revisited Rigidity of inversive distance circle packings Advances in Mathematics

Contents lists available atScienceDirect

Advances in Mathematics

www.elsevier.com/locate/aim

Rigidity of inversive distance circle packings revisited

Xu Xu^a,b

a SchoolofMathematicsandStatistics,WuhanUniversity,Wuhan430072, PR China

b HubeiKeyLaboratoryofComputationalScience(WuhanUniversity),Wuhan 430072,PRChina

a r t i c l e i n f o a bs t r a c t

Article history:

Received10May2017

Receivedinrevisedform18October 2017

Accepted3May2018 Availableonline26May2018 CommunicatedbytheManaging Editors

Keywords:

Inversivedistance Circlepacking Rigidity

Combinatorialcurvature

Inversivedistancecirclepacking metricwasintroduced byP BowersandKStephenson [7] asageneralizationofThurston’s circlepackingmetric [34].Theyconjecturedthattheinversive distance circle packings are rigid.Fornonnegativeinversive distance,Guo[22] provedtheinfinitesimalrigidityandthen Luo [27] provedtheglobalrigidity.Inthispaper,basedonan observationofZhou [37],weprovethisconjectureforinversive distancein(−1,+∞) byvariationalprinciples.Wealsostudy theglobalrigidityofacombinatorialcurvatureintroducedin [14,16,19] withrespecttotheinversivedistancecirclepacking metricswheretheinversivedistanceisin(−1,+∞).

©2018ElsevierInc.Allrightsreserved.

1. Introduction

1.1. Background

Inhisworkonconstructinghyperbolicstructureon3-manifolds,Thurston([34],Chap- ter 13) introduced the notion of circle packing metric on triangulated surfaces with

E-mailaddress:[email protected].

https://doi.org/10.1016/j.aim.2018.05.026 0001-8708/©2018ElsevierInc.Allrightsreserved.

non-obtuse intersection angles. The requirement of prescribed intersection angles cor- responds to the fact that the intersection angle of two circles is invariant under the Möbius transformations. Fortriangulated surfaces with Thurston’scircle packing met- rics,there aresingularities at thevertices. Theclassical combinatorialcurvature Ki is introduced to describe the singularity at the vertex v_i, which is defined as the angle deficit at v_i. Thurston’swork generalizedAndreev’s workon circle packing metricson a sphere [1,2] and gave a complete characterization of the space of the classical com- binatorial curvature.As acorollary,he obtainedthecombinatorial-topologicalobstacle fortheexistenceofaconstantcurvaturecircle packingwithnon-obtuseintersectionan- gles,whichcouldbewrittenascombinatorial-topologicalinequalities.Zhou [37] recently generalizedAndreev–Thurston Theorem to the casethatthe intersectionangles arein [0,π).ChowandLuo [9] introducedacombinatorialRicciflow,acombinatorialanalogue ofthesmoothsurfaceRicciflow,fortriangulatedsurfaceswithThurston’scirclepacking metricsand establishedthe equivalence between the existence of aconstant curvature circlepackingmetricandtheconvergenceofthecombinatorialRicciflow.

Inversive distance circle packing on triangulated surfaces was introduced by Bow- ers and Stephenson [7] as a generalization of Thurston’scircle packing. Different from Thurston’scirclepacking,adjacentcirclesininversivedistancecirclepackingareallowed tobedisjointandtherelativedistanceoftheadjacentcirclesismeasuredbytheinversive distance,whichisageneralizationofintersectionangle.SeeBowers–Hurdal [6],Stephen- son [33] andGuo[22] formoreinformation.Theinversivedistance circlepackingshave practicalapplications inmedical imagingand computergraphics, see [24,35,36] forex- ample.BowersandStephenson[7] conjecturedthattheinversivedistancecirclepackings arerigid. Guo [22] provedtheinfinitesimalrigidityand thenLuo [27] solvedaffirmably the conjecture fornonnegative inversive distance with Euclidean and hyperbolic back- ground geometry. For the spherical background geometry, Ma and Schlenker [29] had acounterexampleshowingthatthere is ingeneralnorigidityand JohnC. Bowersand Philip L. Bowers [4] obtained a new construction of their counterexample using the inversive geometry of the 2-sphere. John Bowers, Philip Bowers and Kevin Pratt [5]

