Size Competitive Meshing without Large Angles

12 July 2007 Gary MillerOverlay Stitch Meshing

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Size Competitive Meshing Size Competitive Meshing without Large Angleswithout Large Angles

Gary L. MillerGary L. MillerCarnegie Mellon Carnegie Mellon

Computer ScienceComputer ScienceJoint work with Todd Joint work with Todd

Phillips and Don SheehyPhillips and Don Sheehy


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The ProblemThe ProblemInput: A Planar Straight Line Input: A Planar Straight Line

GraphGraph


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GraphGraph

Output: A Conforming TriangulationOutput: A Conforming Triangulation


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GraphGraph



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GraphGraph


QualityQuality


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What is a quality triangle?What is a quality triangle?


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No Small AnglesNo Small Angles No Large AnglesNo Large Angles


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Implies triangles Implies triangles havehave

bounded bounded aspect aspect ratioratio..


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bounded bounded aspect aspect ratioratio..Implies triangles Implies triangles havehave

bounded bounded largest largest anglesangles..


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Can be Can be efficientlyefficiently computed by computed by

Delaunay Delaunay RefinementRefinement


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Sufficient Sufficient for many for many applications.applications.


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Can be asymptotically Can be asymptotically smallersmaller then Delaunay then Delaunay

Refinement Refinement triangulations.triangulations.


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Can be asymptotically Can be asymptotically smallersmaller then Delaunay then Delaunay

Refinement Refinement triangulations.triangulations.

More More difficultdifficult to to analyze.analyze.


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In Defense of QualityIn Defense of Quality


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What went wrong?What went wrong?


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What went wrong?What went wrong?


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Interpolation ProblemInterpolation Problem

No Large AnglesNo Large Angles Large AnglesLarge Angles

Large angles give large HLarge angles give large H11 errors that FEMs try to minimize. errors that FEMs try to minimize.


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Paying for the spreadPaying for the spread


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LL

ss

Spread = L/sSpread = L/s


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Optimal No-Large-Angle TriangulationOptimal No-Large-Angle Triangulation


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Paying for the spreadPaying for the spreadWhat if we don’t allow small angles?What if we don’t allow small angles?


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Paying for the spreadPaying for the spreadWhat if we don’t allow small angles?What if we don’t allow small angles?


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O(L/s)O(L/s) triangles! triangles!

What if we don’t allow small angles?What if we don’t allow small angles?


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O(L/s)O(L/s) triangles! triangles!

What if we don’t allow small angles?What if we don’t allow small angles?


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Delaunay RefinementDelaunay Refinement


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Theorem: Delaunay Refinement on Theorem: Delaunay Refinement on point point setssets terminates and returns a terminates and returns a triangulation with triangulation with

• all angles at least 30-all angles at least 30- degrees degrees

• O(n log L/s) trianglesO(n log L/s) triangles

On Point Sets, we only pay for the log of the spread.On Point Sets, we only pay for the log of the spread.


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Paterson’s ExamplePaterson’s ExampleRequires O(nRequires O(n22) points.) points.

O(n) O(n) pointpoint

ss

O(n) O(n) lineline

ss


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Paterson’s ExamplePaterson’s ExampleRequires Requires (n(n22) points.) points.


ss

O(n) O(n) lineline

ss


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ss

O(n) O(n) lineline

ss


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ss

O(n) O(n) lineline

ss


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ss

O(n) O(n) lineline

ss


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ss

O(n) O(n) lineline

ss


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Past ResultsPast Results• O(n) triangles with 90O(n) triangles with 90oo largest angles largest angles

for polygons with holes. for polygons with holes. [Bern, Mitchell, [Bern, Mitchell, Ruppert, 95]Ruppert, 95]

• ((nn22) lower bound for arbitrary ) lower bound for arbitrary PLSGs. PLSGs. [Paterson][Paterson]

• O(nO(n22) triangles with 132) triangles with 132oo angles on angles on PSLGs. PSLGs. [Tan, 96][Tan, 96]


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Past ResultsPast ResultsDelaunay Refinement MethodsDelaunay Refinement Methods No-Large-Angle MethodsNo-Large-Angle Methods

ProsPros ProsProsConsCons ConsCons

Good TheoryGood Theory

Optimal Optimal RuntimeRuntime

Graded MeshGraded Mesh

Simple to Simple to ImplementImplement

Esthetically Esthetically NiceNice

Huge Meshes Huge Meshes O(L/s)O(L/s)

Require Hacks to Require Hacks to handle small input handle small input

angles.angles.

