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Thư viện số Văn Lang: Advances in Discrete Differential Geometry

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Differentials on Planar Triangular Meshes

Wai Yeung Lam and Ulrich Pinkall

Abstract Given a triangulated region in the complex plane, a discrete vector field Y assigns a vectorYi ∈Cto every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that preserves length cross ratios. We show that each holomorphic vector field can be constructed based on a discrete harmonic function in the sense of the cotan Laplacian. Moreover, to each holomorphic vector field we associate in a Möbius invariant fashion a certain holomorphic quadratic differential. Here a quadratic differential is defined as an object that assigns a purely imaginary number to each interior edge. Then we derive a Weierstrass representation formula, which shows how a holomorphic quadratic differential can be used to construct a discrete minimal surface with prescribed Gauß map and prescribed Hopf differential.

1 Introduction

Consider an open subset U in the complex plane C∼=R2 with coordinates z= x+i ytogether with a holomorphic vector field

Y = f

∂x.

HereY is a real vector field. It assigns to eachp∈R2the vector f(p)∈C∼=R2. We do not consider objects like z which are sections of the complexified tangent bundle(TR2)C.

W.Y. Lam (

B

)·U. Pinkall

Technische Universität Berlin, Inst. Für Mathematik, Straße des 17. Juni 136, 10623 Berlin, Germany e-mail: lam@math.tu-berlin.de

U. Pinkall

e-mail: pinkall@math.tu-berlin.de

© The Author(s) 2016

A.I. Bobenko (ed.),Advances in Discrete Differential Geometry, DOI 10.1007/978-3-662-50447-5_7

241

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Note f :U→Cis a holomorphic function, i.e.

0= fz¯ =1 2

∂f

∂x +i∂f

∂y

.

Lettgtdenote the local flow ofY (defined for smallton open subsets ofUwith compact closure inU). Then the euclidean metric pulled back undergt is confor- mally equivalently to the original metric:

gt, =e2u,

for some real-valued functionu. The infinitesimal change in scaleu˙is given by

˙ u = 1

2divY =Re(fz) . Note thatu˙is a harmonic function:

˙ uzz¯ =0.

On the other hand, differentiatingu˙twice with respect tozyields one half the third derivative of f:

˙ uzz= 1

2 fzzz.

It is well-known that the vector fieldYcorresponds to an infinitesimal Möbius trans- formation of the extended complex planeCif and only if f is a quadratic polyno- mial. In this sense fzzz measures the infinitesimal “change in Möbius structure”

under Y (Möbius structures are sometimes also called “complex projective struc- tures” [6]). Moreover, the holomorphic quadratic differential

q := fzzzd z2

is invariant under Möbius transformationsΦ. This is equivalent to saying thatq is unchanged under a change of variableΦ(z)=w=ξ+wheneverΦis a Möbius transformation. This is easy to see ifΦ(z)=az+bis an affine transformation. In this case

dw=a d z d

dw = 1 a

d d z and therefore

Y = ˜f

∂ξ

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with

f˜=a f.

Thus we indeed have

f˜wwwdw2= fzzzd z2.

A similar argument applies toΦ(z)= 1z and therefore to all Möbius transforma- tions.

For realizations from an open subsetUof the Riemann sphereCP1the vanishing of the Schwarzian derivative characterizes Möbius transformations. The quadratic differentialq plays a similar role for vector fields. We callq theMöbius derivative ofY.

An important geometric context where holomorphic quadratic differentials arise comes from the theory of minimal surfaces: Given a simply connected Riemann surface M together with a holomorphic immersiong:MS2⊂R3 and a holo- morphic quadratic differential q on M, there is a minimal surface F: M →R3 (unique up to translations) whose Gauß map is g and whose second fundamental form is Req.

In this paper we will provide a discrete version for all details of the above story.

Instead of smooth surfaces we will work with triangulated surfaces of arbitrary combinatorics. The notion of conformality will be that of conformal equivalence as explained in [3]. Holomorphic vector fields will be defined as infinitesimal con- formal deformations.

