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DiscreteMappings
Yaron Lipman
WeizmannInstituteofScience
12018AMSshortcourse
DiscreteDifferentialGeometry
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Surfacesastriangulations
2
• Trianglesstitchedtobuildasurface.
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Surfacesastriangulations
3
• Trianglesstitchedtobuildasurface.
• ! = ($, &, ');• $ = )* , & = +*, , ' = -*,. .
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Surfacesastriangulations
4
• ! = ($, &, ')
• Rulesforstitchingtriangles:
1. ! isasimplicialcomplex.
2. link )* =∪ 789:∈<+,. isasimpleclosedpolygon.
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Surfacesastriangulations
5
• Definition. Asurfacetriangulation isatriplet! = ($, &, ') satisfyingthestitchingrules.
• Forsurfacetriangulationwithboundary,replacestitchingrule2with
2.link()*) isasimple(closedornot)polygon.
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Surfacesastriangulations
6
• Theboundaryofsurfacetriangulationisa1Dsimplicialcomplex!= = ($=, &=).
• Theinteriorverticesaredenote$> = $ ∖ $=.
• Definition.! = ($, &, ') isconnectedifitisconnectedasagraph($, &).Itis@-connectedifitcannotbedisconnectedbyremoving@ − 1 vertices.
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Surfacesastriangulations
7
• Lemma[Floater’03].If! = ($, &, ') is3-connectedthenanyinteriorvertexcanbeconnectedtoanyothervertex(includingboundary)withaninteriorpath.
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Discretemappings
8
• Definition. Asimplicialmap-:! → ℝE istheuniquepiecewise-linearextensionof
avertexmap-F: $ → ℝE.
• WhenG = 1 wecall- asimplicialfunction.
H =IJ*)*
�
*
↦ - H =IJ*N*
�
*
J* ≥ 0,IJ*
�
*
= 1
H-(H)
)*
N*
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Thediscretemappingproblem
9
• Problem. Giventwotopologicallyequivalentsurfacetriangulations!Q,!R anda
setofcorrespondinglandmarks H*, S* *∈> ⊂ !Q×!R computea“nice”
simplicialhomeomorphism-:!Q → !R.
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Thediscretemappingproblem
10
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Thediscretemappingproblem
11
• Difficulty. Requiresfindingacommonisomorphiccommontriangulation,a
combinatorialproblem!
• Idea. Considermapping!Q,!R toacanonicaldomainV,
-Q:!Q → V and-R:!R → V andconstruct- as- = -RWQ ∘ -Q.
V
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Thediscretemappingproblem
12
• Twoquestions:
• HowtochooseV?
• HowtocomputethesimplicialmapontoV?
V
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Convexcombinationmappings
13
• AtechniquetomapasurfacetriangulationtoℝR.
• Definition. GivenaselectionofaweightperedgeY*, > 0,aconvexcombination
mapping -:! → ℝR isasimplicialmapmappingeachinteriorvertex)* ∈ $> toaplanarpointN* ∈ ℝR sothat
∑ Y*, N, − N* = 0�,∈\8 ,
where\* = ] +*, ∈ & .
• Thispropertyiscalledconvexcombinationproperty↔meanvalueproperty.
N*N,
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Discretemaximumprinciple
14
• Theconvexcombinationpropertyisadiscreteversionofthemeanvalueproperty
ofhamornic functions.
• Theorem(Discretemaximumprinciple).Letℎ:! → ℝ beaconvexcombination
functionand! a3-connectedsurfacetriangulation.Let)* ∈ $>.Then,Ifℎ* = min
,ℎ, orℎ* = max
,ℎ, thenℎ isconstant.
Inparticularℎ achievesitsextremepointontheboundary.
ℎ*
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Convexcombinationmappings
15
• CCMareingeneralnothomeomorphisms,e.g.,theconstantCCM.
• However,withcertainboundaryconditionsandtargetdomainsc CCMare
guaranteedtobehomeomorphic.
• Wewillexploreafamilyofsuchtargetdomains:
ℱ = V
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ℱ forhomeomorphicCCM
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Firstmembersofℱ17
• V isaconvexpolygonaldomaininℝR.
• Hintfromanalysis:
Theorem[Rado-Kneser-Choquet]:Let-: e → ℝR beahamornic mapwhere
-|gh isahomeomorphismontotheboundaryofaconvexregion.Then,- is
homeomorphism.
-e
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CCMintoconvexpolygonaldomain
18
• Theorem(Tutte,Floater).Let! = ($, &, ') bea3-connectedsurfacetriangulationhomeomorphictoadisk.Let-:! → ℝR beaCCMsuchthat-|ij
isa
homeomorphismtoaconvexpolygonenclosingadomainΩ.Then,-:! → Ω isahomeomorphism.
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ComputingCCM
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!
∑ Y*, N, − N*�,∈\8 = 0,)* ∈ $>
)*
N*
N* = l*,)* ∈ $=
)*N*
)*
N*
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Uniqueness
20
• Proposition. Thereisauniquesolutiontothelinearsystem.
• Proof. Considerasolutiontothehomogeneoussystem:
ConsiderfirstcoordinateH* ofN* = (H*, S*).ThisisaCCFhencesatisfiesdiscretemaximumprinciple.
IfH* ≠ 0 thereisanon-zerovalueattheboundary,contradiction.
∑ Y*, N, − N*�,∈\8 = 0,)* ∈ $>
N* = 0,)* ∈ $=
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?
Othermembersofℱ = V ?
TopologyTargetdomain
21
?
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conesector
open
disk
Euclideanconesurfaces
22
• Definition.acompactsurfaceV isaeuclidean conesurfaceifitisametricspace
locallyisometrictoanopendisk,acone,orasectorandthenumberofcone
pointsisfinite.
