重庆大学机自实验班计算机辅助设计与制造总复习课件

TotalReviewofComputer-aidedDesignandManufacturing

TotalReviewofComputer-aided1ScoreAssessmentAttendance(10%)Rollcall5times(2markseachtime)courseerercises(15%)Courseexercises3times(5markseachtime)Termpaper(25%)Examination

(50%)2-houropenbookpaper(CAD90%plusCAM10%),CalculationproblemsandnounsexplainScoreAssessmentAttendance(102ExaminationMaterial

LecturenotesTutorialsandexercisesTeachingMaterial

(MECHANICALENGINEERINGCAD/CAM)

ReferencesbooksSurfacemodellingforCAD/CAM,Chapter1-5,7Geometricmodelling,chapter9-10.

TheCNCWorkshop(ver2),chapter1ExaminationMaterialLecturen3Chapter1:InstructionWhatisCAD/CAM/CAE/CAPP?

Howistherelationshipamongthem?)WhatistheHISTORYofCAD/CAM?HardwareandsoftwareofCAD/CAMsystem?WhatisGeometricModellinganditstypicalapplications?Chapter1:InstructionWhatis4Chapter2:CurvesFourcurvemodelsStandardpolynomial

curve

Ferguson

curve

Bezier

curve

B-spline

curveCurvefittingChapter2:CurvesFourcurvemo5PolynomialCurveModelsCurveSegmentDefinition:

Acubicpolynomialcurvemodel:

r

(u)=a

+bu

+cu2

+

du3

usedinrepresentingacurvesegmentisspecifiedbyits

endconditions,

e.g.,

(a)

4points

(P0,

P1,P2

and

P3)

or

(b) twoendpoints

P0

and

P1;

twoendtangents

t0

and

t1.P0P1P2P3Ingeneral,adegree-npolynomialcurvecanbeusedtofit(n+1)datapoints.

PolynomialCurveModelsCurveS6FergusonCurveModel

Constructingacurvesegment:JoiningtwoendpointsP0andP1;

Havingspecifiedendtangentst0andt1

i.e.,

P0=r(0); P1

=r(1);

t0=r’(0); t1=r’(1)

P1P0t1t0r(u)r(u)=UA=UMV

with0

u

1FergusonCurveModelConstruct7BezierCurveModel

with0

u

1

OnevaluatingtheBezierequationanditsderivativeatu=0,1

r(0)=V0r(1)=Vnr’(0)=n(V1–V0)r’(1)=n(Vn–Vn-1)

Bezierfoundafamilyoffunctionscalled

BernsteinPolynomials

thatsatisfytheseconditions:BezierCurveModelwith08BezierCurveModelCubic(n=3)BeziercurvemodelV0V1V2V3V3V2V1V0V2V1V0V3r(u)=(1–u)3

V0+3u(1–u)2

V1+3u2(1–u)V2+u3

V3

r(u)=

=UMR

r(0)=V0 r’(0)=3

(V1–V0)r(1)=V3

r’(1)=3

(V3–V2)

Theshapeofthecurveresemblesthatofthecontrolpolygon.BezierCurveModelCubic(n=39B-splineModel

with0

u

1

Ni,n(u)

=

TheprimaryfunctionB-splineModeldefinedbyn+1pointsViisgivenbytheWhereB-splineModelwith0u10B-splineModelQuadratic

uniformB-splinemodelwithcontrolpointsV0,V1,andV2

r(t)=

½

[t2

t1]

=U3

M3

P3

0≤t≤1

CubicuniformB-splinemodelwithcontrolpointsV0,V1,V2,andV3

r(t)=1/6

[u3

u2

u1]

=U4

M4

P4

0≤t≤1B-splineModelQuadraticunifor11ParametricContinuityCondition

Twocurvesegmentsra(u)andrb(u)

ra(1)=P1=rb(0)

(C0-continuous)

ra’(1)=t1=rb’(0)

(C1-continuous)

ra’’(1)=rb’’(0)

(C2-continuous)

Collectivelycalleda

parametricC2-condition.