There is also a completely parallel discrete story where conformal equivalence of planar triangulations is replaced by preserving intersection angles of circumcircles.

To some extent we also tell this parallel story that belongs to the world of circle patterns.

The results on planar triangular meshes in this paper are closely related to isother- mic triangulated surfaces in Euclidean space [8].

2 Discrete Conformality

In this section, we review two notions of discrete conformality for planar triangular meshes. We first start with some notations of triangular meshes.

Definition 2.1 A triangular mesh M is a simplicial complex whose underlying topological space is a connected 2-manifold (with boundary). The set of vertices (0-cells), edges (1-cells) and triangles (2-cells) are denoted asV,EandF.

We denote Ei nt the set of interior edges and Vi nt the set of interior vertices.

Without further notice we will assume that all triangular meshes under consideration are oriented.

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Fig. 1 Two neighboring and oriented triangles

Definition 2.2 Arealization z:V →C of a triangular mesh M in the extended complex plane assigns to each vertex iV a point zi ∈C in such a way that for each triangle {i j k} ∈F the points corresponding to its three vertices are not collinear.

Given two complex numbersz1,z2∈Cwe write z1,z2 :=Re(¯z1z2).

We are looking for suitable definitions of conformal structure of a realizationz.

In particular, we want z to be conformally equivalent togzwhenever g:C→ Cis a Möbius transformations. This requirement will certainly be met if we base our definitions on complex cross ratios: Given a triangular mesh z:V →C, we associate a complex number to each interior edge{i j} ∈Ei nt, namely thecross ratio of the corresponding four vertices (See Fig.1)

crz,i j= (zjzk)(zizl) (zkzi)(zlzj).

Notice that crz,i j =crz,j i and hence crz :Ei nt→C is well defined. It is easy to see that two realizations differ only by a Möbius transformation if and only if their corresponding cross ratios are the same. In order to arrive at a more flexible notion of conformality we need to relax the condition that demands the equality of all cross ratios. Two natural ways to do this is to only require equality of either the norm or alternatively the argument of the cross ratios. This leads to two different notions of discrete conformality: conformal equivalence theory[9, 13] andcircle pattern theory[11].

Note that for the sake of simplicity of exposition we are ignoring here realizations inCwhere one of the vertices is mapped to infinity.

2.1 Conformal Equivalence

The edge lengths of a triangular mesh realized in the complex plane provide a dis- crete counterpart for the induced Euclidean metric in the smooth theory. A notion

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of conformal equivalence based on edge lengths was proposed by Luo [9]. Later Bobenko et al. [3] stated this notion in the following form:

Definition 2.3 Two realizations of a triangular meshz, w:V →Careconformally equivalentif the norm of the corresponding cross ratios are equal:

|crz| ≡ |crw|, i.e. for each interior edge{i j}

|(zjzk)||(zizl)|

|(zkzi)||(zlzj)| =|(wjwk)||(wiwl)|

|(wkwi)||(wlwj)|.

This definition can be restated in an equivalent form that closely mirrors the notion of conformal equivalence of Riemannian metrics:

Theorem 2.4 Two realizations of a triangular mesh z, w:V →Care conformally equivalent if and only if there exists u:V →Rsuch that

|wjwi| =eui+uj2 |zjzi|.

Proof It is easy to see that the existence ofu implies conformal equivalence. Con- versely, for two conformally equivalent realizations z, w, we define a function σ : E→Rby

|wjwi| =eσi j|zjzi|.

Sincez, ware conformally equivalentσsatisfies for each interior edge{i j}

σj kσki+σilσlj =0.

For any vertexi and any triangle{i j k}containing it we then define eui :=eσki+σi jσj k.

Note the vertex star of i is a triangulated disk if i is interior, or is a fan ifi is a boundary vertex. Hence the valueui defined in this way is independent of the

chosen triangle.