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Euclideanorbifolds
23
• Asubfamilyofeuclidean conesurfaces.
• Definition. Aeuclidean orbifoldV isasurfacedefinedasthequotientofℝR byasymmetrywallpapergroupn,thatis
V= ℝR/n.
• ThepointofV aretheorbitsofn,thatis N = p(N) p ∈ n .
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Euclideanorbifolds
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Symmetryofthings[Strauss,Burgiel,Conway]
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25
Euclideanorbifolds andtheir
fundamentaldomains
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CCMintoeuclidean orbifolds
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• Theorem(orbifold Tutte).Let! = ($, &, ') bea3-connectedsurfacetriangulationhomeomorphictooneoftheeuclidean orbifoldsV withq cones.
Letr = {)t} ⊂ $ beasetofq distinctvertices.
Let-:! → V beaCCMsuchthatthe-|v isabijectionbetweenr andthecones
ofV.Then,-:! → Ω isahomeomorphism.
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ComputingCMMintoanorbifold
• First,cut! = ($, &, ') toadisk-typetriangulation!w = $w, &w, 'w .• Second,computeasimplicialmapx:!w → ℝR asfollows.
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Computing
CMMinto
anorbifold
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I Y*, N, − N*
�
,∈y8
= 0
Nt = lt
I Y*, N, − N*
�
,∈y8
+ I Y*w,{**w N, − N*w
�
,∈y8|
= 0
N* − lt = {**w N*w − lt
)t Nt
)*
N*
N*w
{**w
lt
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ComputingCMMintoanorbifold
• Lastly,themap-:! → V isdefinedby- H = [x H ].
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Exampleoforbifold CCM
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Homeomorphism
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• Wewilloutlinetheideaoftheproof.
• Letx:!w → ℝR bethesolutiontothelinearsystempreviousdescribed.
• Step1.Buildabranchedcover!′′ to! bystitchingcopiesof!′ accordingtothegroupn.Considertheextensionx:!ww → ℝR.
• Step2.Showx:!ww → ℝR doesnotdegenerateandmaintainstheorientationof
at-leastonetriangle.
• Step3.Ifx doesnotdegenerateandmaintainsorientationofatriangle,itwillalso
notdegeneratenorfliporientationofanyneighbortriangle.
• Step4.Ifx:!ww → ℝR maintainsorientationofalltrianglesitisahomeomorphism.
Consequently-:! → V isahomeomorphism.
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Homeomorphism
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• Step1.Buildabranchedcover!′′ to! bystitchingcopiesof!′ accordingtothegroupn.Considertheextensionx:!ww → ℝR.AllverticessatisfytheCCP.
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Homeomorphism
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• Step2.Wewillshowastrongerclaim.Every(generic)pointintheplaneiscovered
byat-leastonepositivelyorientedtriangle.
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Homeomorphism
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• Step3.Ifx doesnotdegenerateandmaintainsorientationofatriangle,itwillalso
notdegeneratenorflipanyneighbortriangle.
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Homeomorphism
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• Step4.Ifx:!ww → ℝR maintainsorientationofalltrianglesitisahomeomorphism.
Consequently-:! → V isahomeomorphism.
• Repeatwindingnumberargumentbutnowweknowthatalltrianglesare
positivelyoriented.
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ComparisonofCCM
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Variational principle
39
• WhenY*, = Y,* thereexistsavariational form
min12IY*, N, − N*
R�
Ä89
s.t. boundaryconditions
• ThisenergyiscalleddiscreteDirichlet energy,&h(N).
• Apopularchoiceofweightscomesfromaskingthat&h N = ∫ |ã-|�i .
• Theseweightsarecalledcotan weightsandY*, = cot å*, + cot ç*,.
• ThemeshisDelaunay(å*, + ç*, < è) iff Y*, > 0.
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Conformality
40
• TheDirichlet energysatisfies:
&h N = &v N + &ê N
where&ê istheareafunctionalsummingpositiveareasoftriangles.
• Theorbifold Tutte theoremimpliesthat&ê N = ë{+ë(V) constant.• Sincethenumberofpointconstraintsmatchesthedegreesoffreedomin
conformalmapwecanask:
Does-:! →V convergetoaconformalmapunderrefinementof!?
• Theorem. ConvergenceiníQ holdsforV atriangleorbifold.If! isDelaunay
uniformconvergencehold.
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Discreteuniformization
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[Springborn etal.08] Orbifold-Tutte
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Discretemappingofsurfaces
44
• Backtothediscretemappingproblem:wegotasolutionforup-to4
landmarkconstraints.
• Discreteextremalquasiconformal maps….?
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Openproblems
45
• Problem. Canℱ beenlarged?
• Iamnotawareofsuchresult.
• Problem. Canℱ beenlargedunderextraconditions?
• Severalinterestingsuchresults.Seenotes.
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Beyondeuclidean
46
• CCMcanbegeneralizedtohyperbolicplane.
• Basicresults(Tutte,Orbifold Tutte)stillholds.• Allowsinfinitenumberofcones.
• Drawback: nolongeralinearmodel
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Beyondeuclidean
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Higherdimensions?
48
• CounterexampletoTutte exists.Thefollowingexampleby[Floater,Pham-Trong].
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Theend
49
• Funding:• ThisworkwassupportedbytheIsraelScienceFoundation(grantNo.ISF1830/17).
• Thanks:• Courseteammates:Keenan,Max,Justin,andJohannes.
• AMSpeople:Lori,Tom
• You!
• Proofreadingandgoodadvice:• NoamAigerman,Nadav Dym
• Code:• https://github.com/noamaig/euclidean_orbifolds