recently proved the global rigidity of convex inversive distance circle packings in the Riemannsphere.GeandJiang [12,13] recentlystudiedthedeformationofcombinatorial curvature and found away to searchfor inversive distance circle packingmetrics with constantcone angles. They also obtained someresults onthe image of curvature map for inversive distance circle packings. Ge and Jiang [14] and Ge and the author [19]

furtherextendedacombinatorialcurvatureintroducedbyGeand theauthor in [16–18]

to inversive distance circle packings and studied the rigidity and deformation of the curvature.

Inthispaper,basedonanobversionofZhou [37],weproveBowersandStephenson’s rigidityconjectureforinversivedistancein(−1,+∞).Themaintoolsarethevariational principle established by Guo [22] for inversive distance circle packings and the exten- sionof locally convexfunctionintroduced byBobenko,Pinkall and Springborn [3] and systematicallydeveloped by Luo [27]. We referto Glickenstein [20] for anice geomet-

ric interpretation of the variational principle in [22]. There are many other works on variationalprinciplesoncirclepackings.SeeBrägger [8],Rivin [31],Leibon [25],Chow–

Luo [9], Bobenko–Springborn [7], Marden–Rodin [30],Spingborn [32], Stephenson [33], Luo [28],Guo–Luo [23],Dai–Gu–Luo [10],Guo [21] andothers.

1.2. Inversivedistancecirclepackings

In this subsection, we briefly recall the inversive distance circle packing introduced by Bowers and Stephenson [7] in Euclidean and hyperbolic background geometry. For more information on inversive distance circle packing metrics, the readerscan referto Stephenson [33],BowersandHurdal [6] andGuo [22].

Suppose M is a closed surface with a triangulation T = {V,E,F}, where V,E,F represent the sets of vertices, edges and faces respectively. Let I : E → (−1,+∞) be afunction assigning eachedge {ij} aninversive distance Iij ∈ (−1,+∞), which is denoted as I >−1 inthe paper.The triple(M,T,I) willbe referredto as aweighted triangulationofMbelow.Alltheverticesareorderedonebyone,markedbyv1,· · ·,vN, where N =|V|is thenumberof vertices,andwe oftenuseitodenote thevertexv_i for simplicitybelow.Weusei∼jtodenotethattheverticesiandj areadjacent,i.e.,there is an edge {ij}∈ E with i, j as end points. Allfunctions f :V →Rwill be regarded as column vectors inR^N and fi =f(vi) is thevalue of f at vi. And we use C(V) to denotethesetoffunctionsdefinedonV.R>0denotesthesetofpositivenumbersinthe paper.

Each mapr:V →(0,+∞) isacirclepacking, whichcouldbetakenas theradiusr_i ofacircleattachedtothevertex i.Given (M,T,I),weassigneachedge{ij}thelength

