High quality local interpolation by composite parametric surfaces

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by  Michele Antonelli

High quality local interpolation by composite parametric surfaces
MicheleAntonellia,CarolinaVittoriaBeccarib,GiulioCasciolab
aDepartmentofMathematics,UniversityofPadova,ViaTrieste63,35121Padova,Italy
bDepartmentofMathematics,UniversityofBologna,PiazzadiPortaSanDonato5,40126Bologna,Italy
Abstract
InCAGDthedesignofasurfacethatinterpolatesanarbitraryquadrilateralmeshisdefinitelyachallengingtask. The
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basicrequirementistosatisfybothcriteriaconcerningtheregularityofthesurfaceandaestheticconcepts.
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Withregardtotheaestheticquality,itiswellknownthatinterpolatorymethodsoftenproduceshapeartifactswhen
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2 thedatapointsareunevenlyspaced. Intheunivariatesetting,thisproblemcanbeovercome,oratleastmitigated,by
exploitingapropernon-uniformparametrization,thataccountsforthegeometryofthedata. Moreover,recently,the
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same principle has been generalized and proven to be effective in the context of bivariate interpolatory subdivision
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J schemes.

7 In this paper, we propose a construction for parametric surfaces of good aesthetic quality and high smoothness
thatinterpolatequadrilateralmeshesofarbitrarytopology.

] In the classical tensor product setting the same parameter interval must be shared by an entire row or column
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of mesh edges. Conversely, in this paper, we assign a different parameter interval to each edge of the mesh. This
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particular structure, which we call an augmented parametrization, allows us to interpolate each section polyline of
h. themeshatparametersvaluesthatpreventwigglingoftheresultingcurveorotherinterpolationartifacts. Thisyields
t highqualityinterpolatorysurfaces.
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m TheproposedsurfacesareageneralizationofthelocalunivariatesplineinterpolantsintroducedinBeccarietal.
(2013)andAntonellietal. (2014),thatcanhavearbitrarycontinuityandarbitraryorderofpolynomialreproduction.
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In particular, these surfaces retain the same smoothness of the underlying class of univariate splines in the regular

1 regions of the mesh (where, locally, all vertices have valence 4). Moreover, in mesh regions containing vertices of
v valenceotherthan4,wesuitablydefineG1-orG2-continuoussurfacepatchesthatjointheneighboringregularones.
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3 Keywords: Quadrilateralmesh,Localinterpolation,Non-uniformparametrization,Surface,Arbitrarytopology,
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Curvenetwork
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2010MSC: 65D05,65D07,65D17
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.
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1 1. Introduction
:
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A central topic in computer-aided geometric design is the construction of parametric curves and surfaces that
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X interpolate a given set of points. In 2D, these points are the vertices of a control polygon, whereas in 3D they are
r thevertices ofa controlpolyhedron (alsocalled amesh). In thiscontext, a “good”interpolant isone thatfaithfully
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mimicstheshapesuggestedbytheinputdata: thismeans,e.g.,thatitdoesnotpresentselfintersections,noritbends
tomuchoritistootightwithrespecttothegivencontrolpolygonorpolyhedron.
Concerningunivariateinterpolation, itiswellknownthataproperchoiceofparametrizationiscrucialtoobtain
goodqualitycurvesthatinterpolateunevenlyspaceddata(basicreferencesare[1,2]). Inparticular,auniformpara-
metrizationmaygiverisetonoticeableinterpolationartifacts,sinceitdoesnotaccountforthegeometryofthedata.
Theseartifactsoftendisappear,oraregreatlymitigated,whenasuitablenon-uniformparametrizationisused(Figure
1isanexample).
Emailaddresses:[email protected](MicheleAntonelli),[email protected](CarolinaVittoriaBeccari),
[email protected](GiulioCasciola)
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(a) Uniformparametrization (b) Centripetalparametrization
Figure1:Cubic(global)splineinterpolationofunevenlyspaceddata.Theuniformparametrizationfailswhereasapropernon-uniformparametri-
zationyieldsagoodinterpolant.
Although there is probably no “best” parametrization, since any method can be defeated by a suitably chosen
dataset,sometechniquesproducegoodresultsinthemajorityofcriticalcases. Oneofthemosteffective[3–5],and
probablythemostused, isthecentripetalparametrization. Alongsidethis, itisworthmentioningtheNielson-Foley
parametization[6]andtherecentmethodproposedin[7],which,insomecases,showsbetterperformancethaneither
oftheprevious.
The possibility to manage a non-uniform parametrization is even more crucial when using local interpolation
methods. Thelatteraremoreefficientthanglobalapproaches,becausetheydonotinvolvesolvingsystemsoflinear
equations, which can readily be of large dimension when complex surfaces have to be generated. Moreover, local
modificationofthedatacanbehandledbylocalupdatingoftheinterpolationmodelanddoesnotrequirerecomputing
the entire model from scratch. However, local interpolation techniques are more prone to generating interpolation
artifactsthanglobalonesandthereforetheysufferevenmorefromapoorchoiceofparametrization.
