angle Guarantees
A Bad Intersection
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Angle Guarantees
How Bad can it be?
A Bad Intersection
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Angle Guarantees
How Bad can it be?
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Angle Guarantees
How Bad can it be?
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Angle Guarantees
How Bad can it be?
About 10o
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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
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Size Bounds
How big is the resulting triangulation?
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Size Bounds
How big is the resulting triangulation?
Goal log(L/s)-competitive with optimal
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Size Bounds
How big is the resulting triangulation?
Goal log(L/s)-competitive with optimal
Overlay Phase O(n log L/s) points added
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Size Bounds
How big is the resulting triangulation?
Goal log(L/s)-competitive with optimal
Overlay Phase O(n log L/s) points
Stitching Phase O(sE lfs0-1(z)dz)
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Rupperts Idea

• r Q(lfs(c))
• area(D) Q(lfs(c)2)
• of triangles ?(ss lfs(x,y)-2 dxdy)

r
c
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Rupperts 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 well-graded meshes with
bounded smallest angle.
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Extending Rupperts Idea
of triangles ?(ss lfs(x,y)-2 dxdy)
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Extending Rupperts Idea
of triangles ?(ss lfs(x,y)-2 dxdy)
An input edge e intersecting the overlay mesh
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Extending Rupperts 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
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Extending Rupperts 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 compare these integrals with
optimal.
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Competitive Analysis
Warm-up
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Competitive Analysis
Warm-up
90o
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Competitive Analysis
Warm-up
?o
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Competitive Analysis
Warm-up
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Competitive Analysis
Warm-up
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Competitive Analysis
Warm-up
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Competitive Analysis
Warm-up
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Competitive Analysis
Warm-up
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Competitive Analysis
Warm-up
Suppose we have a triangulation with all angles
at least 170o.
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Competitive Analysis
Warm-up
Suppose we have a triangulation with all angles
at least 170o.
an optimal
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Competitive Analysis
Warm-up
Suppose we have a triangulation with all angles
at least 170o.
an optimal
Every input edge is covered by empty lenses.
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Competitive Analysis
This is what we will integrate over to get a
lower bound.
Suppose we have a triangulation with all angles
at least 170o.
the optimal
Every input edge is covered by empty lenses.
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Competitive Analysis
This is what we will integrate over to get a
lower bound.
We show that in each lens, we put O(log (L/s))
points on the edge.
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Competitive Analysis
This is what we will integrate over to get a
lower bound.
e
We show that in each lens, we put O(log (L/s))
points on the edge.
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Conclusion

• A new algorithm for no-large-angle triangulation.
• The output has bounded degree triangles.
• The first log-competitive analysis for such an
  algorithm.

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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 simple calculus to bound mesh sizes.

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Where to go from here?

• 3D?

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Where to go from here?

• 3D?
• Better angle bounds?

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Where to go from here?

• 3D?
• Better angle bounds?
• Find a constant competitive algorithm.

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Where to go from here?

• 3D?
• Better angle bounds?
• Find a constant competitive algorithm.

Thanks
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Competitive Analysis
sz2 e 1/lfs(z) dz O(log (L/s))
e
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Competitive Analysis
sz2 e 1/lfs(z) dz O(log (L/s))
e
Parametrize e as 0, l where l length(e)
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Competitive Analysis
sz2 e 1/lfs(z) dz O(log (L/s))
e
Parametrize e as 0, l
1st trick lfs \ge s everywhere
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Competitive Analysis
sz2 e 1/lfs(z) dz O(log (L/s))
t
e
Parametrize e as e(t) for t2 0, l l
length(e)
1st trick lfs s everywhere
2nd trick lfs ct for t2 0,l/2
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Competitive Analysis
sz2 e 1/lfs(z) dz O(log (L/s))
t
e
Parametrize e as e(t)
? z2 e 1/lfs(z) dz 2 ?s 0 1/s 2 ? sl/2 1/x
dx O(1) O(log(l/2) log s) O(log l/s)
O(log L/s)
1st trick lfs s everywhere
2nd trick lfs ct for t2 0,l/2