l_ij =

r_i²+r_j²+ 2r_ir_jI_ij (1.1) forEuclidean backgroundgeometryand

l_ij = cosh⁻¹(cosh(r_i) cosh(r_j) +I_ijsinh(r_i) sinh(r_j)) (1.2) forhyperbolicbackgroundgeometry,whereIij istheEuclideanandhyperbolicinversive distanceofthetwocirclescenteredatviandvjwithradiiriandrjrespectively.Notethat thelengthl_ij in(1.1) and(1.2) iswell-defined forallr_i >0,r_j>0 under thecondition I_ij >−1.IfI_ij ∈(−1,0),thetwocirclesattachedtotheverticesiandjintersectwithan obtuseangle.IfIij ∈[0,1],thetwocirclesintersectwithanon-obtuseangle.Wecantake Iij = cos Φij with Φij ∈[0,^π₂] andthen theinversivedistance circle packingis reduced to Thurston’scircle packing. If Iij ∈ (1,+∞), the two circles attached to the vertices i and j are disjoint. See Fig. 1for possible arrangements of the circles. Guo [22] and Luo [27] systematicallystudiedthe rigidityof inversive distance circle packing metrics for nonnegative inversive distance I ≥ 0, i.e. I_ij ≥ 0 forevery edge {ij}∈ E. In this paper, wefocusonthecasethatI >−1.

Fig. 1.Inversive distance circle packings.

Thefollowingisourmain result,whichsolvesBowersandStephenson’s rigiditycon- jectureforinversive distancein(−1,+∞).

Theorem 1.1. Given a closed triangulated surface (M,T,I) with inversive distance I : E→(−1,+∞) satisfying

I_ij+I_ikI_jk≥0, I_ik+I_ijI_jk≥0, I_jk+I_ijI_ik≥0 (1.3) forany topologicaltriangleijk∈F.

(1) A Euclideaninversivedistancecirclepackingon(M,T)isdeterminedbyitscombi- natorial curvature K:V →R uptoscaling.

(2) A hyperbolicinversivedistancecirclepackingon (M,T)isdeterminedbyitscombi- natorial curvature K:V →R.

Remark1.ForI∈[0,1],theaboveresultwasAndreevandThurston’srigidityforcircle packingwithintersectionanglesin[0,^π₂].ForI∈(−1,1],theaboveresultwastherigidity for circlepacking with intersectionangles in[0,π) recently obtainedby Zhou [37]. For I≥0,theaboveresultwastherigidityforinversive distancecirclepackingobtainedby Guo [22] andLuo[27].Ourresultunifiestheseresultsandallowstheinversivedistances totakevaluesinalargerdomain.

Remark2.ItisinterestingtonotethatinTheorem1.1,foratopologicaltriangleijk∈ F,ifoneof I_ij,I_ik,I_jk is negative,the othertwo mustbe nonnegative.Soat mostone ofI_ij,I_ik,I_jk isnegative.

We further extend the rigidity to combinatorial α-curvature introduced in [14–19], whichisdefinedas

Rα,i= Ki

s^α_i

for α∈R, where s_i =r_i for theEuclidean background geometry and s_i = tanh^r₂ⁱ for thehyperbolicbackgroundgeometry.

Theorem 1.2.Given a closed triangulated surface (M,T,I) with inversive distance I : E →(−1,+∞)satisfying

Iij+IikIjk≥0, Iik+IijIjk≥0, Ijk+IijIik≥0

for any topological triangle ijk ∈F. R is a givenfunction defined on thevertices of (M,T).

(1) IfαR≡0,thereexistsatmostoneEuclideaninversivedistancecirclepackingmetric withcombinatorialα-curvatureR uptoscaling. IfαR≤0andαR≡0,there exists at most one Euclidean inversive distance circle packing metric with combinatorial α-curvatureR.

(2) IfαR≤0,thereexistsatmostonehyperbolicinversivedistancepackingmetricwith combinatorialα-curvatureR.

1.3. Planof paper

The paper is organized as follows. In Section 2, we study the Euclidean inversive distance circle packingmetricsand proveTheorem1.1 and1.2 forthe Euclideanback- groundgeometry.InSection3,westudythehyperbolicinversivedistancecirclepacking metricsandproveTheorem1.1,1.2 forthehyperbolic backgroundgeometry.

