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Thepointisthatcubicsplinesstrikeagoodbalancebetweeneciencyandaccuracy,andarereasonablystraightforwardtoconstructtheyarethereforeagood ‘defaultchoice’forinterpolatingdataSplineCurvesAsplinecurveisamathematicalrepresentationforwhichitiseasytobuildaninterfacethatwillallowauserto designandcontroltheshapeofcomplexcurvesandsurfacesRecallA=(aij)isstrictlydiagonallydominantifjaiij>Xnj=1j6=ijaijjforalli=1;;nSpline segments–howtodefineapolynomialon[0,1]–thathasthepropertiesyouwant–andiseasytocontrolSplinecurves–howtochaintogetherlotsofsegments–sothatthewholecurvehasthepropertiesyouwant–andiseasytocontrolRefinementandevaluation–howtoadddetailtosplinesOverviewThepoint:An introductiontosplinesandasampleofthevariousapproachesSinceforn+given2AnExampleThedefaultRfunctionforttingasmoothingsplineiscalledThe syntaxis(x,y,cv=FALSE)wherexshouldbeavectorofvaluesforinputvariable,yisavectorofvaluesforForclampedsplines,therearetwoadditional equationsthatinvolveaandanCondition(*)explainswhythesplinehasadditionalinectionpoints.Ifwenowtakeanythreepoints[x0;y0];[x1;y1]and[x2;y2], wecansubstitutethenintotheequationtogetthreesimultaneousequationswhichwecansolvefortheunknownsa2,a1andaWenowhavetheequationofa curveinterpolatingthethreepointsTomLyche,CarlaManni,andHendrikSpeleersB-SplinesSplinesegments–howtodefineapolynomialon[0,1]–thathas thepropertiesyouwant–andiseasytocontrolSplinecurves–howtochaintogetherlotsofhereasplinetobeapiecewisepolynomial,andwediscussbothspline functionsandsplinecurvesandsurfacesTherearetwocommonapproachestoconstructingaThereisnolocalcontrol(changeofonecontrolpointaffectsthe wholecurve)DegreeofcurveisfixedbythenumberofcontrolpointsCurvesWestartwiththedefinitionofB-splinesbymeansofarecurrencerelation,and deriveseveraloftheirThegeneralapproachisthattheuserentersasequenceofpoints,andacurveisconstructedwhoseshapecloselyfollowsthissequence
FigureAnon-parametricspline.SeethehandoutTomLyche,CarlaManni,andHendrikSpeleers.EachcontrolpointisPrintedintheUnitedKingdomatthe UniversityPress,CambridgeAbstractThischapterpresentsanoverviewofpolynomialsplinetheory,withspe-cialemphasisontheB-splinerepresentation,spline approximationproperties,andhierarchicalsplinerefinementTheForclampedsplines,therearetwoadditionalequationsthatinvolveaandanCondition(*) explainswhythesplinehasadditionalinectionpointsLibraryofCongressCataloginginAsplinecurveisamathematicalrepresentationforwhichitiseasytobuild aninterfacethatwillallowausertodesignandcontroltheshapeofcomplexcurvesandsurfacesSeethehandoutaboutnaturalcubicsplineinterpolationAbstract Thischapterpresentsanoverviewofpolynomialsplinetheory,withspe-cialemphasisontheB-splineThenamesplinecomesfromthephysical(instrument)spline draftsmenusetoproducecurvesAgeneralcubicpolynomialisrepresentedby:y=Ax+Bx+Cx+DCubicSplineMimickingtheformofthepiecewiselinear interpolant,inthiscasewerequirethatoneachsubinterval[xi,xi+1]thepiecewiseinterpolantssatisfiess(x)=si(x)=ai+bi(xxi)+ci(xxi)2+di(xxi)3, whereai,bi,ci,anddiarecoefficientstobedeterminedAcatalogrecordforthispublicationisavailablefromtheBritishLibrary