Size depends on Size depends on smallest angle.smallest angle.

Smaller MeshesSmaller Meshes

Worst-Case Worst-Case OptimalOptimal


Not well-gradedNot well-graded

Complicated to Complicated to ImplementImplement


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angles.angles.







OUROUR


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angles.angles.




Only Worst-Case Only Worst-Case BoundsBounds



OUROUR


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angles.angles.


Smaller Smaller MeshesMeshes

Worst-Case Worst-Case Optimal sizeOptimal size




OUROUR

Graded on AverageGraded on Average

Log L/s -competitiveLog L/s -competitive


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angles.angles.


Smaller Smaller MeshesMeshes

Worst-Case Worst-Case Optimal sizeOptimal size




OUROUR

Graded on AverageGraded on Average

Log L/s -competitiveLog L/s -competitive

Our Angle bounds are not as good, 170Our Angle bounds are not as good, 170oo versus ~140 versus ~140oo


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Local Feature SizeLocal Feature Size

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

xxlfs(x)lfs(x)Note: lfs is defined on the whole plane.Note: lfs is defined on the whole plane.


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The OSM AlgorithmThe OSM Algorithm(Overlay Stitch Meshing)(Overlay Stitch Meshing)


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• An Overlay Edge is An Overlay Edge is keptkept if if1. It does not intersect the input,1. It does not intersect the input,

OROR


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OROR

2. It forms any 2. It forms any goodgood intersection with the intersection with the input.input.


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OROR

2. It forms any 2. It forms any goodgood intersection with the intersection with the input.input.

at least 30at least 30oo


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Angle GuaranteesAngle Guarantees


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We want to prove We want to prove that the stitch that the stitch

vertices that we vertices that we keep do not form keep do not form

bad angles.bad angles.


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An Overlay TriangleAn Overlay Triangle


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An Overlay EdgeAn Overlay Edge


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An Overlay EdgeAn Overlay Edge

Gap BallGap Ball


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A Good IntersectionA Good Intersection


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A Bad IntersectionA Bad Intersection


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How Bad can it How Bad can it be?be?


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Angle GuaranteesAngle GuaranteesHow Bad can it How Bad can it

be?be?


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be?be?


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be?be?

About 10About 10oo


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be?be?

About 10About 10oo

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


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Size BoundsSize BoundsHow big is the resulting triangulation?How big is the resulting triangulation?


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


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Overlay Phase: O(n log L/s) Overlay Phase: O(n log L/s) points addedpoints added


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Overlay Phase: O(n log L/s) Overlay Phase: O(n log L/s) pointspoints

Stitching Phase: O(Stitching Phase: O(ssEE lfs lfs00-1-1(z)dz) (z)dz)



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Ruppert’s IdeaRuppert’s Idea

rr

cc

r = r = (lfs(c))(lfs(c))

area(area() = ) = (lfs(c)(lfs(c)22))

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


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Ruppert’s IdeaRuppert’s Idea

rr

cc

r = r = (lfs(c))(lfs(c))

area(area() = ) = (lfs(c)(lfs(c)22))

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

CaveatCaveat: Only Works for well-graded meshes : Only Works for well-graded meshes

with bounded with bounded smallestsmallest angle. angle.