Thecompositecurvetopassthrough

P0,P1,P2,andthetangentst0andt2areassumedtobe

given.Thus,theproblemhereistodeterminethe

unknownt1sothatthetwocurvesegmentsareC2-continuousatthecommonjoinP1.P0P1P2t2t0t1=?ra(u)rb(u)ParametricContinuityConditio12CubicSplineFitting(FergusonModel)

EmployingFergusoncurvemodel

ra(u)=UCSa

rb(u)=UCSb

with0u1

U=[u3

u2

u1]C=

Sa

=

[P0P1t0t1]T

Sb

=

[P1P2t1t2]TApplyingC2continuity:ra’’(1)=6P0–6P1+2t0+4t1rb’’(0)=

-6P1+6P2-4t1-2t2C0-continuityandC1-continuityalreadyappliedCubicSplineFitting(Ferguson13CubicSplineFitting（FergusonModel)ApplyingparametricC2-condition

t0+4t1+t2=3(P2–P0)

Now,considerconstructingaC2-continuouscurvepassingthroughasequenceof

n+1

(P0

toPn)

pointsEndtangents

t0

and

tn

aregiven,inadditiontothe

(n+1)

points{Pi

}.

(Howmanycurvesegments???)

Therearetotally

n

curvesegments.Foreachpairofneighbouringcurvesegments

ri-1(u)

and

ri(u),wehave

ti-1+4ti+ti+1=3(Pi+1–Pi-1)

fori=1,2,…,n–1

CubicSplineFitting（Ferguson14B-splineModelOnevaluatingthecubicB-spline(k=4) anditsderivativeatt=1,0,

r

(0)=[4V1+(V0+V2)]/6 r(1)=[4V2+(V1+V3)]/6

r’(0)=(V2–V0)/2 r’(1)=(V3–V1)/2B-splinecurvesandBeziercurveshavemanyadvantagesincommonControlpointsinfluencecurvesegmentshapeinapredictable,naturalway,makingthemgoodcandidatesforuseinaninteractivedesignenvironment.Bothtypesofcurveareaxisindependent,multivalued,andbothexhibittheconvexhullproperty.B-splinecurveshaveadvantagesoverBeziercurves:Localcontrolofcurveshape.Theabilitytoaddcontrolpointswithoutincreasingthedegreeofthecurve.V0V1V3V2B-splineModelOnevaluatingth15CubicSplineFittingEstimationofendtangents,t0andtnCircularendcondition

Polynomialendcondition

Freeendcondition

CubicSplineFittingEstimation16Chapter3:SurfacesFoursurfacepatchmodelsStandardpolynomial

surfacepatch

Ferguson

surfacepatch

Bezier

surfacepatch

B-spline

surfacepatch

ThreeSurfaceConstructionMethodsTheFMILLmethod

Fergusonfittingmethod

B-splinefittingmethodCurvedBoundaryInterpolatingSurfacePatchesChapter3:SurfacesFoursurfac17StandardPolynomialPatchModel

Consideravector-valuedpolynomialfunction

r(u,v)whosedegreesarecubicinbothuandvwithcoefficients

dijfor(ui,vj).Thatisabi-cubic(standard)polynomialpatchdefinedas

r(u,v)

=

with0

u,v

1

whichcanbeexpressedinamatrixformas

r(u,v)

=

UDVT

where,

U=[u3

u2

u1],V=[v3

v2

v1],

andthe

coefficientsmatrix

D=

StandardPolynomialPatchMode18FergusonSurfacePatchModelSolvingthe16linearequationsfortheunknowncoefficientsdij

givesusaFergusonpatchequation:

r(u,v)=UDVT=UCQCTVT for0

u,v

1

C=

Q=FergusonSurfacePatchModelSo19BezierSurfacePatchModel

r(u,v)==UMBMTVT

0

u,v

1

Where

M=

B=

ThematrixMiscalleda(cubic)Beziercoefficientmatrix,andB

iscalledaBeziercontrolpointnetwhichformsacharacteristicpolyhedron.BezierSurfacePatchModelr(20BezierSurfacePatchModelBezierpatchvs.FergusonPatch