The construction of local spline interpolants with specific properties and non-uniform parametrization has long
been an open question. This is probably one of the reasons why, so far, local interpolation methods have had low
uptakeincomputerdesignapplications.Intheunivariatesetting,tworecentworkshavedevelopedageneralapproach
forsolvingtheproblem[8,9]. Thetechniqueproposedinthosepapersallowsustochooseanarbitrarysupportwidth
and,correspondingly,constructvariousclassesoflocalsplineinterpolantsthatdifferonefromanotherintheirdegree,
continuityandapproximationorder. Inthisway,onecanpicktheinterpolatingsplinewhosepropertiesbestsuitthe
contextofapplication. Moreover,thepopularCatmull-Romsplines[10],aswellasothertypesoflocalinterpolants
previouslyappeared(anon-exhaustivelistincludes[11–15])arespecialinstancesoftheconstruction.
Inthispaperwegeneralizetheunivariatelocalinterpolatorysplinesin[8,9]tothebivariatesetting. Inparticular,
by introducing an appropriate new parametrization technique, we will obtain parametric surfaces of good aesthetic
qualityandhighsmoothness.
We shall start by considering the case of a regular mesh, which is one where each vertex has exactly 4 incident
edges. The main hurdle is represented by the need of defining a non-uniform parametrization that preserves the
qualityofthelocalinterpolants,whengeneralizingfromtheunivariatetothebivariatesetting. Thestandardapproach
is to construct a tensor product surface from the univariate splines that one wishes to use. Being uv the parametric
domain, the tensor product requires the generation of one set of parameter values for all isoparametric curves in
the u-direction; the same holds for the v-direction. For finding such a parametrization, the most reasonable way of
proceedingistocreateagoodparametrizationforeachisoparametriccurveusingasuitablemethod,suchas,e.g.the
centripetal parametrization, and then average each of these parameterizations to yield one. This approach will only
produceacceptableresultsifalltheisoparametriccurvesessentiallyyieldthesameparametrization.Conversely,when
thedatapointssignificantlydeviatefromaregulargrid,itwillresultinapoorchoiceinparameters.Theisoparametric
curveswillunnaturallywiggle,andthisdefectwillbereflectedinthesurface(see,e.g.[2,§7.5.1]).
Theabovediscussionsuggeststhattheclassicalideaofaveragingtheparametrizationshouldbeavoided. Hence
wewilltakeanotheravenue. Inparticularweassigntoeachmeshedgeaparameterinterval, withoutrequiringthat
thesameparameterintervalissharedbyanentireroworcolumnofmeshedges. Wecallsuchaparametrizationan
augmentedparametrization.
A similar parametrization concept has previously been exploited in the context of subdivision schemes. It was
initiallyproposedinconnectionwithCatmull-Clarksurfaces[16–19],toenrichthemethodwithgreatercontrolover
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theshapeoffree-formobjectsandlocalshapeeffectssuchascreaseedges. Infact,theterm“augmented”wasfirstly
introducedin[18]withasimilarmeaningtotheoneweusehere. Morerecently,aninterpolatorysubdivisionscheme
withaugmentedparametrizationhasbeendiscussedin[20].Theschemeallowsforinterpolatingeachsectionpolyline
ofthemeshatindependentparametervaluesandthereforethemostappropriateparametrizationisusedtoconstruct
eachsectioncurveofthelimitsurface.Thisapproachyieldssurfacesoffarsuperioraestheticqualitycomparedtotheir
uniformortensor-productcounterparts. Unfortunately,itissubjecttothetypicalissuesofinterpolatorysubdivision
schemes. Inparticulartheresultingsurfacescannotbeevaluatedatarbitraryparametersandhavesmoothnesslimited
toC1.
The construction of the local interpolatory surfaces proposed in this paper proceeds as follows. We start by
assumingthatagoodparametrizationhasbeencomputedindependentlyforeachsectionpolylineofthemeshusing
a suitable method. Accordingly, a different parameter interval is assigned to each mesh edge, which gives rise to
an augmented parametrization. Next, we choose a class of local univariate spline interpolants based on the desired
smoothness,orderofpolynomialreproductionorsupportwidth. Wethensuitablyconstructacompositesurfacethat
containsalltheinterpolatorycurvesofthisclassgeneratedfromtheindividualsectionpolylinesofthemeshwiththe
associatedparameterizations.Inthisway,eachsectionpolylineofthemeshisinterpolatedattheparametervaluesthat
allowthebestqualityoftheresultingsectioncurve. Moreover,thesectioncurvesare“featurecurves”ofthesurface,
which,inturn,isagoodqualityinterpolanttotheinputmesh. Wealsoprovethat,iftheunderlyingunivariatesplines
areCk continuous, thenthecompositesurfaceisGk (meaningthatderivativesagreeaftersuitablereparametrization
[21]). Asaconsequence,thecompositesurfaceretainsthegoodpropertiesofthecorrespondingunivariatesplines.
Oppositelytoglobaltechniques,theconsideredapproachdoesnotrequiretosolvelargesystemsoflinearequa-
tionsandallowsforlocaleditingofthegeneratedsurfaces. Moreover,otherlocalmethodstypicallyrequiretosupply
additional input data, such as cross-boundary derivatives and twist vectors (an example is given by the well-known
Coonspatches[1]). Amajordrawbackisthatthesequantitiesneedtobeheuristicallyestimatedandthisislikelyto
beoneofthereasonswhy,tothebestofourknowledge,onlyG1 orG2 Coonspatcheshavebeenconsideredsofar.