2. Euclideaninversivedistancecirclepackings

2.1. Admissiblespace ofEuclidean inversivedistancecirclepackingmetrics forasingle triangle

Givenaweightedtriangulatedsurface(M,T,I) withweightI >−1.Supposeijkis atopologicaltriangleinF.Hereandinthefollowing,tosimplifynotations,whenweare discussingatriangleijk,weuseli todenotethelengthoftheedge{jk}anduseIi to denote theinversive distanceofthetwo circlesatthevertices jandk.IntheEuclidean backgroundgeometry, thelengthli oftheedge{jk}isthen definedby

li=

r²_j+r²_k+ 2rjrkIi. (2.1) ForI >−1,inorderthatthelengthsli,lj,lk forΔijk∈F satisfythetriangleinequali- ties, thereare somerestrictionsontheradii.Denote theadmissiblespace oftheradius vectorsforafaceΔijk∈F as

Ω^E_ijk:={(ri, rj, rk)∈R³>0|li< lj+lk, lj < li+lk, lk< li+lj}. (2.2)

In thecase of I ∈[0,1], as noted by Thurston [34], Ω^E_ijk = R³>0. However,in general, Ω^E_ijk=R³>0 forI∈(−1,+∞).It isproved [22] thattheadmissiblespaceΩ^E_ijk forI≥0 isasimplyconnectedopen subsetofR³>0 andΩ^E_ijk maynotbeconvex. Set

Ω^E=∩Δijk∈FΩ^E_ijk (2.3)

to be thespace of admissibleradius function on the surface. Ω^E is obviously anopen subsetofR^N>0.Everyr∈Ω iscalledaninversivedistance circlepackingmetric.

Asnotedin [22],inorderthattheedgelengthsli,lj,lksatisfythetriangleinequalities, wejustneed

0<(li+lj+lk)(li+lj−lk)(li+lk−lj)(lj+lk−li)

= 4l²_il_k²−(l_i²+l²_k−l_j²)²

= 2l²_il_j²+ 2l²_il²_k+ 2l²_jl_k²−l⁴_i −l_j⁴−l_k⁴.

(2.4)

Substituting thedefinition of edgelength (2.1) in theEuclidean backgroundgeometry into(2.4),bydirectcalculations,wehave

1

4(l_i+l_j+l_k)(l_i+l_j−l_k)(l_i+l_k−l_j)(l_j+l_k−l_i)

=r_i²r_j²(1−I_k²) +r_i²r_k²(1−I_j²) +r²_jr²_k(1−I_i²)

+ 2r_i²rjrk(Ii+IjIk) + 2rir_j²rk(Ij+IiIk) + 2rirjr_k²(Ik+IiIj)>0.

Denote

γijk:=Ii+IjIk, γjik :=Ij+IiIk, γkij:=Ik+IiIj, (2.5) thenwehavethefollowingresultonEuclideantriangleinequalities.

Lemma2.1 ([22]).Suppose (M,T,I)isaweighted triangulated surfacewith weightI >

−1 and ijk is a topological triangle in F. The edge lengths li,lj,lk defined by (2.1) satisfy thetriangleinequalitiesifandonly if

r²_ir²_j(1−I_k²) +r²_ir²_k(1−I_j²) +r_j²r_k²(1−I_i²) + 2r_i²rjrkγijk+ 2rir²_jrkγjik

+ 2rirjr²_kγkij>0. (2.6)

Wehavethefollowingdirectcorollaryobtainedin [37] byLemma2.1.

Corollary 2.2. If Ii,Ij,Ik ∈(−1,1] and γijk ≥ 0, γjik ≥ 0, γkij ≥0, then the triangle inequalitiesare satisfiedforany(r_i,r_j,r_k)∈R³>0.

Remark3.Specially,ifIi= cos Φi,Ij = cos Φj,Ik= cos ΦkwithΦi,Φj,Φk∈[0,^π₂],then wehaveIi,Ij,Ik∈(−1,1] andγijk≥0,γjik≥0,γkij≥0.Sothetriangleinequalitiesare satisfiedforallradiusvectorsinR³>0,whichwasobtainedbyThurstonin [34].However, ifweonlyrequireΦi,Φj,Φk∈[0,π),then(2.6) isequivalentto

r²_ir²_jsin²Φk+r_i²r_k²sin²Φj+r_j²r²_ksin²Φi+ 2r²_irjrk(cos Φi+ cos Φjcos Φk) + 2r_ir_j²r_k(cos Φ_j+ cos Φ_icos Φ_k) + 2r_ir_jr_k²(cos Φ_k+ cos Φ_icos Φ_j)>0.

