ERROR TERM IN POINTWISE APPROXIMATION OF THE CURVATURE OF A CURVE

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Niveau: Supérieur, Doctorat, Bac+8
ERROR TERM IN POINTWISE APPROXIMATION OF THE CURVATURE OF A CURVE VINCENT BORRELLI AND FABRICE ORGERET .Let P be a polygonal line approximating a planar curve curve ?, the discrete curvature kd(P) at a vertex P ? P is (usually) defined to be the quotient of the angle between the normals of the two segments with vertex P by the average length of these segments. In this article we give an explicit upper bound of the difference |k(P)? kd(P)| between the curvature k(P) at P of the curve and the discrete curvature in terms of the polygonal line's data, the supremums over ? of the curvature function k and its derivative k?, and a new ge- ometrical invariant, the return factor ??. One consequence of this upper bound is that it is not needed to know precisely which curve is passing through the vertices of the polygonal line P to have a pointwise information on its curvature. Keywords: Smooth curves; Discrete curvatures, Approximation, Differential Geometry. 1. INTRODUCTION 1.1. Generalities. Curvature estimates are often needed in applications from computer vi- sion, computer graphics, geometric modeling and computer aided design. Indeed, they are of a fundamental importance for the intelligence of the geometry of meshes and there is, in applied geometry, a vast literature devoted to their study ([2],[3],[5],[9],[10], for instance).

  • any reasonable curve

  • passing through

  • such points exist

  • average length

  • point de ?

  • ciently small neighborhood

  • going through every

  • flat point

  • small compared


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ERRORTERMINPOINTWISEAPPROXIMATIONOFTHECURVATUREOFACURVEVINCENTBORRELLIANDFABRICEORGERET.LetPbeapolygonallineapproximatingaplanarcurvecurveΓ,thediscretecurvaturekd(P)atavertexPPis(usually)definedtobethequotientoftheanglebetweenthenormalsofthetwosegmentswithvertexPbytheaveragelengthofthesesegments.Inthisarticlewegiveanexplicitupperboundofthedifference|k(P)kd(P)|betweenthecurvaturek(P)atPofthecurveandthediscretecurvatureintermsofthepolygonalline’sdata,thesupremumsoverΓofthecurvaturefunctionkanditsderivativek0,andanewge-ometricalinvariant,thereturnfactorΩΓ.OneconsequenceofthisupperboundisthatitisnotneededtoknowpreciselywhichcurveispassingthroughtheverticesofthepolygonallinePtohaveapointwiseinformationonitscurvature.Keywords:Smoothcurves;Discretecurvatures,Approximation,DifferentialGeometry.1.INTRODUCTION1.1.Generalities.Curvatureestimatesareoftenneededinapplicationsfromcomputervi-sion,computergraphics,geometricmodelingandcomputeraideddesign.Indeed,theyareofafundamentalimportancefortheintelligenceofthegeometryofmeshesandthereis,inappliedgeometry,avastliteraturedevotedtotheirstudy([2],[3],[5],[9],[10],forinstance).Neverthelessandasfarasweknow,thisliteratureonlyfocussesontheconvergenceissue:givenasequenceofmeshesapproximatingasurface,dodiscretecurvaturesconvergeto-wardscurvaturesofthesurface?Yetthereisanotherquestion,atleastasimportant,theapproximationissue:givenonemeshapproximatingasurface,canweevaluatetheerrorbetweendiscretecurvaturesandcurvaturesofthesurface?Thegoalofthisarticleistobeginthestudyofthisapproximationissueinaone-dimensionalsetting,forcurves.1.2.ApproximationProblemforCurves.Therearemanywaystotacklewiththeprob-lemofpointwiseapproximationofcurvature.ThemostnaturaloneisprobablytostartwithacurveΓandasequenceofpolygonallines(Pn)nNwhichinterpolatethecurvemoreandmoreclosely,thentodefineadiscretecurvaturekd(Pin)oneveryvertexPinPnsuchthatthenumbers{kd(Pin)}nNtendtowardthecurvaturek(Pi)ofΓatPi=limnPin.Thisapproachturnsouttobeboth,relativelyeasyandefficient:ifPi1,Pi,Pi+1arethreeconsecutiveverticesofapolygonallinePthenthefollowingnumberγπkd(Pi)=i,ηiwithγi=(PiPi1,PiPi+1)[0,π[1andηi=(|PiPi1|+|PiPi+1|)2doestheworkunderveryweakhypothesisonthepolygonalsequence(see[2]forinstanceortheproofof[8]).Nevertheless,althoughnatural,thisprocessisinadequateforconcrete1