Conversely,thesurfacesproposedinthispapercanhavehighercontinuity.
An important consequence of the locality of the proposed construction is that the method can be extended to
handlemeshescontainingextraordinaryvertices(namelyverticeswithvalencedifferentthan4). Anysuchmeshcan
bepartitionedintoregularandextraordinaryregions. Looselyspeaking,regularregionsarethosewhereallvertices
areregularandthereforethelocalinterpolationschemecanbeapplied. Patchingtheregularregionswillgeneratea
surfacewith“holes”correspondingtotheextraordinaryregions. Acrosstheboundaryofsuchholes,thetangentfield
(and higher order derivative fields) are determined by the surrounding regular patches and, as a consequence of the
augmentedparametrization, theychangeateveryboundarypointinapeculiarway. Themissingpartofthesurface
shallbedefinedinsuchawaytointerpolatetheexistingderivativefields,uptoareasonableorderofcontinuity.
The general problem of how to fill the hole around an extraordinary vertex is an active research topic even in
thecaseoftheuniformparametrization. Recentdevelopmentsinclude[22–24],wheretransfinitemulti-sidedpatches
are generated by interpolating derivative ribbons. Our main point here is not to provide a one-stop solution, but to
demonstratehowtoobtaingoodqualitylocalinterpolatingsurfacesbyexploiting,insynergy,theaugmentedparame-
trization,theregularpatchesdescribedaboveandalocalpatchingschemeforextraordinaryvertices. Wewilladdress
thisissuebymansofeitherG1Coons-Gregorypatches[1,25,26]andtheirG2version[27,28].Inbothcases,wewill
seehowtosuitablytweakthedefinitionofthepatches,inordertointerpolatethedesiredboundaryinformation. This
willallowustogenerateaugmentedsurfacesthatinterpolateameshcontainingextraordinaryvertices,havearbitrary
smoothnessintheregularregions,andareG1orG2continuousintheextraordinaryregions.
We would like to remark that the research reported in this paper was carried out under the European Eurostars
projectNIIT4CAD,aimedatthedevelopmentofnewtechnologies,mainlybasedonsubdivisionmethods,formodel-
ingarbitrarytopologysurfaceswithinaCADsystem. Oneoftheadvancedobjectivesoftheprojectwastoconstruct
localinterpolatorysurfacessuitableforintegrationinaCADsystemandthispaperisapartofsuchastudy.
Theremainderofthepaperisorganizedasfollows. InSection2werecalltheunivariatelocalsplineinterpolants
presentedin[8,9]. Basedonthese,inSection3weintroduceourlocalmethodtogenerateinterpolatorysurfacesof
goodqualityandhighorderofcontinuity, givenasinputaregularmesh. Wealsodiscussalltherelevantproperties
of the constructed surfaces. Section 3.1 presents some application examples to demonstrate the effectiveness of the
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methodanditssuperiorperformancewithrespecttotheclassicaltensorproductapproach. InSection4weaddress
the problem of patching the extraordinary regions of the mesh by G1 and G2 augmented Coons-Gregory patches.
The latter patches are constructed to interpolate boundary curves and cross-boundary derivative fields. When this
informationisnotavailable,itneedstobesuitablygeneratedandSection4.3illustratesapossibleapproachtohandle
thisissue. Section4.4presentsexamplesofaugmentedsurfacescontainingextraordinaryverticesandfinallySection
5summarizestheresultsachievedandsuggestssometopicsforfutureresearch.
2. Local,non-uniform,univariatesplineinterpolation
Suppose we are given a sequence of points p ,p ,...,p in Rn, n (cid:62) 2, where p and p are distinct and
0 1 N j j+1
p = p andanassociatedsequenceofparametervalues x < x < ··· < x . Weconsidertheperiodicsplinecurve
N 0 0 1 N
F:[x ,x ]→Rndefinedas
0 N (cid:88)
F(x)= p ψ(x), (1)
i i
i
whereψ : [x ,x ] → Rarefundamentalsplinefunctions,namelytheyarepiecewisefunctionsonthepartition{x }
i 0 N j
suchthatψ(x ) = δ . Sinceweareinterestedinlocalinterpolation, werequirethateveryψ hascompactsupport,
i j i,j i
namelythatitisnonzeroinafinitenumberofparametricintervals.Inthisway,whenthedatapointsareinRn,n=2,3,
formula(1)representsaparametriccurvethatlocallyinterpolatesthegivendata.Itisevidentthatthisapproachallows
ustoovercometheneedforsolvinglinearsystemsofequationsthatarisewhenglobalsplineinterpolationisused.
Intheremainderofthepaper,wesupposethatthefundamentalfunctionsψ haveevensupportwidthw,namely
i
that
ψi(x)=0, x(cid:60)[xi−w,xi+w].
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Thisassumptionguaranteesthataninterpolatingspline(1)preservespossiblesymmetriesinthedataandallowsusto
consistentlysimplifythenotationthatwillbeintroducedlateron.
If we limit ourselves to considering polynomial splines, the salient properties that characterize the fundamental
functionsaredegree,continuity,maximumdegreeofpolynomialsthatcanbereproduced(thisisequaltotheapprox-
imationorderminusone)andsupportwidth. Inthisrespect,itiseasilyseenthattheparametricinterpolant Fin(1)
hasthesamepropertiesofthefundamentalfunctionsψ.