Specially, ifΦ_i+ Φ_j ≤ π,Φ_i+ Φ_k ≤π,Φ_j + Φ_k ≤π [37], or Φ_i = Φ_j ∈[0,^π₂] [37], or Φ_i= Φ_j = Φ_k ∈[0,π),thetriangleinequalitiesaresatisfied.

ByLemma2.1,theadmissiblespaceΩ^E_ijk forthetopologicaltriangleijk∈F may not be the whole space R³>0. Furthermore, it is not always convex for all I_i,I_j,I_k ∈ (−1,+∞).However,wehavethefollowingusefullemma onthestructure oftheadmis- sible spaceΩ^E_ijk.

Lemma2.3. Givenaweightedtriangulatedsurface(M,T,I)with I >−1.Foratopolog- ical triangleijk∈F,if

γ_ijk≥0, γ_jik≥0, γ_kij≥0, (2.7) then the admissiblespace Ω^E_ijk is asimply connectedopen subset of R³>0.Furthermore, foreach connectedcomponent V of R³>0\Ω^E_ijk,theintersection V ∩Ω^E_ijk isaconnected component of Ω^E_ijk\Ω^E_ijk,on whichθi isaconstant function.

Proof. Define

F :R³>0→R³>0

(r_i, r_j, r_k)→(r²_j+r²_k+ 2r_jr_kI_i, r_i²+r²_k+ 2r_ir_kI_j, r_i²+r_j²+ 2r_ir_jI_k) and

G:R³>0→R³>0

(li, lj, lk)→(l²_i, l²_j, l_k²),

then Gis adiffeomorphismofR³>0 andH =G⁻¹◦F is themap sending (ri,rj,rk) to (li,lj,lk).

We first prove that H is injective. To prove this, we just need to prove that F is injective.Notethat

∂(Fi, Fj, Fk)

∂(r_i, r_j, r_k) = 2

⎛

⎜⎝

0 r_j+r_kI_i r_k+r_jI_i r_i+r_kI_j 0 r_k+r_iI_j r_i+r_jI_k r_j+r_iI_k 0

⎞

⎟⎠,

whichimpliesthat ∂(F_i, F_j, F_k)

∂(ri, rj, rk)

= 8(r_j+r_kI_i)(r_k+r_iI_j)(r_k+r_iI_j) + 8(r_k+r_jI_i)(r_i+r_kI_j)(r_j+r_iI_k)

= 16rirjrk(1 +IiIjIk) + 8riγijk(r²_j+r²_k) + 8rjγjik(r²_i +r²_k) + 8rkγkij(r²_i +r²_j).

Bythecondition(2.7) andtheCauchyinequality,wehave ∂(F_i, F_j, F_k)

∂(ri, rj, rk)

≥16rirjrk(1 +IiIjIk+γijk+γjik+γkij)

= 16r_ir_jr_k(1 +I_i)(1 +I_j)(1 +I_k).

BytheconditionthatI_i,I_j,I_k∈(−1,+∞),wehave ^∂(Fi,Fj,Fk)

∂(ri,rj,rk)

>0 foranyr∈R³>0.If thereare r= (ri,rj,rk)∈R³>0 and r= (r_i,r_j,r_k)∈R³>0 satisfyingF(r)=F(r), then wehave

0 =F(r)−F(r) = ∂(Fi, Fj, Fk)

∂(r_i, r_j, r_k)

r+θ(r−r)·(r−r)^T, θ∈(0,1), which implies r = r by the nondegeneracy of ^∂(F_∂(r^i,Fj,Fk)

i,rj,rk) on R³>0. So the map F is injectiveonR³>0,whichimpliesthatH isinjectiveonR³>0.