2.2 Circle Patterns

Given a triangular mesh realized in the complex plane we consider the circum- scribed circles of its triangles. These circles inherit an orientation from their tri- angles. The intersection angles of these circles from neighboring triangles (Fig.2)

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Fig. 2 The intersection angle of two neighboring circumscribed circles

define a functionφ: Ei nt → [0,2π)which is related to the argument of the corre- sponding cross ratio via

φi j =Arg(crz,i j). (1) Based on these angles we obtain another notion of discrete conformality which reflects the angle-preserving property that we have in the smooth theory.

Definition 2.5 Two realizations of a triangular meshz, w:V →Chave the same pattern structure if the corresponding intersection angles of neighboring circum- scribed circles are equal:

Arg(crz,i j)=Arg(crw,i j), i.e. for each interior edge{i j}

Arg(zjzk)(zizl)

(zkzi)(zlzj) =Arg(wjwk)(wiwl) (wkwi)(wlwj).

Just as conformal equivalence was related to scale factorsu at vertices, having the same pattern structure is related to the existence of certain angular velocitiesα located at vertices:

Theorem 2.6 Two realizations of a triangular mesh z, w:V →Chave the same pattern structure if and only if there existsα:V → [0,2π)such that

wjwi

|wjwi| =eiαi+αj2 zjzi

|zjzi|.

Proof The argument is very similar to the one for Theorem2.4. In particular, the existence of the function αeasily implies equality of the pattern structures. Con- versely, assuming identical pattern structures we take anyω: E→Rthat satisfies

wjwi

|wjwi| =eiωi j zjzi

|zjzi|.

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For any vertexiand any triangle{i j k}containing it we defineαi∈ [0,2π)such that eiαi =eiki+ωi jωj k).

Note the vertex star of i is a triangulated disk if i is interior, or is a fan ifi is a boundary vertex. Hence having the same pattern structure implies that the valueαi

is independent of the chosen triangle.

3 Infinitesimal Deformations and Linear Conformal Theory

We will linearize both of the above notions of discrete conformality by considering infinitesimal deformations. This will allow us to relate them to linear discrete com- plex analysis, based on a discrete analogue of the Cauchy Riemann equations [4,5, 10] (See the survey [12]).

Definition 3.1 Aninfinitesimal conformal deformationof a realizationz:V →C of a triangular mesh is a mapz˙:V →Csuch that there existsu:V →Rsatisfying

Rez˙j− ˙zi

zjzi

= ˙zj− ˙zi,zjzi

|zjzi|2 =ui+uj

2 .

We calluthescale changeat vertices.

Definition 3.2 Aninfinitesimal pattern deformationof a realizationz:V →Cof a triangular mesh is a map˙z:V →Csuch that there existsα:V →Rsatisfying

Imz˙j− ˙zi

zjzi = ˙zj− ˙zi,i(zjzi)

|zjzi|2 =αi+αj

2 .

We callαtheangular velocitiesat vertices.

Example 3.3 The infinitesimal deformationsz˙:=az2+bz+c, wherea,b,c∈C are constants, are both conformal and pattern deformations since

˙ zj− ˙zi

zjzi =(azi+b/2)+(azj+b/2).

Infinitesimal conformal deformations and infinitesimal pattern deformations are closely related:

Theorem 3.4 Suppose z:V →Cis a realization of a triangular mesh. Then an infinitesimal deformationz˙:V →Cis conformal if and only if iz is a pattern defor-˙ mation.

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Proof Notice

˙zj− ˙zi,zjzi

|zjzi|2 = iz˙jiz˙i,i(zjzi)

|zjzi|2 .

and the claim follows from Definition3.1and3.2.

3.1 Infinitesimal Deformations of a Triangle

Letz:V →Cbe a realization of a triangulated mesh andz˙an infinitesimal defor- mation. Up to an infinitesimal translationz˙is completely determined by the infini- tesimal scalings and rotations that it induces on each edge. These infinitesimal scal- ings and rotations of edges satisfy certain compatibility conditions on each triangle.

These conditions involve the cotangent coefficients well known from the theory of discrete Laplacians. As we will see in Sect.3.2, for conformal deformations (as well as for pattern deformations) the infinitesimal scalings and rotations of edges are indeed discrete harmonic functions.