ByevaluatingthecornerconditionsoftheBezierpatch, wehavethefollowingrelationships:

Atu=0,v=0,

r(0,0)=V00 s00=3(V10–V00) t00=3(V01–V00) x00=9(V00–V01

–V10+V11)BezierSurfacePatchModelBezi21B-splineSurfacePatchModelConsidera44arrayofcontrolvertices{Vij}.

r(u,v)=

=UNBNTVT

for0

u,v

1

N=

B-splineSurfacePatchModelCo22SurfaceConstructionMethodsItisdesiredtouselowdegree(usuallycubic)polynomialpatchmodeltoformacompositesurface.Threemethodstobeintroduced:TheFMILLmethod

Fergusonfittingmethod

B-splinefittingmethod

SurfaceConstructionMethodsIt23B-SplineSurfaceFittingComparisonbetweenFergusonfittingandB-splinefittingSamecompositesurfaceresultedWhenmakingfurtherchanges,localchangeforB-splinesurface,globalchangeforFergusonsurface.Question:Whenonecontrolpointischanged,howmanypatchesareaffected?B-SplineSurfaceFittingCompar24CurvedBoundaryInterpolatingSurfacePatches

Methodsofconstructingasurfacepatchinterpolatingtoasetofboundarycurves:Ruledsurfaces

Loftedsurfaces

Coonssurfaces

Twotypesofsweepsurfacepatches:Translationalsweeppatches

RotationalsweeppatchesCurvedBoundaryInterpolating25RuledSurfaces

Considertwoparametriccurves,

r0(u)andr1(u)with0

u

1(seefigure).Alinearblendingofthe2curvesdefinesasurfacepatchcalledaruledsurface

r

(u,

v)=r0(u)+v

(r1(u)-r0(u));0

u,v

1Avectorinthedirectionofr1(u)-r0(u)iscalledarulingvector

t(u).

RuledSurfacesConsidertwop26TranslationalSweepSurfacePatches

InputSummaryTwoparametricspacecurves,g(u)andd(v).

Atranslationalsweepsurfaceisdefinedbythe trajectoryofthecurveg(u)

sweptalongthesecondcurved(v).Themovingcurveg(u)iscalledagenerator

curveTheguidingcurved(v)iscalledadirector

curve

r(u,v)=g(u)+d(v)-d(0)0

u,v

1

r(u,v)g(u)TranslationalSweepSurfacePa27RotationalSweepSurfacePatches

Alsoknownas

surfaceofrevolution

Considerasectioncurves(u)onthex-zplane

s(u)=x(u)i+z(u)k=

(x(u),0,z(u))

Rotatethesectioncurves(u)aboutthez-axis,theresultingsweepsurfacecanbeexpressedasanparametricequationas:

r(u,)=(x(u)cos,x(u)sin,z(u))

r(u,)RotationalSweepSurfacePatch28Chapter4:SolidModellingTwosolidmodelrepresentationschemesGraph-basedmodel(B-reps)Booleanmodel(CSG)EulerFormulaChapter4:SolidModellingTwo29Graph-BasedModelsForsolidsrepresentedasplanar-facedpolyhedron,manysimplerepresentationschemesareavailable,e.g.,connectivitymatrixforpolyhedron.Connectivitymatrix(oradjacencymatrix):Abinarymatrix

0-elementindicatesnoconnectivityexists

1-elementsindicateconnectivityexistsbetweenthepairofelements (vertices,edges,orfaces).

Graph-BasedModelsForsolidsr30BooleanModelsThebinarytreeforthismodelTheleafnodesaretheprimitivesolids,withBooleanoperatorsateachinternalnodeandtheroot.Eachinternalnodecombinesthetwoobjectsimmediatelybelowitinthetree,and,ifnecessary,transformstheresultinreadinessforthenextoperation.