Notethat

F_i=r²_j+r²_k+ 2r_jr_kI_i≥2r_jr_k(1 +I_i), F_j =r²_i +r²_k+ 2r_ir_jI_k ≥2r_ir_k(1 +I_j), Fk =r²_i +r²_j+ 2rirjIk ≥2rirj(1 +Ik).

By the condition thatIi,Ij,Ik ∈ (−1,+∞), ifF is bounded, we have rirj, rirk, rjrk

arebounded, whichimpliesthatr_i²+r²_j,r²_i+r_k²,r²_j+r²_karebounded.Similarly,wehave F_i ≤(1+|I_i|)(r²_j+r_j²). ThisimpliesthatF is aproper mapfrom R³>0 to R³>0.Bythe invarianceofdomaintheorem,wehaveF isadiffeomorphismbetweenR³>0andF(R³>0).

AndthenH isadiffeomorphismbetweenR³>0 andH(R³>0).

Set

L={(li, lj, lk)|li+lj > lk, li+lk> lj, lj+lk > li},

thenΩ^E_ijk=H⁻¹(H(R³>0)∩ L).ToprovethatΩ^E_ijk issimplyconnected,wejustneedto provethatH(R³>0)∩ Lisacone.NotethatLisaconeinR³>0boundedbythreeplanes

Li={(li, lj, lk)∈R³>0|li=lj+lk}, Lj={(li, lj, lk)∈R³>0|lj=li+lk}, Lk={(li, lj, lk)∈R³>0|lk=li+lj}.

Note thatH isadiffeomorphismbetweenR³>0 andH(R³>0),H(R³>0) isaconebounded bythree quadraticsurfaces

Σ_i={(l_i, l_j, l_k)∈R³>0|l_i²=l²_j+l²_k+ 2l_jl_kI_i}, Σ_i={(l_i, l_j, l_k)∈R³>0|l_j²=l²_i +l²_k+ 2l_il_kI_j}, Σi={(li, lj, lk)∈R³>0|l_k²=l_i²+l²_j+ 2liljIk}.

In fact, ifri = 0, then lj = rk,lk =rj and l²_i = r_j²+r²_k+ 2rjrkIi =l²_j+l²_k+ 2ljlkIi. Σ_i isinfacttheimageof r_i = 0 underH. BythediffeomorphismofH,Σ_i,Σ_j, Σ_k are mutually disjoint.Furthermore, if Ii ∈ (−1,1], we have(lj−lk)² < l_i² ≤(lj +lk)² on Σ_i. And if I_i ∈ (1,+∞), we have l_i² > (l_j +l_k)² on Σ_i. This implies that Σ_i ⊂ L if Ii∈(−1,1] andΣi∩ L=∅ifIi∈(1,+∞).SimilarresultsholdforΣj and Σk.

To provethatH(R³>0)∩ Lis acone, wejustneed toconsider thefollowing casesby thesymmetrybetweeni,j,k.

IfIi,Ij,Ik∈(−1,1],H(R³>0)∩ LisaconeboundedbyΣi,Σj,Σk andH(R³>0)∩ L= H(R³>0).

IfIi,Ij ∈(−1,1] andIk ∈(1,+∞),H(R³>0)∩ LisaconeboundedbyΣi,Σj andLk. IfI_i∈(−1,1] andI_j,I_k ∈(1,+∞),H(R³>0)∩ LisaconeboundedbyΣ_i,L_j andL_k. IfIi,Ij,Ik∈(1,+∞),H(R³>0)∩ Lisaconebounded byLi,Lj and Lk.Inthiscase, H(R³>0)∩ L=L.