Consider three pairwise distinct pointsz1,z2,z3∈Cthat do not lie on a line. In the followingi,j,kdenotes any cyclic permutation of the indexes 1,2,3. The trian- gle angle at the vertexiis denoted byβi. We adopt the convention that allβ1, β2, β3

have positive sign if the trianglez1,z2,z3is positively oriented and a negative sign otherwise. Suppose we have an infinitesimal deformation of this triangle. Then there existsσi j, ωi j∈Rsuch that

˙

zj− ˙zi =i j+i j)(zjzi). (2) The scalarsσi jandωi jdescribe the infinitesimal scalings and rotations of the edges.

They satisfy the following compatibility conditions:

Lemma 3.5 Givenσi j, ωi j∈Rthe following statements are equivalent:

(a) There exist˙zisuch that (2) holds.

(b) We have

0=12+12)(z2z1)+23+23)(z3z2)+31+31)(z1z3). (3) (c) There existsω∈Rsuch that

=23+icotβ131σ12)

=31+icotβ212σ23)

=12+icotβ323σ31).

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(d) There existσ ∈Rsuch that

σ =σ23+icotβ1(iω3112)

=σ31+icotβ2(iω1223)

=σ12+icotβ3(iω2331).

Proof The relation between (a) and (b) is obvious. We show the equivalence between (b) and (c). WithAdenoting the signed triangle area we have the following identities:

0= i(zjzi),zjzi, 2A= i(zjzi),zkzj, i(zjzi),i(zjzi) = zjzi,zjzi.

Using these identities andz3z2∈spanR{i(z1z3),i(z2z1)}we obtain z3z2=cot3)i(z2z1)−cot2)i(z1z3). (4) Cyclic permutation yields

z1z3=cot1)i(z3z2)−cot3)i(z2z1), z2z1=cot2)i(z1z3)−cot1)i(z3z2).

Substituting these identities into Equation (3) we obtain 0= σ1

cot3)i(z2z1)−cot2)i(z1z3)

+ω23i(z3z2) +σ2

cot1)i(z3z2)−cot3)i(z2z1)

+ω31i(z1z3) +σ3

cot2)i(z1z3)−cot1)i(z3z2)

+ω12i(z2z1)

=

ω1+cotβ12σ3)

i(z3z2) +

ω2+cotβ23σ1)

i(z1z3) +

ω3+cotβ31σ2)

i(z2z1).

Now we use thatλ1, λ2, λ3∈Csatisfy

λ1i(z3z2)+λ2i(z1z3)+λ3i(z2z1)=0,

if and only if λ1 =λ2=λ3. This establishes the equivalence of (b) and (c). The equivalence of (b) and (d) is seen in a similar fashion by eliminatingi(zjzi)in

(3) instead of(zjzi).

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The quantityωabove describes the average rotation speed of the triangle. Simi- larly, it can be verified that the aboveσ satisfies

σ = R˙ R

where Rdenotes the circumradius of the triangle. Thusσ signifies an average scal- ing of the triangle.

3.2 Harmonic Functions with Respect to the Cotangent Laplacian

In smooth complex analysis conformal maps are closely related to harmonic func- tions. If a conformal map preserves orientation it is holomorphic and satisfies the Cauchy Riemann equations. In particular, its real part and the imaginary part are conjugate harmonic functions. Conversely, given a harmonic function on a simply connected surface then it is the real part of some conformal map.

A similar relationship manifests between discrete harmonic functions (in the sense of the cotangent Laplacian) and infinitesimal deformations of triangular meshes. Discrete harmonic functions can be regarded as the real part of holomorphic functions which satisfies a discrete analogue of the Cauchy Riemann equations. In particular, a relation between discrete harmonic functions and infinitesimal pattern deformations was found by Bobenko, Mercat and Suris [2]. Integrable systems were involved in this context. We extend their result to include the case of infinitesimal conformal deformations.