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Ageneralmethodforconstructinglocalinterpolatorysplineswithspecificpropertiesandnon-uniformparametri-
zationhasbeendevelopedinthetworecentpapers[8,9].Inthatframework,wecanchooseanarbitrarysupportwidth
and,correspondingly,constructvariousclassesoflocalsplineinterpolantsoftheform(1)thatdifferonefromanother
intheirdegree,continuityandapproximationorder. SpecialinstancesofsuchsplinesareCatmull-Romsplines[10],
aswellasothertypesoflocalinterpolantsincludingthosein[11–15].
Adopting the notation in [8, 9], we indicate a class of splines having Degree g, Continuity order k, Polynomial
reproductiondegreemandSupportwidthwbytheshorthandnotationDgCkPmSw. Tables1,2,and3in[8]andTable
2in[9]summarizethedifferentclassesthatwecanconstructforthemostusualchoicesofsupportwidth,namely4,
6and8. Someexampleswithw=4includethefamiliesD3C0P3S4,D3C1P2S4,D4C2P1S4,D5C2P2S4from[8]and
D2C1P2S4orD3C2P2S4from[9]. Evenmoreclassesofsplinescanbefoundwideningthesupportwidthto6or8.
Remark1. Thelocalsplineinterpolantsin[8]and[9]differinthefollowingway. In[8]afundamentalfunctionis
formedbyonepolynomialsegmentineachinterval[xi,xi+1]. Thework[9]generalizestheconstructioninsuchaway
that, inaparametricinterval, afundamentalfunctioncanbecomposedofmorethanonepolynomialsegment, with
propercontinuityatthejoinbetweenonesegmentandanother. SuchsplinesarecalledBr-splines, wherer denotes
thenumberofdifferentpolynomialpiecescontainedineachparameterinterval. ComparedtoB1splines,Brsplines
can have lower degree, at equal support width, continuity and order of approximation. For example, the classes
D2C1P2S4orD3C2P2S4mentionedabovearerespectivelyofB2andB3type.
Wenowintroducealocalnotationfortherestrictionofasplineinterpolant Ftoaparametricinterval[xs,xs+1].
This will allow us to emphasize the local dependence on data and parameters and will be useful to generalize the
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(a) (b)
Figure2: (a)Fundamentalfunctionψi oftheclass D3C1P2S4, withthedifferentpiecesrepresentedindifferentcolors. (b)Thefundamental
functionsoftheclassD3C1P2S4thatarenonzerointheinterval[xs,xs+1].
aboveframeworktothebivariatecase. Tothisaim,weshallobservethatineachparametricintervalthereareexactly
wnonzerofundamentalfunctionsandeachofthemdependsonasequenceofw−1parameterintervalsoftheform
di = xi+1−xi. (2)
Therefore,wecanconvenientlydefinethevector
(cid:16) (cid:17)
d= ds−w+1,...,ds,...,ds+w−1 , (3)
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which contains all the parameter intervals necessary to evaluate F in [xs,xs+1]. We call such a vector the local
parameter vector relative to [xs,xs+1] (or, equivalently, relative to psps+1). If we now map the interval [xs,xs+1] to
[0,d ],wecanwritethesegmentofFboundedby p and p as
s s s+1
s+w
(cid:12) (cid:88)2
F(x)(cid:12)(cid:12)[0,ds] = piψi(x,d), (4)
i=s−w+1
2
whereψi(x,d)istherestrictionofψi(x)to[xs,xs+1]expressedintermsofthesequenceofparameterintervals(3).
Asanexampleofoursettingandnotation,weprovideinAppendixAtheexpressionsforthewell-knownCatmull-
Rom splines [10], corresponding to the class D3C1P2S4 in [8]. The fundamental functions of this class are also
illustratedinFigure2.
Moreover, to facilitate the reader in reproducing the examples proposed in the following sections, we also list
in appendix the fundamental functions D5C2P2S4. The latter are an interesting application of the approach in [8],
since they have high continuity, in spite of a very limited support. Moreover we remark that, given our interest in
interpolating 3-dimensional data, it may be also useful to consider splines with continuityC3, or higher, and some
graphical examples will be presented in the forthcoming sections. As it can reasonably be expected, in this case
theexplicitexpressionofthefundamentalfunctionsismorecomplicated. Nevertheless, theevaluationofthespline
interpolantcanstillbeperformedinacomputationallyefficientwayasdiscussedin[8,9].
Asrecalledintheprevioussection, thechoiceoftheparametersequence{x }hasalargeinfluenceontheshape
j
ofaninterpolatingsplinecurveandinthisrespectvariouseffectivemethodscanbeconsidered,suchas[3,6,7,29].
Sincethefocusofthispaperisnotoncomparingdifferenttechniques,wewilltakeasarunningexamplethecentripetal
parametrization, which isacknowledged toproduce goodresults forthemajority ofcritical datasets. Accordingto
thecentripetalparametrization,fixedx ,theparametervaluesx , j=1,...,N arecomputedthrough
0 j
xi+1 = xi+(cid:13)(cid:13)(cid:13)pi+1− pi(cid:13)(cid:13)(cid:13)α2, (5)
with α = 1 and (cid:107)·(cid:107) denoting the Euclidean norm. It is therefore clear that the parameter values depend on the
2 2
geometryoftheinterpolationpoints. Inaddition,from(5),thechordalanduniformparameterizationscanbederived,
settingrespectivelyα=1andα=0[1,2].