For any case,H(R³>0)∩ Lisacone inR³>0.BythefactthatH is adiffeomorphism between R³>0 and H(R³>0), we havethe admissible space Ω^E_ijk =H⁻¹(H(R³>0)∩ L) is simplyconnected.

By the analysis above, if H(R³>0) ⊂ L, then Ω^E_ijk = H⁻¹(H(R³>0)∩ L) = R³>0. If H(R³>0)\ L=∅,thenΩ^E_ijkisapropersubsetofR³>0.IfIi >1,theboundarycomponent Σi={l_i²=l²_j+l²_k+2ljlkIi}isoutofthesetL.BythefactthatΩ^E_ijk =H⁻¹(H(R³>0)∩L) andH :R³>0→H(R³>0) isadiffeomorphism,wehaveH⁻¹(L_i) isaconnectedboundary component of Ω^E_ijk, on which θi = π,θj = θk = 0. This completes the proof of the lemma. 2

Corollary 2.4. For atopological triangle ijk ∈F with inversive distance I >−1 and γijk ≥0,γjik≥0,γkij≥0,thefunctionsθi,θj,θk defined onΩ^E_ijk could becontinuously extended byconstant toθi,θj,θk defined onR³>0.

Remark4.

(1) IfIi,Ij,Ik ∈[0,+∞),obviouslywehaveγijk≥0,γjik ≥0,γkij ≥0.SoLemma2.3 generalizesLemma3in [22] obtainedbyGuo.

(2) IfIi,Ij,Ik ∈(−1,1] andγijk ≥0,γjik ≥0, γkij ≥0, bythe proof of Lemma2.3, Ω^E_ijk =R³>0, whichisobtainedbyZhou [37].

(3) TheconditionIi,Ij,Ik ∈(−1,+∞) and γijk ≥0,γjik ≥0,γkij ≥0 containsmore cases,forexample,Ii=−¹₂,Ij = 1 and Ik = 2,inwhichcase theadmissiblespace Ω^E_ijk isstillsimplyconnected.

2.2. Infinitesimalrigidity ofEuclidean inversivedistancecirclepackings

Setui = lnri, then we haveUijk^E := ln(Ω^E_ijk) isa simplyconnected subset of R³ by Lemma 2.3. If (ri,rj,rk) ∈ Ω^E_ijk, li,lj,lk satisfy the triangle inequalities and forms a Euclidean triangle.Denote theinner angle at thevertex ias θi. Wehavethefollowing usefullemma.

Lemma2.5. Forany topologicaltriangleijk∈F,wehave

∂θ_i

∂uj

= ∂θ_j

∂ui

= 1 Al_k²

r²_ir²_j(1−I_k²) +r_i²r_jr_kγ_ijk+r_ir_j²r_kγ_jik

(2.8) onUijk^E ,whereA=l_jl_ksinθ_i.

Proof. Bythecosinelaw, wehavel²_i =l²_j+l²_k−2ljlkcosθi.Taking thederivativewith respect to li, we have ^∂θ_∂lⁱ

i = ^l_Aⁱ, where A=ljlksinθi is two times ofthe areaof ijk.

Similarly,wehave ^∂θ_∂lⁱ

j = ^−licos_A ^θk,^∂θ_∂lⁱ

k = ^−licos_A ^θj.Bythedefinitionofl_i,l_j,l_k,wehave

∂li

∂r_j = rj+rkIi

l_i ,∂lj

∂r_j = 0,∂lk

∂r_j = rj+riIk

l_k . Then

∂θi

∂uj

=rj

∂θi

∂rj

=rj(∂θi

∂li

∂li

∂rj

+∂θi

∂lk

∂lk

∂rj

)

=rj

rj+rkIi

A −licosθj(rj+riIk) Al_k

= 1 Alk

lk(r_j²−rjrkIi)−l²_i +l²_k−l²_j 2lk

(r_j²+rirjIk)