Theorem 3.6 Let z:V →Cbe a simply connected triangular mesh realized in the complex plane and h:V →Rbe a function. Then the following are equivalent:

(a) h is a harmonic function with respect to the cotangent Laplacian, i.e. using the notation of Fig.1, for all interior vertices iVi ntwe have

j

(cotβi jk +cotβlj i)(hjhi)=0. (5)

(b) There exists an infinitesimal conformal deformation z˙:V →Cwith scale factors given by h. It is unique up to infinitesimal rotations and translations.

(c) There exists an infinitesimal pattern deformation iz˙:V →Cwith h as angu- lar velocities. It is unique up to infinitesimal scalings and translations.

Proof We show the equivalence of the first two statements. The equivalence of the first and the third follows similarly.

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Supposehis a harmonic function. Since the triangular mesh is simply connected, equation (5) implies the existence of a functionω˜ :F→Rsuch that for all interior edges{i j}we have

˜i j k˜j il =i(cotβi jk +cotβlj i)(hjhi).

Hereω˜is unique up to an additive constant and called theconjugate harmonic func- tionofh. Usingω˜ we define a functionω:E →Rvia

i j =˜i j kicotβi jk(hjhi).

Lemma3.5now implies that there exists˙z:V →Csuch that (˙zj− ˙zi)=

hi+hj

2 +i j

(zjzi).

This gives us the desired infinitesimal conformal deformation ofzwithh as scale factors.

To show uniqueness, supposez,˙ z˙are infinitesimal conformal deformations with the same scale factors. Thenz˙− ˙zpreserves all the edge lengths of the triangular mesh and hence is induced from an Euclidean transformation.

Conversely, given an infinitesimal conformal deformationz˙with scale factorsh.

We write

˙

zj− ˙zi=

hi+hj

2 +i j

(zjzi)

for someω:E →R. Lemma3.5implies that there is a functionω˜ :F →Rsuch that

˜i j k =i j+icotβi jk(hjhi).

We have

˜i j k˜j il =i(cotβi jk +cotβlj i)(hjhi) and

j

(cotβi jk+cotβlj i)(hjhi)=0.

Thereforehis harmonic.

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4 Holomorphic Quadratic Differentials

In this section, we introduce a discrete analogue of holomorphic quadratic differen- tials. We illustrate their correspondence to discrete harmonic functions. It reflects the property in the smooth theory that holomorphic quadratic differentials parame- trize Möbius structures on Riemann surfaces ([6, Chap. 9]).

To simplify the notation, we make use of discrete differential forms. We denoteE the set of oriented edges andEi ntthe set of oriented interior edges. Given an oriented triangular mesh M, a complex-valued functionη: E →Cis called adiscrete 1- formif

η(ei j)= −η(ej i)ei jE.

It isclosedif for every face{i j k}

η(ei j)+η(ej k)+η(eki)=0.

It isexactif there exists a function f :V →Csuch that η(ei j)=d f(ei j):= fjfi.

Similarly, we can consider discrete 1-forms on the dual graphMofMand these are calleddual 1-forms. Given an oriented edgee, we denoteeits dual edge oriented from the right face ofeto its left face. The set of oriented dual edges is denoted byE.

Definition 4.1 Given a triangular meshz:V →Crealized on the complex plane, a functionq :Ei ntiRdefined on interior edges is adiscrete holomorphic quadratic differentialif it satisfies for every interior vertexiVi nt

j

qi j =0,

j

qi j/d z(ei j)=0.

Theorem 4.2 Let q:Ei ntiRbe a holomorphic quadratic differential on a real- ization z :V →Cof a triangular mesh. SupposeΦ :C→Cis a Möbius transfor- mation which does not map any vertex to infinity. Then q is again a holomorphic quadratic differential onw:=Φz.

Proof Since Möbius transformations are generated by Euclidean transformations and inversions, it suffices to consider the inversion in the unit circle at the origin

w:=Φ(z)=1/z.

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We have

j

qi j/dw(ei j)=

j

zizjqi j/d z(ei j)= −zi

j

qi jzi2

j

qi j/d z(ei j)=0.

Hence the claims follow.