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Inthefollowingsectionwegeneralizetothebivariatesettingtheconsideredlocal,non-uniform,univariatespline
interpolants. Before proceeding, we need to remark that our assumption to work with periodic data has the sole
purposeofsimplifyingthepresentation. Opencurvescansimilarlybedefined, providedthat“special”fundamental
functionsareusedinordertoevaluate(1)intheboundaryintervals. Thesefundamentalfunctionsaredefinedsoasto
interpolatethederivativesof Fuptosuitableorderat x and x andwereferthereaderto[9]formoredetails. The
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surfaceconstructiondevelopedinthefollowingsectioncanbeadaptedtohandleopendatasets,alongthesamelines
oftheunivariatecase.
3. Localinterpolationofregularmeshesbyhighqualitysurfacesofarbitrarycontinuity
InthissectionweassumethataregularmeshofpointsinR3 isgivenandwedevelopalocalmethodtogenerate
aninterpolatorysurfaceofgoodqualityandhighorderofcontinuity,basedontheunivariatefundamentalfunctions
introducedintheprevioussectionandonapropertechniqueofparametrization. By“goodquality”wemeanthatthe
interpolatingsurfacewillfittheshapesuggestedbytheinputdatainafaithfulmannerandwillnotpresentundesired
interpolationartifacts. Moreover,wewillprovethatthesurfacehasthesamesmoothnessoftheunderlyingclassof
fundamentalfunctions,whichcanhavearbitrarycontinuityinthegeneralframework[8,9].
Aregularmeshisonewhereeveryvertexbelongstofouredges(andfaces)andcanthusbeseenasarectangular
grid of 3D points, whose edges are associated with two independent domain directions. We say that two edges are
opposite when they have no vertex in common and they belong to the same face, whereas we say that two edges
are adjacent when they have one vertex in common and do not belong to the same face. We call an edge ribbon
anyorderedsequenceofpairwiseoppositeedgesandasectionpolylineapolylineformedbyasequenceofpairwise
adjacentedges. Moreover, wecallasectioncurveanycurvethatinterpolatestheverticesofasectionpolyline. For
simplicity, the discussion will be limited to meshes without boundary and therefore we can assume that all section
polylinesandcurvesareclosed.
In the introduction to this work, we have drawn the reader’s attention the well-known fact that a tensor-product
surfaceisnot,ingeneral,aninterpolantofgoodquality. Thisisbecauseallisocurvesofsuchasurfacemusthavethe
sameparametrization.Asrecalled,thelatterrequirementmayresultinanunnaturalwigglingofthesectioncurvesthat
isverylikelytohappenwhenthedatapointsareunevenlyspacedandisevenmoreevidentwhenlocalinterpolation
methodsareused.
In contrast to the tensor product technique, we wish to construct a surface where the vertices of each section
polylineareinterpolatedattheparametervaluesthatallowforthebestqualityoftheresultingsectioncurve.
Tothisaim,foreverysectionpolyline,wederiveanon-uniformparametersequencebyexploitinganappropriate
data-dependent technique of parametrization. Then we assign to the section polyline edges the resulting parameter
intervals.Forinstance,labeledby p themeshpoints,thecentripetalparametrizationappliedtoeachsectionpolyline
i,j
willyieldtheedgeparameterintervals
di,j (cid:66)(cid:13)(cid:13)(cid:13)pi+1,j− pi,j(cid:13)(cid:13)(cid:13)α2 and ei,j (cid:66)(cid:13)(cid:13)(cid:13)pi,j+1− pi,j(cid:13)(cid:13)(cid:13)α2, α= 21. (6)
Thiscanbeeasilyseencomparingtheaboveexpressionswithformulae(2)–(5).
Wecalltheresultingconfigurationofparametersanaugmentedparametrization. Theaugmentedparametrization
isfeaturedbythefactthattheparameterintervalsallocatedtotheedgesofonemeshfacewillnotform,ingeneral,a
rectangle.Theterminologywasfirstlyusedin[18],dealingwithnon-uniformsubdivisionschemes,whereaugmented
facesofamesharedefinedwithasimilarmeaning. Conversely,inatensorproductsurface,theparameterintervals
mustbeequalforalledgesbelonginginthesameedgeribbon,or,equivalently,theintervalsallocatedtotheedgesof
everymeshfacemustformarectangle.