BooleanModelsThebinarytree31BasicConceptsofSolidModelEuler’slaw(orEuler’sformula)Foravalidsolid(polyhedron),thefollowingrelationshipmustbesatisfied:

V–E+F-(L–F)=2–2H

V=NumberofverticesE=NumberofedgesF=NumberoffacesL=NumberofedgeloopsH=Numberofthroughholes

Thisexpressioncanalsobere-writtenas:

V–E+F-

R=2–2H

WhereR=L–Fisthenumberofinterioredgeloops.

ExternaledgeloopInterioredgeloopBasicConceptsofSolidModelE32Chapter7:PartProgrammingandManufacturingWhatisCNC/NC?

Howabouttheircharacteristics?)WhatisCNC/MC/FMS/CIMS?

Howistherelationshipamongthem?)WhatisthebasicconstructionforNCprogramming?HowtodeterminethecocrdinatesystemsofNCmachinetools?WhatisRP/RE?

Howabouttheircharacteristics?)Chapter7:PartProgrammingan33TipYoushouldpreparesufficientmaterials.Youshouldbringyourscientificcalculator,notyouriPhone.Youmayneedaruler.

Alloftheseformthescopeoftestinthefinalexam.TipYoushouldpreparesufficie34ThefinaltipPractice,practice,andpractice…Thefinaltip35ThankyouWishyouforthebestgrades!Thankyou36TotalReviewofComputer-aidedDesignandManufacturing

TotalReviewofComputer-aided37ScoreAssessmentAttendance(10%)Rollcall5times(2markseachtime)courseerercises(15%)Courseexercises3times(5markseachtime)Termpaper(25%)Examination

(50%)2-houropenbookpaper(CAD90%plusCAM10%),CalculationproblemsandnounsexplainScoreAssessmentAttendance(1038ExaminationMaterial

LecturenotesTutorialsandexercisesTeachingMaterial

(MECHANICALENGINEERINGCAD/CAM)

ReferencesbooksSurfacemodellingforCAD/CAM,Chapter1-5,7Geometricmodelling,chapter9-10.

TheCNCWorkshop(ver2),chapter1ExaminationMaterialLecturen39Chapter1:InstructionWhatisCAD/CAM/CAE/CAPP?

Howistherelationshipamongthem?)WhatistheHISTORYofCAD/CAM?HardwareandsoftwareofCAD/CAMsystem?WhatisGeometricModellinganditstypicalapplications?Chapter1:InstructionWhatis40Chapter2:CurvesFourcurvemodelsStandardpolynomial

curve

Ferguson

curve

Bezier

curve

B-spline

curveCurvefittingChapter2:CurvesFourcurvemo41PolynomialCurveModelsCurveSegmentDefinition:

Acubicpolynomialcurvemodel:

r

(u)=a

+bu

+cu2

+

du3

usedinrepresentingacurvesegmentisspecifiedbyits

endconditions,

e.g.,

(a)

4points

(P0,

P1,P2

and

P3)

or

(b) twoendpoints

P0

and

P1;

twoendtangents

t0

and

t1.P0P1P2P3Ingeneral,adegree-npolynomialcurvecanbeusedtofit(n+1)datapoints.

PolynomialCurveModelsCurveS42FergusonCurveModel

Constructingacurvesegment:JoiningtwoendpointsP0andP1;

Havingspecifiedendtangentst0andt1

i.e.,

P0=r(0); P1

=r(1);

t0=r’(0); t1=r’(1)

P1P0t1t0r(u)r(u)=UA=UMV

with0

u

1FergusonCurveModelConstruct43BezierCurveModel

with0

u

1

OnevaluatingtheBezierequationanditsderivativeatu=0,1

r(0)=V0r(1)=Vnr’(0)=n(V1–V0)r’(1)=n(Vn–Vn-1)

Bezierfoundafamilyoffunctionscalled

BernsteinPolynomials

thatsatisfytheseconditions:BezierCurveModelwith044BezierCurveModelCubic(n=3)BeziercurvemodelV0V1V2V3V3V2V1V0V2V1V0V3r(u)=(1–u)3