We are going to show that on a simply connected triangular mesh, there is a correspondence between discrete holomorphic quadratic differentials and discrete harmonic functions.

We first show how to construct a discrete holomorphic quadratic differential from a harmonic function. Given a functionu:V →Ron a realization ofz:V →Cof a triangular mesh M. If we interpolate it piecewise-linearly over each triangular face, its gradient is constant on each face and we have gradzu :F →Cgiven by

gradzui j k =iuid z(ej k)+ujd z(eki)+ukd z(ei j) 2Ai j k

.

Note that we ignore here the non-generic case (which leads to the vanishing of the area) where the triangle degenerates in the sense that its circumcircle passes through the point at infinity. Also note that for a non-degenerate triangle that is mapped byz inCin an orientation reversing fashion the areaAi j kis considered to have a negative sign. Granted this, one can verify that the gradient ofusatisfies

gradzui j k,d z(ei j) =ujui ∀{i j} ⊂ {i j k} ∈ F.

We defineuz :F →Cby

uz := 1 2gradzu. and the dual 1-formduz : Ei nt →ConMby

duz(ei j):=(uz)i j k(uz)j il

where{i j k}is the left face and{j il}is the right face of the oriented edgeei j. Lemma 4.3 Given a function u:V →R on a realization of a triangular mesh z:V →C, we have

duz(ei j)d z(ei j)

=−i 2

cotβij k(ukuj)+cotβkij (ukui)+cotβilj(ului)+cotβl ji(uluj)

which is purely imaginary (Fig.1).

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Proof Since

gradzui j k,d z(ei j) =ujui = gradzuj kl,d z(ei j),

we have

Re(duz(ei j)d z(ei j))=0. On the other hand, using equation (4) we get

Re(duz(ei j)i d z(ei j))

=Re(((uz)i j k(uz)j il)i d z(ei j))

=(gradzui j k,cotβj ki d z(ej k)−cotβkij d z(eki) + gradzuj il,cotβiljd z(eil)−cotβl jid z(elj))/2

=1 2

cotβj ki (ukuj)+cotβkij (ukui)+cotβilj(ului)+cotβlji(uluj) .

Hence the claim follows.

Lemma 4.4 Given a realization z:V →Cof a triangular mesh. A function u: V →Ris harmonic if and only if the function q: Ei ntiRdefined by

qi j :=duz(ei j)d z(ei j) is a holomorphic quadratic differential.

Proof Noteqis well defined since

qi j=duz(ei j)d z(ei j)=duz(ej i)d z(ej i)=qj i. It holds for general functionsu:V →Rthat

Re(q)≡0

j

qi j/d z(ei j)=

j

duz(ei j)=0 ∀iVi nt.

We know from Lemma4.3that for every interior vertexiVi nt

j

qi j=

j

duz(ei j)d z(ei j)= i 2

j

(cotβi jk +cotβlj i)(ujui).

Hence,uis harmonic if and only ifqis a holomorphic quadratic differential.

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Lemma 4.5 Let z:V →Cbe a realization of a simply connected triangular mesh.

Given a function q: Ei ntiRsuch that for every interior vertex iVi nt

j

qi j/d z(ei j)=0,

there exists a function u:V →Rsuch that for every interior edge{i j}

qi j =duz(ei j)d z(ei j).

Proof We consider a dual 1-formτ onMdefined by τ(ei j)=qi j/d z(ei j).

SinceM is simply connected and

j

τ(ei j)=

j

qi j/d z(ei j)=0,

there exists a functionh:F →Csuch that

dh(ei j):=hi j khj il =τ(ei j).

It implies we have Re(dh(e)d z(e))=Re(q)≡0 and

ω(ei j):= 2h¯i j k,d z(ei j) = 2h¯j il,d z(ei j)

is a well-definedR-valued 1-form. Since the triangular mesh is simply connected and for every face{i j k}

ω(ei j)+ω(ej k)+ω(eki)=0,

there exists a functionu :V →Rsuch that for every oriented edgeei j

du(ei j)=ujui =ω(ei j).