= 1 Al_k²

r²_ir²_j(1−I_k²) +r²_irjrkγijk+rir_j²rkγjik

,

where the cosine law is used inthe third line and the definitionof thelength (2.1) is used inthefourthline.Thisalsoimplies _∂u^∂θi

j =_∂u^∂θj

i. 2 Remark5.Theequation _∂u^∂θi

j = ^∂θ_∂u^j

i hasbeen obtainedunderdifferent conditionsin [9, 11,22] andtheformulas for ^∂θ_∂lⁱ

j and ^∂θ_∂lⁱ

i wasobtainedbyChow andLuo [9]. Ingeneral, for Ii,Ij,Ik ∈(−1,+∞), _∂u^∂θi

j havenosign. However,ifIi,Ij,Ik ∈(−1,1] and γijk≥0, γjik ≥0,γkij ≥0,by(2.8),wehave _∂u^∂θi

j ≥0.Furthermore, _∂u^∂θi

j = 0 ifandonlyifIk = 1 andI_i+I_j= 0.Especially,ifI_i= cos Φ_i,I_j= cos Φ_j,I_k = cos Φ_kwithΦ_i,Φ_j,Φ_k∈[0,^π₂], we have _∂u^∂θi

j ≥0,and _∂u^∂θi

j = 0 ifandonlyifΦk= 0 andΦi = Φj =^π₂.

Remark 6.Geometrically, the three circles at thevertices have apowercenter O. It is known [35,36] that _∂u^∂θi

j = ^h_l^k

k, where hk isthesigned distanceof thepowercenterO to theedge{ij},whichispositiveifOisintheinteriorofthetriangleijkandnegativeif thepowercenterOisoutofthetriangleijk.SoundertheconditionI_i,I_j,I_k ∈(−1,1]

and γ_ijk ≥0,γ_jik≥0,γ_kij≥0,thepowercenterO isinthetriangleijk.

Lemma2.5showsthatthematrix

Λ^E_ijk:= ∂(θi, θj, θk)

∂(u_i, u_j, u_k) =

⎛

⎜⎜

⎝

∂θi

∂ui

∂θi

∂uj

∂θi

∂uk

∂θj

∂ui

∂θj

∂uj

∂θj

∂uk

∂θk

∂ui

∂θk

∂uj

∂θk

∂uk

⎞

⎟⎟

⎠

is symmetriconUijk^E .ForthematrixΛ^E_ijk, wehavethefollowingusefulproperty.

Lemma 2.6. For any topological triangle ijk ∈ F with inversive distance Ii,Ij,Ik ∈ (−1,+∞)andγ_ijk ≥0,γ_jik ≥0,γ_kij≥0,thematrixΛ^E_ijk isnegativesemi-definitewith rank2and kernel{t(1,1,1)^T|t∈R}onUijk^E .

Proof. TheproofisparalleltothatofLemma6in [22] withsomemodifications.Bythe calculationsinLemma2.5, foratriangleijk∈F,wehave

⎛

⎜⎝ dθ_i dθj

dθk

⎞

⎟⎠=−1 A

⎛

⎜⎝

l_i 0 0 0 lj 0 0 0 lk

⎞

⎟⎠

⎛

⎜⎝

−1 cosθ_k cosθ_j cosθk −1 cosθi

cosθj cosθi −1

⎞

⎟⎠

×

⎛

⎜⎜

⎜⎝

0 ^l

2 i+r_j²−r²k

2lirj

l²_i+r_k²−r²j

2lirk

l²_j+r_i²−r²k

2ljri 0 ^l

2 j+r_k²−r²i

2ljrk

l²_k+r²_i−r²j

2lkri

l²_k+r²_j−r²i

2lkri 0

⎞

⎟⎟

⎟⎠

⎛

⎜⎝

ri 0 0 0 rj 0 0 0 rk

⎞

⎟⎠

⎛

⎜⎝ dui

duj

duk

⎞

⎟⎠.

Writetheaboveformulaas