We now aim to construct a composite surface, which we call an augmented surface, where every section curve
is a local, univariate, spline of class DgCkPmSw and has an independent parametrization determined by the edge
parameter intervals of the corresponding section polyline. This means that the section curve piece bounded by p
i,j
and p shouldhaveanassociatedparametricintervaloflengthequaltod . Analogously, theparametricinterval
i+1,j i,j
betweenanytwovertices, p and p ofthecorrespondingsectioncurveshouldhavelengthe . Inthisway,each
i,j i,j+1 i,j
section polyline of the mesh can be interpolated at the parameter values that allow the best quality of the resulting
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d d d d
−2,2 −1,2 0,2 1,2
p p p p p
−2,2 −1,2 0,2 1,2 2,2
e e e e e
−2,1 −1,1 0,1 1,1 2,1
d d d d
−2,1 −1,1 0,1 1,1
p p p p p
−2,1 −1,1 0,1 1,1 2,1
e−2,0 e−1,0 S˜ e0,0 S e1,0 e2,0
v v
d d d d
−2,0 u −1,0 u 0,0 1,0
p p p p p
−2,0 −1,0 0,0 1,0 2,0
e e e e e
−2,−1 −1,−1 0,−1 1,−1 2,−1
d d d d
−2,−1 −1,−1 0,−1 1,−1
p p p p p
−2,−1 −1,−1 0,−1 1,−1 2,−1
Figure3:LabelingofverticesandparameterintervalsfortheconstructionoftheaugmentedsurfacepatchShavingtheexpression(10).
sectioncurve.Moreover,suchcurvesserveasaguidefortheshapeofthesurface,whichwillbeinturnagoodquality
interpolantoftheinputmesh.
It is sufficient to illustrate the construction for a generic surface patch interpolating the points p ,p ,p ,
s,t s+1,t s,t+1
p andinparticular,withoutlossofgenerality,wecanconsiderthepatchwhichinterpolates p ,p ,p ,p ,
s+1,t+1 0,0 1,0 0,1 1,1
depictedinFigure3. (Asofnow,theneighboringpatchS˜ inFigure3shallbeoverlooked. Itwillbeusedlaterinthe
proofofProposition2.) Anyotherpatchcanbeanalogouslyderivedafterproperindexshift.
Let [0,1]2 be the parametric domain associated with the considered patch. The first stage of the construction
is to choose a class of local, univariate, spline interpolants DgCkPmSw, based on the properties that we seek in the
finalsurface. Hence, inviewoftheCk continuityofthefundamentalfunctions, weconsiderthepolynomialsδ (v)
i,j
i = −w +1,...,w −1, j = 0and(cid:15) (u)i = 0, j = −w +1,...,w −1,ofdegree2k+1,uniquelydeterminedbythe
2 2 i,j 2 2
followingconditions:
δi,j :[0,1]→[di,j,di,j+1],
δi,j(0)=di,j, and δi,j(1)=di,j+1, (7a)
d(r)(0)=d(r)(1)=0, r=1,...,k,
i,j i,j
and
(cid:15)i,j :[0,1]→[ei,j,ei+1,j],
(cid:15)i,j(0)=ei,j, and (cid:15)i,j(1)=ei+1,j, (7b)
(cid:15)(r)(0)=(cid:15)(r)(1)=0, r=1,...,k.
i,j i,j
The above polynomials, which we call the local parametrization functions, interpolate the two parameter intervals
correspondingtooppositeedgesofameshfaceandthereforehavetheeffectof“blending”suchintervals. Moreover,
the requirement that δ (v) and (cid:15) (u) have vanishing derivatives up to order k at 0 and 1 guarantees that they are
i,j i,j
monotonicfunctionswithpositivefirstderivative.
We use the local parametrization functions to associate with any couple of coordinates (u,v) ∈ [0,1]2 two local
parametervectorsdandeasfollows:
(cid:16) (cid:17)
d= d(v)= δ−w+1,0(v),...,δ0,0(v),...δw−1,0(v) , (8a)
2 2
and
(cid:16) (cid:17)
e=e(u)= (cid:15)0,−w+1(u),...,(cid:15)0,0(u),...,(cid:15)0,w−1(u) . (8b)
2 2
In this way, at the patch boundary described by (u,0), u ∈ [0,1], d = (cid:0)d−w/2+1,0,...,d0,0,...,dw/2−1,0(cid:1) is the local
parameter vector relative to p p (see definition (3)). In addition, at the opposite boundary (u,1), d is the local
0,0 1,0
7
parameter vector relative to p p . Analogously, for any point (0,v) or (1,v), v ∈ [0,1], e is the local parameter
0,1 1,1
vectorassociatedrespectivelywith p p or p p .
0,0 0,1 1,0 1,1
Wealsoconsiderthetwolocalvariables
x=uδ (v), and y =v(cid:15) (u), (9)
0,0 0,0
suchthatxandyspantheintervals[0,δ (v)]and[0,(cid:15) (u)]whilerespectivelyuandvvaryin[0,1].
0,0 0,0
Finally,forany(u,v)∈[0,1]2,wedefinetheaugmentedsurfacepatchSby
w w
(cid:88)2 (cid:88)2 (cid:16) (cid:17)
S(u,v)= p Ψ x,y,d,e , (10)
i,j i,j
i=−w+1j=−w+1
2 2
where
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Ψ x,y,d,e =ψ x,d ψ y,e , (11)
i,j i j
withlocalparametervectorsd,egivenby(8a)–(8b)andx,ycomputedthrough(9).
