DiscreteMappings

Yaron Lipman

WeizmannInstituteofScience

12018AMSshortcourse

DiscreteDifferentialGeometry

Surfacesastriangulations

2

• Trianglesstitchedtobuildasurface.

Surfacesastriangulations

3

• Trianglesstitchedtobuildasurface.

• ! = ($, &, ');• $ = )* , & = +*, , ' = -*,. .

Surfacesastriangulations

4

• ! = ($, &, ')

• Rulesforstitchingtriangles:

1. ! isasimplicialcomplex.

2. link )* =∪ 789:∈<+,. isasimpleclosedpolygon.

Surfacesastriangulations

5

• Definition. Asurfacetriangulation isatriplet! = ($, &, ') satisfyingthestitchingrules.

• Forsurfacetriangulationwithboundary,replacestitchingrule2with

2.link()*) isasimple(closedornot)polygon.

Surfacesastriangulations

6

• Theboundaryofsurfacetriangulationisa1Dsimplicialcomplex!= = ($=, &=).

• Theinteriorverticesaredenote$> = $ ∖ $=.

• Definition.! = ($, &, ') isconnectedifitisconnectedasagraph($, &).Itis@-connectedifitcannotbedisconnectedbyremoving@ − 1 vertices.

Surfacesastriangulations

7

• Lemma[Floater’03].If! = ($, &, ') is3-connectedthenanyinteriorvertexcanbeconnectedtoanyothervertex(includingboundary)withaninteriorpath.

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*

Thediscretemappingproblem

9

• Problem. Giventwotopologicallyequivalentsurfacetriangulations!Q,!R anda

setofcorrespondinglandmarks H*, S* *∈> ⊂ !Q×!R computea“nice”

simplicialhomeomorphism-:!Q → !R.

Thediscretemappingproblem

10

Thediscretemappingproblem

11

• Difficulty. Requiresfindingacommonisomorphiccommontriangulation,a

combinatorialproblem!

• Idea. Considermapping!Q,!R toacanonicaldomainV,

-Q:!Q → V and-R:!R → V andconstruct- as- = -RWQ ∘ -Q.

V

Thediscretemappingproblem

12

• Twoquestions:

• HowtochooseV?

• HowtocomputethesimplicialmapontoV?

V

Convexcombinationmappings

13

• AtechniquetomapasurfacetriangulationtoℝR.

• Definition. GivenaselectionofaweightperedgeY*, > 0,aconvexcombination

mapping -:! → ℝR isasimplicialmapmappingeachinteriorvertex)* ∈ $> toaplanarpointN* ∈ ℝR sothat

∑ Y*, N, − N* = 0�,∈\8 ,

where\* = ] +*, ∈ & .

• Thispropertyiscalledconvexcombinationproperty↔meanvalueproperty.

N*N,

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*

Convexcombinationmappings

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• CCMareingeneralnothomeomorphisms,e.g.,theconstantCCM.

• However,withcertainboundaryconditionsandtargetdomainsc CCMare

guaranteedtobehomeomorphic.

• Wewillexploreafamilyofsuchtargetdomains:

ℱ = V

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

CCMintoconvexpolygonaldomain

18

• Theorem(Tutte,Floater).Let! = ($, &, ') bea3-connectedsurfacetriangulationhomeomorphictoadisk.Let-:! → ℝR beaCCMsuchthat-|ij

isa

homeomorphismtoaconvexpolygonenclosingadomainΩ.Then,-:! → Ω isahomeomorphism.

ComputingCCM

19

!

∑ Y*, N, − N*�,∈\8 = 0,)* ∈ $>

)*

N*

N* = l*,)* ∈ $=

)*N*

)*

N*

Uniqueness

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• Proposition. Thereisauniquesolutiontothelinearsystem.

• Proof. Considerasolutiontothehomogeneoussystem:

ConsiderfirstcoordinateH* ofN* = (H*, S*).ThisisaCCFhencesatisfiesdiscretemaximumprinciple.

IfH* ≠ 0 thereisanon-zerovalueattheboundary,contradiction.

∑ Y*, N, − N*�,∈\8 = 0,)* ∈ $>

N* = 0,)* ∈ $=

?

Othermembersofℱ = V ?

TopologyTargetdomain

21

?

conesector

open

disk

Euclideanconesurfaces

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• Definition.acompactsurfaceV isaeuclidean conesurfaceifitisametricspace

locallyisometrictoanopendisk,acone,orasectorandthenumberofcone

pointsisfinite.

Euclideanorbifolds

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• Asubfamilyofeuclidean conesurfaces.

• Definition. Aeuclidean orbifoldV isasurfacedefinedasthequotientofℝR byasymmetrywallpapergroupn,thatis

V= ℝR/n.

• ThepointofV aretheorbitsofn,thatis N = p(N) p ∈ n .

Euclideanorbifolds

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Symmetryofthings[Strauss,Burgiel,Conway]

25

Euclideanorbifolds andtheir

fundamentaldomains

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.

ComputingCMMintoanorbifold

• First,cut! = ($, &, ') toadisk-typetriangulation!w = $w, &w, 'w .• Second,computeasimplicialmapx:!w → ℝR asfollows.

Computing

CMMinto

anorbifold

28

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

ComputingCMMintoanorbifold

• Lastly,themap-:! → V isdefinedby- H = [x H ].

30

Exampleoforbifold CCM

31

Homeomorphism

32

• 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.

Homeomorphism

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• Step1.Buildabranchedcover!′′ to! bystitchingcopiesof!′ accordingtothegroupn.Considertheextensionx:!ww → ℝR.AllverticessatisfytheCCP.

Homeomorphism

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• Step2.Wewillshowastrongerclaim.Every(generic)pointintheplaneiscovered

byat-leastonepositivelyorientedtriangle.

35

Homeomorphism

36

• Step3.Ifx doesnotdegenerateandmaintainsorientationofatriangle,itwillalso

notdegeneratenorflipanyneighbortriangle.

Homeomorphism

37

• Step4.Ifx:!ww → ℝR maintainsorientationofalltrianglesitisahomeomorphism.

Consequently-:! → V isahomeomorphism.

• Repeatwindingnumberargumentbutnowweknowthatalltrianglesare

positivelyoriented.

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.

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.

41

42

Discreteuniformization

43

[Springborn etal.08] Orbifold-Tutte

Discretemappingofsurfaces

44

• Backtothediscretemappingproblem:wegotasolutionforup-to4

landmarkconstraints.

• Discreteextremalquasiconformal maps….?

Openproblems

45

• Problem. Canℱ beenlarged?

• Iamnotawareofsuchresult.

• Problem. Canℱ beenlargedunderextraconditions?

• Severalinterestingsuchresults.Seenotes.

Beyondeuclidean

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• CCMcanbegeneralizedtohyperbolicplane.

• Basicresults(Tutte,Orbifold Tutte)stillholds.• Allowsinfinitenumberofcones.

• Drawback: nolongeralinearmodel

Beyondeuclidean

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Higherdimensions?

48

• CounterexampletoTutte exists.Thefollowingexampleby[Floater,Pham-Trong].

Theend

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• 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