V0+3u(1–u)2

V1+3u2(1–u)V2+u3

V3

r(u)=

=UMR

r(0)=V0 r’(0)=3

(V1–V0)r(1)=V3

r’(1)=3

(V3–V2)

Theshapeofthecurveresemblesthatofthecontrolpolygon.BezierCurveModelCubic(n=345B-splineModel

with0

u

1

Ni,n(u)

=

TheprimaryfunctionB-splineModeldefinedbyn+1pointsViisgivenbytheWhereB-splineModelwith0u46B-splineModelQuadratic

uniformB-splinemodelwithcontrolpointsV0,V1,andV2

r(t)=

½

[t2

t1]

=U3

M3

P3

0≤t≤1

CubicuniformB-splinemodelwithcontrolpointsV0,V1,V2,andV3

r(t)=1/6

[u3

u2

u1]

=U4

M4

P4

0≤t≤1B-splineModelQuadraticunifor47ParametricContinuityCondition

Twocurvesegmentsra(u)andrb(u)

ra(1)=P1=rb(0)

(C0-continuous)

ra’(1)=t1=rb’(0)

(C1-continuous)

ra’’(1)=rb’’(0)

(C2-continuous)

Collectivelycalleda

parametricC2-condition.

Thecompositecurvetopassthrough

P0,P1,P2,andthetangentst0andt2areassumedtobe

given.Thus,theproblemhereistodeterminethe

unknownt1sothatthetwocurvesegmentsareC2-continuousatthecommonjoinP1.P0P1P2t2t0t1=?ra(u)rb(u)ParametricContinuityConditio48CubicSplineFitting(FergusonModel)

EmployingFergusoncurvemodel

ra(u)=UCSa

rb(u)=UCSb

with0u1

U=[u3

u2

u1]C=

Sa

=

[P0P1t0t1]T

Sb

=

[P1P2t1t2]TApplyingC2continuity:ra’’(1)=6P0–6P1+2t0+4t1rb’’(0)=

-6P1+6P2-4t1-2t2C0-continuityandC1-continuityalreadyappliedCubicSplineFitting(Ferguson49CubicSplineFitting（FergusonModel)ApplyingparametricC2-condition

t0+4t1+t2=3(P2–P0)

Now,considerconstructingaC2-continuouscurvepassingthroughasequenceof

n+1

(P0

toPn)

pointsEndtangents

t0

and

tn

aregiven,inadditiontothe

(n+1)

points{Pi

}.

(Howmanycurvesegments???)

Therearetotally

n

curvesegments.Foreachpairofneighbouringcurvesegments

ri-1(u)

and

ri(u),wehave

ti-1+4ti+ti+1=3(Pi+1–Pi-1)

fori=1,2,…,n–1

CubicSplineFitting（Ferguson50B-splineModelOnevaluatingthecubicB-spline(k=4) anditsderivativeatt=1,0,

r

(0)=[4V1+(V0+V2)]/6 r(1)=[4V2+(V1+V3)]/6

r’(0)=(V2–V0)/2 r’(1)=(V3–V1)/2B-splinecurvesandBeziercurveshavemanyadvantagesincommonControlpointsinfluencecurvesegmentshapeinapredictable,naturalway,makingthemgoodcandidatesforuseinaninteractivedesignenvironment.Bothtypesofcurveareaxisindependent,multivalued,andbothexhibittheconvexhullproperty.B-splinecurveshaveadvantagesoverBeziercurves:Localcontrolofcurveshape.Theabilitytoaddcontrolpointswithoutincreasingthedegreeofthecurve.V0V1V3V2B-splineModelOnevaluatingth51CubicSplineFittingEstimationofendtangents,t0andtnCircularendcondition