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ExtendingExtending Ruppert’s Idea Ruppert’s Idea# of triangles = # of triangles = ((ssss lfs(x,y) lfs(x,y)-2-2 dxdy) dxdy)


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ExtendingExtending Ruppert’s Idea Ruppert’s Idea

An input edge An input edge e e intersecting intersecting

the overlay meshthe overlay mesh



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An input edge An input edge e e intersecting intersecting

the overlay meshthe overlay mesh

# of triangles along # of triangles along ee = = ((sszz22 e e lfs(z) lfs(z)-1-1 dz) dz)



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

integrals with optimal.integrals with optimal.

# of triangles along # of triangles along ee = = ((sszz22 e e lfs(z) lfs(z)-1-1 dz) dz)



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Competitive AnalysisCompetitive AnalysisWarm-upWarm-up


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9090oo


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??oo


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90



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


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an optimalan optimal


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an optimalan optimal Every input edge is Every input edge is covered by empty covered by empty

lenses.lenses.


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Competitive AnalysisCompetitive Analysis


the optimalthe optimal Every input edge is Every input edge is covered by empty covered by empty

lenses.lenses.

This is what we will integrate over to get a lower bound.This is what we will integrate over to get a lower bound.


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

edge.edge.



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

edge.edge.

ee’’

In other words:In other words:



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ConclusionConclusion• A new algorithm for no-large-angle A new algorithm for no-large-angle

triangulation.triangulation.• The output has bounded degree The output has bounded degree

triangles.triangles.• The first log-competitive analysis for The first log-competitive analysis for

such an algorithm.such an algorithm.


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ConclusionConclusion• A new algorithm for no-large-angle A new algorithm for no-large-angle

triangulation.triangulation.• The output has bounded degree The output has bounded degree

triangles.triangles.• The first log-competitive analysis for The first log-competitive analysis for

such an algorithm.such an algorithm.• We used simple calculus to bound We used simple calculus to bound

mesh sizes.mesh sizes.


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Where to go from here?Where to go from here?• 3D?3D?


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Where to go from here?Where to go from here?• 3D?3D?• Better angle bounds?Better angle bounds?


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Where to go from here?Where to go from here?• 3D?3D?• Better angle bounds?Better angle bounds?• Find a constant competitive Find a constant competitive

algorithm.algorithm.


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Where to go from here?Where to go from here?• 3D?3D?• Better angle bounds?Better angle bounds?• Find a constant competitive Find a constant competitive

algorithm.algorithm.

ThanksThanks


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ee’’

sszz22 e’ e’ 1/lfs(z) dz = O(log (L/s)) 1/lfs(z) dz = O(log (L/s))


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ee’’

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



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ee’’

Parametrize e’ as [0, Parametrize e’ as [0, ll]]

11stst trick: lfs \ge s everywhere trick: lfs \ge s everywhere



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Competitive AnalysisCompetitive Analysissszz22 e’ e’ 1/lfs(z) dz = O(log (L/s)) 1/lfs(z) dz = O(log (L/s))

ee’’

Parametrize e’ as e’(t) for tParametrize e’ as e’(t) for t22 [0, [0, ll] ] ll = length(e’) = length(e’)

11stst trick: lfs trick: lfs ¸̧ s everywhere s everywhere

22ndnd trick: lfs trick: lfs ¸̧ ct for t ct for t22 [0, [0,ll/2]/2]

tt


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Competitive AnalysisCompetitive Analysissszz22 e’ e’ 1/lfs(z) dz = O(log (L/s)) 1/lfs(z) dz = O(log (L/s))

ee’’

Parametrize e’ as Parametrize e’ as e’(t)e’(t)11stst trick: lfs trick: lfs ¸̧ s everywhere s everywhere

22ndnd trick: lfs trick: lfs ¸̧ ct for ct for tt22 [0, [0,ll/2]/2]

tt

zz22 e’ e’ 1/lfs(z) dz 1/lfs(z) dz ·· 2 2 ss 00 1/s + 2 1/s + 2 ssll/2/2

1/x dx 1/x dx

= O(1) + O(log(= O(1) + O(log(ll/2) – log s)/2) – log s)

= O(log = O(log ll/s)/s)

= O(log L/s)= O(log L/s)

Size Competitive Meshing without Large Angles

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