It can be verified that

h= 1

2gradzu =uz. Hence we obtain

qi j =τ(ei j)d z(ei j)=dh(ei j)d z(ei j)=duz(ei j)d z(ei j)

for every interior edge{i j}.

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Theorem 4.6 Suppose z:V →Cis a realization of a simply connected triangular mesh. Then any holomorphic quadratic differential q: Ei ntiRis of the form

qi j =duz(ei j)d z(ei j)ei jEi nt

for some harmonic function u:V →R.

Furthermore, the space of holomorphic quadratic differentials is a vector space isomorphic to the space of discrete harmonic functions module linear functions.

Proof The first part of the statement follows from Lemmas4.4and4.5. In order to show the second part, it suffices to observe that

duz ≡0 ⇐⇒ gradua ⇐⇒ du= a,d z ⇐⇒ u = a,z +b

for somea,b∈C.

In previous sections, we showed that every harmonic function corresponds to an infinitesimal conformal deformation. The following shows that discrete holo- morphic quadratic differentials are the change in the intersection angles of circum- scribed circles.

Theorem 4.7 Let z:V →C be a realization of a simply connected triangular mesh. Suppose u :V →Ris a discrete harmonic function andz is an infinitesi-˙ mal conformal deformation with u as scale factors. Then we have

duzd z= −1 2

˙ crz

crz

= −i 2φ˙

whereφ˙ :Ei nt→Rdenotes the change in the intersection angles of neighboring circumscribed circles.

Proof We write(˙zj− ˙zi)=(hi+2hj +i j)(zjzi). Applying Lemma4.3we have

˙

crz,i j/crz,i j =j kki+illj

=i

cotβij k(ukuj)+cotβkij (ukui)+cotβilj(ului)+cotβlji(uluj)

= −2duz(ei j)d z(ei j).

The equality

˙ crz

crz

=˙

follows from Equation (1).

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5 Conformal Deformations in Terms of End( C

2

)

In this section we show how an infinitesimal conformal deformation gives rise to a discrete analogue of a holomorphic null curve in C3. Later we will see that the real parts of such a “holomorphic null curve” can be regarded as the Weierstrass representation of a discrete minimal surface.

Up to now we have mostly treated the Riemann sphere CP1 as the extended complex planeC=C∪ {∞}. In this section we will take a more explicitly Möbius geometric approach: We will represent fractional linear transformations of C by linear transformations ofC2with determinant one. Actually, the group of Möbius transformations is

Möb(C)∼= PSL(2,C)∼= SL(2,C)/(±I). (6) However, since we are mainly interested in infinitesimal deformations and any map into PSL(2,C)whose values stay close to the identity admits a canonical lift to SL(2,C), we can safely ignore the difference between PSL(2,C)and SL(2,C).

Given a realization z:V →C of a triangular mesh we consider its lift ψ:V →C2

ψ:=

z 1

and regard the realization as a mapΨ :V →CP1where Ψ :=C

z 1

= [ψ].

The action of a Möbius transformation on the Riemann sphere is given by a matrix A∈SL(2,C), which is unique up to sign:

[ϕ] → [].

Before we investigate infinitesimal deformations we first consider finite defor- mations of a triangular meshΨ :V →CP1. Given such a finite deformation, the change in the positions of the three vertices of a triangle{i j k}can be described by a Möbius transformation, which is represented byGi j k ∈SL(2,C). They satisfy a compatibility condition on each interior edge{i j}(see Fig.1):

[Gi j kψi] = [Gj ilψi], [Gi j kψj] = [Gj ilψj].

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Suppose now that the mesh is simply connected. Then up to a global Möbius trans- formation the mapG:F →SL(2,C)can be uniquely reconstructed from themul- tiplicative dual 1-formdefined as

G(ei j):=Gj il1Gi j k.

G(ei j)is defined whenever{i j}is an interior edge and we have G(ei j)=G(ej i)1.

Moreover, for every interior vertexi we have

j

G(ei j)=I.