NotethatSdependsonalocalgridofmeshverticesofsizeequaltow×w,wherewisthesupportwidthofthe
underlyingfunctionsψ andψ . Forbrevity,wewillsometimessaythatthepatchhassupportw.
i j
Remark2. Forsimplicityofpresentationwehaveassumedthatthefunctionsψ(x,d)andψ (y,e)in(11)belongto
i j
the same family DgCkPmSw. However, it is nor difficult to see that the construction of an augmented patch can be
adaptedforthecasewherethefundamentalfunctionsbelongtotwodifferentclassesDgCkPmSw.
WeremarkthattheaugmentedpatchSisnotatensorproductsurface. Moreprecisely,wecaninterprettheabove
constructionasfollows. Foragiven(u¯,v¯)∈[0,1]2,letd¯ ande¯ bethevectors
d¯ = d(v¯), e¯ =e(u¯),
computedthrough(8a)–(8b)andletd¯ = δ (v¯)ande¯ = (cid:15) (u¯). ThenthevalueofSat(u¯,v¯)isequaltothevalueat
0,0 0,0
(x¯,y¯)ofthetensorproductpatch
w w
(cid:88)2 (cid:88)2 (cid:16) (cid:17) (cid:16) (cid:17)
T (x,y)= p ψ x,d¯ ψ y,e¯ , (x,y)∈[0,d¯]×[0,e¯].
(u¯,v¯) i,j i j
i=−w+1j=−w+1
2 2
i.e.S(u¯,v¯) = T (x¯,y¯). Inthisview,adifferenttensor-productpatchT determinesthevalueofSateachdomain
(u¯,v¯) (u¯,v¯)
point(u¯,v¯)∈[0,1]2.
TheconstructionoftheaugmentedsurfacepatchSisschematizedinFigure4foraclassoffundamentalfunctions
having support width w = 4 and, more generally, can be summarized as follows. Every mesh face gives rise to a
surfacepatch,parameterizedoverthedomain[0,1]2. Foragiven(u¯,v¯) ∈ [0,1]2 welocallyinterpolatetheparameter
intervals,separatelyintheuandvdirection,inordertogeneratetwolocalparametervectors d¯ = d(v¯)ande¯ = e(u¯).
We also map (u¯,v¯) into a couple of values (x¯,y¯). This mapping is such that, at the boundaries (u,0) and (u,1),
u ∈ [0,1], x spans respectively the entire intervals [0,d ] and [0,d ]. Analogously, at the boundaries (0,v) and
0,0 0,1
(1,v), v ∈ [0,1], y spans respectively the intervals [0,e ] and [0,e ]. These are precisely the parameter intervals
0,0 1,0
allocatedtotheedgesofthegivenmeshface. Finally,weconsiderthesurfaceT (x,y)definedastensorproductof
(u¯,v¯)
thelocalunivariatefundamentalfunctionsofclassDgCkPmSwond¯ ande¯ andwesetS(u¯,v¯)=T (x¯,y¯).
(u¯,v¯)
Theremainderofthissectionisdevotedtoshowingthattheresultingcompositesurfacehasallthesoughtprop-
erties. We start by proving that the surface section curves form a network of univariate spline interpolants of class
DgCkPmSw, where each curve has an independent non-uniform parametrization determined by the edge parameter
intervalsofthecorrespondingsectionpolyline. Inotherwords,ifanetworkofsectioncurvesofclassDgCkPmSwwas
independently determined before constructing the surface, then the surface would precisely be the transfinite inter-
polantofthesecurves.
Proposition 1. The section curves of an augmented surface based on a class of fundamental functions DgCkPmSw
arelocalunivariatesplinesinthesameclass.
8
v
1
d
(u¯,v¯) 0,1
e0,1 (cid:15)0,1(u¯) e1,1 u e0,0 e1,0
0 1
d−1,1 d0,1 d1,1 (cid:15)0,1(u¯) S(u¯,v¯)
e0,0 (cid:15)0,0(u¯) e1,0 d
δ−1,0(v¯) δ0,0(v¯) δ1,0(v¯) (cid:15)0,0(u¯) 0,0
d d d y¯ T(x¯,y¯)=S(u¯,v¯)
−1,0 0,0 1,0
e0,−1 (cid:15)0,−1(u¯)e1,−1 (cid:15) (u¯)
0,−1
δ−1,0(v¯) x¯ δ0,0(v¯) δ1,0(v¯)
Figure4:Constructionofanaugmentedpatchinthecasew=4.
Proof. It suffices to show that the section curve segment bounded by p and p (see Figure 3) belongs to the
0,1 1,1
unique local, non-uniform spline curve F of class DgCkPmSwthat interpolates the section polyline with vertices
...,p ,p ,p ,p ... andparameterintervals...,d ,d ,d ,d .... Hencethestatementcanbeextended
−1,1 0,1 1,1 2,1 −1,1 0,1 1,1 2,1
totheentirenetworkofsectioncurvesbylocallyapplyingthesameargument.
Accordingtothelocalrepresentation(4),wehave
w
(cid:88)2
F(x)= p ψ(x,d¯), x∈[0,d ], (12)
i,1 i 0,1
i=−w+1
2
withlocalparametervector
(cid:16) (cid:17)
d¯ = d−w+1,1,...,d0,1,...,dw−1,1 .
