BasictheoryofcurveandsurfaceDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMGeometricrepresentationParametric

Non-parametricExplicit

Implicit

y=f(x)f(x,y)=0x=x(u), y=y(u)DisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMGeometricrepresentationExample-circleParametric

Non-parametricExplicit

Implicit

y=R2–x2

x2+y2–R2=0x=Rcos, y=RsinDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMEachformhasitsownadvantagesanddisadvantages,dependingontheapplicationforwhichtheequationisused.GeometricrepresentationDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMNon-parametric(explicit)OnlyoneyvalueforeachxvalueCannotrepresentclosedormultiple-valuedcurvessuchascircley=f(x)DisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMNon-parametric(implicit)Advantages–canproduceseveraltypeofcurve–setthecoefficientsDisadvantagesNotsurewhichvariabletochooseastheindependentvariablef(x,y)=0ax2+bxy+cy2+dx+ey+f=0DisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMNon-parametric(cont)DisadvantagesNon-parametricelementsareaxisdependant,sothechoiceofcoordinatesystemaffectstheeaseofusingtheelementandcalculatingtheirproperties.Problemifthecurvehasaverticalslope(infinity).Theyrepresentunboundedgeometrye.g ax+by+c=0defineaninfinitelineDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMparametricExpressrelationshipforthex,yandzcoordinatesnotintermofeachotherbutofoneormoreindependentvariable(parameter).AdvantagesOffermoredegreesoffreedomforcontrollingtheshape(non-parametric)y=ax3+bx2+cx+d(parametric)x=au3+bu2+cu+dy=eu3+fu2+gu+hDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMParametric(cont)Advantages(cont)Transformationscanbeperformeddirectlyonparametricequations.Parametricformsreadilyhandleinfiniteslopeswithoutbreakingdowncomputationally dy/dx=(dy/du)/(dx/du)Completelyseparatetherolesofthedependentandindependentvariable.DisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMParametric(cont)Advantages(cont)easytoexpressintheformofvectorsandmatricesInherentlybounded.DisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMParametriccurveUseparametertorelatecoordinatexandy(2D).AnalogyParametert(time)–[x(t),y(t)asthepositionoftheparticleattimet]xyt1t2t3t4t5t6DisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMParametriccurveFundamentalgeometricobjects–lines,raysandlinesegmentablineabrayabLinesegmentAllsharethesameparametricrepresentationDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMParametriclinea=(ax,ay),b=(bx,by)x(t)=ax+(bx-ax)ty(t)=ay+(by-ay)tParametertisvariedfrom0to1todefineallpointalongthelineWhent=0,thepointisat“a”,astincreasestoward1,thepointmovesinastraightlinetob.Forlinesegment:0t1Forline:-tForray:0t

abDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMParametriclineExampleAlinefrompoint(2,3)topoint(-1,5)canberepresentedinparametricformasx(t)=2+(-1–2)t=2–3ty(t)=3+(5–2)t=3+3tDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMParametriclinePositionsalongthelinearebasedupontheparametervalueE.gmidpointofalineoccursatt=0.5Exercise:Findtheparametricformforthesegmentwithendpoints(2,4,1)and(7,5,5).Findthemidpointofthesegmentbyusingt=0.5DisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMParametriclineAnswer:Parametricform:x(t)=2+(7–2)t=2+5ty(t)=4+(5–4)t=4+tz(t)=1+(5–1)t=1+4tDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMAnswerMidpointx(0.5)=2+5(0.5)=5.56Y(0.5)=4+(0.5)=4.55Z(0.5)=1+4(0.5)=33ParametriclineDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMAnotherbasicexampleConicsection-thecurves/portionsofthecurves,obtainedbycuttingaconewithaplane.Thesectioncurvemaybeacircle,ellipse,parabolaorhyperbola.Parametriccurve(conicsection)ellipsehyperbolaparabolaDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMParametriccurve(circle)Thesimplestnon-linearcurve-circle -circlewithradiusRcenteredattheoriginx(t)=Rcos(2t)y(t)=Rsin(2t)0t1DisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMIft=0.125a1/8circleParametriccurve(circle)t=0.25a1/4circlet=0.5a?circlet=1acircle

CirculararcDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMCirclewithcenterat(xc,yc)x(t)=Rcos(2t)+xc,

y(t)=Rsin(2t)+yc,Parametriccurve(circle)DisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMParametriccurveEllipsex(t)=acos(2t)y(t)=bsin(2t)Hyperbolax(t)=asec(t)y(t)=btan(t)parabolax(t)=at2y(t)=2atababDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMControlforthiscurveShape(baseduponparametricequation)Location(baseduponcenterpoint)SizeArc(baseduponparameterrange)Radius(acoefficienttounitvalue)DisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMParametriccurveGenerallyAparametriccurvein3DspacehasthefollowingformF:[0,1](x(t),y(t),z(t))wherex(),y()andz()arethreereal-valuedfunctions.Thus,F(t)mapsarealvaluetintheclosedinterval[0,1]toapointinspaceforsimplicity,werestrictthedomainto[0,1].Thus,foreachtin[0,1],therecorrespondstoapoint(x(t),y(t),z(t))inspace.Ifz()isremoved-?AcurveinacoordinateplaneDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMTangentvectorandtangentlineTangentvectorVectortangenttotheslopeofcurveatagivenpointTangentlineThelinethatcontainsthetangentvectorDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMF(t)=(x(t),y(t),z(t))Tangentvector:F’(t)=(x’(t),y’(t),z’(t))Wherex’(t)=dx/dt,y’(t)=dy/dt,z’(t)=dz/dtMagnitude/lengthIfvectorV=(a,b,c)|V|=a2+b2+c2

UnitvectorUv=V/|V|ComputetangentvectorDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMComputetangentlineTangentlineattiseitherF(t)+uF’(t)orF(t)+u(F’(t)/|F’(t)|)ifpreferunitvectoruisaparameterforlineDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMexampleQuestion:-givenaCircle,F(t)=(Rcos(2t),Rsin(2t)),0t1FindtangentvectorattandtangentlineatF(t).DisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMexampleAnswerdx=Rcos(2t),dy=Rsin(2t)x’(t)=dx/dt=-2Rsin(2t),y’(t)=dy/dt=2Rcos(2t)Tangentvector=(-2Rsin(2t),2Rcos(2t))DisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMexampleAnswerTangentlineF(t)+u(F’(t))(Rcos(2t),Rsin(2t))+u(-2Rsin(2t),2Rcos(2t))(Rcos(2t)+u(-2Rsin(2t))),(Rsin(2t)+u(2Rcos(2t)))DisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMExampleCheck/proveLetsay,t=0, Tangentvector=(-2Rsin(2t),2Rcos(2t)) =(0,2R) tangentline=(Rcos(2t)+u(-2Rsin(2t))),(Rsin(2t)+

u(2Rcos(2t))) =(R,u(2R))RDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMTangentvectorSlopeofthecurveatanypointcanbeobtainedfromtangentvector.Tangentvectoratt=(x’(t),y’(t))Slopeatt=dy/dx=y’(t)/x’(t)DisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMThecurvatureatapointmeasurestherateofcurving(bending)asthepointmovesalongthecurvewithunitspeedWhenorientationischangedthecurvaturechangesitssign,thecurvaturevectorremainsthesameStraightlinecurvature=?curvatureDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMcurvatureCircleistangenttothecurveatPliestowardtheconcaveorinnersideofthecurveatPCurvature=1/r ,rradiusPPDisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMcurvatureThecurvatureatu,k(u),canbecomputedasfollows:k(u)=|f'(u)×f''(u)|/|f'(u)|3

Howaboutcurvatureofacircle?

DisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMCurveuseindesignEngineeringdesignrequiresabilitytoexpresscomplexcurveshapes(beyondconic)andinteractive.Boundingcurvesforturbineblades,shiphulls,etcCurveofintersectionbetweensurfaces.DisediakanolehSuriatibteSadimon,GMM,FSKSM,UTMAdesignis“GOOD”ifitmeetsitsdesignspecifications:Thesemaybeeither:Functional-doesitworks.Technical-isitefficient,doesitmeetcertainbenchmarkorstandard.Aesthetic-doesitlookright,thisisbothsubjectiveandopinionislikelytochangeintimeorcombinationofboth.Curv