The compatibility conditions imply that for interior each edge {i j} there exist λi j,i, λi j,j ∈C\{0}such that

G(ei ji =λi j,iψi

G(ei jj =λi j,jψj. Sinceλi j,iλi j,j =det(G(ei j))=1, we have

λi j :=λi j,i =1i j,j. Because ofG(ei j)=G(ei j)1we know

λi j =λi j,i =1j i,i =λj i.

Henceλdefines a complex-valued function on the setEi ntof interior edges.

We now show that for each interior edgeλi jdetermines the change in the cross ratio of the four points of the two adjacent triangles. Note that the cross ratio of four points inCP1can expressed as

cr([ψj],[ψk],[ψi],[ψl])=detk, ψj)detl, ψi) deti, ψk)detj, ψl).

Lemma 5.1 Suppose we are given four points[ψi],[ψj],[ψk],[ψl] ∈CP1and G∈ SL(2,C)with

i =λ1ψi

j =λψj

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for someλ∈C\{0}. Then the cross ratio of the four transformed points [ ˜ψi] = [i], [ ˜ψj] = [j], [ ˜ψk] = [k], [ ˜ψl] = [ψl] is given by

cr([ ˜ψj],[ ˜ψk],[ ˜ψi],[ ˜ψl])=cr([ψj],[ψk],[ψi],[ψl])/λ2. Proof

cr([ ˜ψj],[ ˜ψk],[ ˜ψi],[ ˜ψl])=det(Gψk,Gψj)detl,Gψi) det(Gψi,Gψk)det(Gψj, ψl)

=cr([ψj],[ψk],[ψi],[ψl])/λ2.

We now can summarize the information about finite deformations of a realization as follows:

Theorem 5.2 LetΨ :V →CP1 be a realization of a simply connected triangu- lar mesh. Then there is a bijection between finite deformations ofΨ inCP1 mod- ulo global Möbius transformations and multiplicative dual 1 forms G: Ei nt → SL(2,C)satisfying for every interior vertex i

j

G(ei j)=I

and for every interior edge

G(ei j)=G(ej i)−1 G(ei ji =λ−1i j ψi

G(ei jj =λi jψj.

Here λ:Ei nt→C\{0}. We denote cr: Ei nt →C the cross ratios of Ψ and cr: Ei nt→Cthe cross ratios of a new realization described by G. Then

cr=cr2. In particular,

|λ| ≡1 =⇒ the deformation is conformal.

Arg(λ)≡0 =⇒ the deformation is a pattern deformation.

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Suppose we have a family of deformations described by dual 1-forms Gt : Ei nt→SL(2,C)withG0I. By consideringη:= dtd|t=0Gt we obtain the fol- lowing description of infinitesimal deformations:

Corollary 5.3 LetΨ :V →CP1be a realization of a simply connected triangular mesh. Then there is a bijection between infinitesimal deformations of Ψ in CP1 modulo infinitesimal Möbius transformations and dual 1 formsη: Ei nt→sl(2,C) satisfying for every interior vertex i

j

η(ei j)=0 (7)

and for every interior edge

η(ei j)= −η(ej i) η(ei ji = −μi jψi

η(ei jj =μi jψj.

Hereμ:Ei nt→C. We denotecr:Ei nt →Cthe cross ratios ofΨ andcr˙ :Ei nt→ C the rate of change in cross ratios induced by the infinitesimal deformation described byη. Then

μ= −1 2

˙ cr cr. In particular,

Re(μ)≡0 =⇒ the infinitesimal deformation is conformal,

Im(μ)≡0 =⇒ the infinitesimal deformation is a pattern deformation.

Note that given a mesh, the 1-formηis uniquely determined by the eigenfunction μ. We now investigate the constraints onμimplied by the closedness condition (7) ofη.

Consider the symmetric bilinear form( , ):C2×C2 →sl(2,C) (φ, ϕ)v:=det(φ, v)ϕ+det(ϕ, v)φ.

Forψi =ψj∈C2we define mi j := 1

deti, ψj)(ψj, ψi)∈sl(2,C).

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