2 2
Thepatchboundarywithendpoints p and p isdescribedby(10)foru∈[0,1],v=1. Atanysuchpoint(u,v),
0,1 1,1
thefundamentalfunctionsψ oftheclass DgCkPmSw, j = −w/2+1,...,w/2−1, havethevaluesψ (y,e) = 1and
j 1
ψ (y,e)=0,forany j(cid:44)1,and,by(9),x=ud . Substitutingin(10)weget
j 0,1
w
(cid:88)2
S(u,1)= p ψ(x,d).
i,1 i
i=−w+1
2
Hence, the statement follows by observing that the local parameter vectors d and d¯ are equal and thus the curve
segmentsrepresentedbytheaboveformulaandby(12)areidentical.
Weshallnowstudythecontinuityofanaugmentedcompositesurface.Preliminarily,weobservethateachsurface
patchisdefinedasacompositionofinfinitelydifferentiablefunctions(weassumethattheedgeparameterintervalsare
nonzero, whenthemeshverticesaredistinct). Moreover,Proposition1guaranteesthecontinuityoftheconstructed
surface and entails that two neighboring augmented patches haveCk continuous derivatives in the direction of their
commonboundary. Thefollowingpropositionshowsthatthesamesmoothnessholdsinthecross-boundarydirection
andinparticularallowsustoconcludethattheconstructedsurfaceisgloballyGk continuous1. Foreaseofnotation,
weformulatethestatementforthetwopatchesSandS˜ representedinFigure3. Itisimmediatetoseethattheresult
holdswhenconsideringanytwoneighboringpatchesandtheircommonboundary, providedappropriateadjustment
ofnotation.
1Gkcontinuityreferstoagreementofderivativesagreeaftersuitablereparametrization[21].
9
Proposition2. LetSandS˜ bethetwoadjacentaugmentedsurfacepatchesdepictedinFigure3,basedonfundamental
functionsofclassDgCkPmSw. Thenthederivativesacrosstheircommonboundarysatisfytherelation
∂r ∂r
S(u,v)| =∆r(v) S˜(u,v)| , r=0,...,k, (13)
∂ur (0,v) ∂ur (1,v)
where
δ (v)
∆(v)= 0,0 , (14)
δ (v)
−1,0
andδ andδ arethelocalparametrizationfunctionsdefinedin(7a).
0,0 −1,0
The above result can be shown by direct verification, and we postpone the proof to the end of this section. For
nowitisimportanttoobservethatrelation(14)providesthescaling∆(v)thatrelatesthecross-boundaryderivatives
ofSandS˜. Ingeneral,∆(v)isdifferentateachboundarypoint,butvariessmoothlyalongtheboundary,beingdefined
astheratiooftwopositivepolynomials. AsaconsequenceofProposition2weobtainthefollowingresult.
Proposition 3. Two augmented surface patches built upon a class of fundamental functions DgCkPmSwjoin along
theircommonboundarywithGk-continuity.
Proof. Withoutlossofgenerality,wecantakethetwopatchesSandS˜ inProposition2andconsiderthemap
(cid:32) (cid:33)
∆(v)u+1
ρ(u,v)= ,
v
where ∆(v) is given by (14). The above function ρ is aCk reparametrization between the two domains of S and S˜.
Moreover,exploitingrelation(13),itisimmediatetoverifythat
∂r ∂r (cid:16) (cid:17)
S(u,v)| = S˜ ◦ρ (u,v)| , r=0,...,k.
∂ur (0,v) ∂ur (0,v)
TheaboveobservationsentailthatthetwoconsideredpatchesjoinalongtheircommonboundarywithGk-continuity.
ToconcludethissectionweprovideaproofofProposition2. Preliminarily,thefollowingLemmaisstatedasan
independentresult,sinceitwillbelaterrecalledinSection4.
Lemma 1. Let S be an augmented surface patch of the form (10). Then its cross-boundary derivatives satisfy the
followingrelations
∂r ∂r (cid:16) (cid:17)
S(u,v)| = S x,y,d,e | δr (v), u¯ =0,1,
∂ur (u¯,v) ∂xr (u¯,v) 0,0
(15)
∂r ∂r (cid:16) (cid:17)
S(u,v)| = S x,y,d,e | (cid:15)r (u), v¯ =0,1,
∂vr (u,v¯) ∂yr (u,v¯) 0,0
whereδ and(cid:15) arethelocalparametrizationfunctionsdefinedin(7a)–(7b).
0,0 0,0
Proof. Theresultimmediatelyfollowsbyusingthechainrule,therelation(9)andthefactthatδ(r)(v) = (cid:15)(r)(u) = 0,
i,0 0,j
r=1,...,k,u,v=0,1andi, j=−w +1,...,w −1.
2 2
We observe that the above Lemma is based on the fact that the local parametrization functions (7a)–(7b) have
vanishingderivativesattheendpointsoftheirintervalofdefinition. Aswewillseeshortly,thispropertyalsoplaysa
prominentroleinprovingProposition2. Aninterestingconsequenceisthat,inordertoprescribeacorrectsetoflocal
parametrizationfunctions,itisnotsufficienttorequirethatthesefunctionsinterpolatetheedgeparameterintervals.In
fact,althoughthisstrategymaystillallowustointerpolateanetworkofindependentlyparameterizedcurves,without
theaforementionedconstraintonthederivatives,wewouldnotbeabletoensurethatthecompositesurfaceisglobally
Gk continuous.
10