Polynomialendcondition

Freeendcondition

CubicSplineFittingEstimation52Chapter3:SurfacesFoursurfacepatchmodelsStandardpolynomial

surfacepatch

Ferguson

surfacepatch

Bezier

surfacepatch

B-spline

surfacepatch

ThreeSurfaceConstructionMethodsTheFMILLmethod

Fergusonfittingmethod

B-splinefittingmethodCurvedBoundaryInterpolatingSurfacePatchesChapter3:SurfacesFoursurfac53StandardPolynomialPatchModel

Consideravector-valuedpolynomialfunction

r(u,v)whosedegreesarecubicinbothuandvwithcoefficients

dijfor(ui,vj).Thatisabi-cubic(standard)polynomialpatchdefinedas

r(u,v)

=

with0

u,v

1

whichcanbeexpressedinamatrixformas

r(u,v)

=

UDVT

where,

U=[u3

u2

u1],V=[v3

v2

v1],

andthe

coefficientsmatrix

D=

StandardPolynomialPatchMode54FergusonSurfacePatchModelSolvingthe16linearequationsfortheunknowncoefficientsdij

givesusaFergusonpatchequation:

r(u,v)=UDVT=UCQCTVT for0

u,v

1

C=

Q=FergusonSurfacePatchModelSo55BezierSurfacePatchModel

r(u,v)==UMBMTVT

0

u,v

1

Where

M=

B=

ThematrixMiscalleda(cubic)Beziercoefficientmatrix,andB

iscalledaBeziercontrolpointnetwhichformsacharacteristicpolyhedron.BezierSurfacePatchModelr(56BezierSurfacePatchModelBezierpatchvs.FergusonPatch

ByevaluatingthecornerconditionsoftheBezierpatch, wehavethefollowingrelationships:

Atu=0,v=0,

r(0,0)=V00 s00=3(V10–V00) t00=3(V01–V00) x00=9(V00–V01

–V10+V11)BezierSurfacePatchModelBezi57B-splineSurfacePatchModelConsidera44arrayofcontrolvertices{Vij}.

r(u,v)=

=UNBNTVT

for0

u,v

1

N=

B-splineSurfacePatchModelCo58SurfaceConstructionMethodsItisdesiredtouselowdegree(usuallycubic)polynomialpatchmodeltoformacompositesurface.Threemethodstobeintroduced:TheFMILLmethod

Fergusonfittingmethod

B-splinefittingmethod

SurfaceConstructionMethodsIt59B-SplineSurfaceFittingComparisonbetweenFergusonfittingandB-splinefittingSamecompositesurfaceresultedWhenmakingfurtherchanges,localchangeforB-splinesurface,globalchangeforFergusonsurface.Question:Whenonecontrolpointischanged,howmanypatchesareaffected?B-SplineSurfaceFittingCompar60CurvedBoundaryInterpolatingSurfacePatches

Methodsofconstructingasurfacepatchinterpolatingtoasetofboundarycurves:Ruledsurfaces

Loftedsurfaces

Coonssurfaces

Twotypesofsweepsurfacepatches:Translationalsweeppatches

RotationalsweeppatchesCurvedBoundaryInterpolating61RuledSurfaces

Considertwoparametriccurves,

r0(u)andr1(u)with0

u

1(seefigure).Alinearblendingofthe2curvesdefinesasurfacepatchcalledaruledsurface

r

(u,

v)=r0(u)+v

(r1(u)-r0(u));0

u,v

1Avectorinthedirectionofr1(u)-r0(u)iscalledarulingvector

t(u).

RuledSurfacesConsidertwop62TranslationalSweepSurfacePatches

InputSummaryTwoparametricspacecurves,g(u)andd(v).

Atranslationalsweepsurfaceisdefinedbythe trajectoryofthecurveg(u)

sweptalongthesecondcurved(v).Themovingcurveg(u)iscalledagenerator

curveTheguidingcurved(v)iscalledadirector

curve

r(u,v)=g(u)+d(v)-d(0)0

u,v

1

r(u,v)g(u)TranslationalSweepSurfacePa63RotationalSweepSurfacePatches

Alsoknownas

sur