《计算机辅助几何造型技术》

1计算机辅助几何造型技术

■主讲教师：秦开怀教授、博导

■akh-dcs@

■所在单位：清华大学计算机科学与技术系

■时间：2007年9月~2008年1月

1

Textbooks/References

■J.Hoschek&D.Lasser,Fundamentalsof

ComputerAidedGeometricDesign,AKPeters,

Ltd,Massachusetts,1993.

■DavidFRogers,IntroductiontoNURBS,Morgan

Kaufmann,2001.

■L.Regl&W.Tiller,TheNURBSBook(2nd

Edition),Springer-VerlagBerlinHeidelberg,New

York,1997.

■CarldeBoor,APracticalGuidetoSplines,New

York,Springer-Verlag,1978.

2

Textbooks/References

।(Continued)

■M.E.Mortenson,GeometricModeling,

JohnWaley&Sons,Inc.,1985.

■G.Farin,CurvesandSurfacesfor

ComputerAidedGeometricDesign(5th

Edition),ElsevierInc.,2002.(李双喜译，

CAGD曲线曲面，科学出版社，2006)

■E.J.Stollnit乙T.DeRose&D.H.Salesin,

WaveletsforComputerGraphics,Theory&

Applications,MorganKaufmannPublishers,

Inc.,SanFrancisco,1996.

3

Textbooks/References

।(Continued)

■DenisZorin&PeterSchroder,Subdivisionfor

ModelingandAnimation,SIGGRAPH2000

CourseNotes#23,2000.

■R.Barzel,Physically-BasedModelingfor

ComputerGraphics,AStructuredApproach,

AcademicPress,Inc.,SanDiego,1992.

■D.N.Metaxas,Physic-BasedDeformable

Models,ApplicationstoComputerVision,

Graphics&MedicalImaging,KluwerAcademic

Publishers,Massachusetts,1997.

4

Textbooks/References

।(Continued)

■DonaldHearn&M.PaulineBaker,

ComputerGraphicswithOpenGL(Third

Hition),PearsonBducation,2004(中译

本：赫恩等著，蔡士杰等译，《计算机图

形学(第三版)》，电子工业出版社，

2005-06.)

■J.D.Foley,etal,ComputerGraphics:

Principles&Practice(2ndEditioninC),

Addison-Wesley,Reading,MA,1996.

5

《ad/77gPolicy

■Threeassignments30%

■Discussions/learninginclassroom5%

■Oneprojectsubstitutingforthefinal

examination65%

6

窄/77arks

>-----------

■Thethreeassignmentistobecompletedindividuallyon

yourself,butdiscussionsamongfellowstudentsare

allowed.

■Theprojectsubstitutesforthefinalexamination.Two

studentscanworktogetherasagroup.

■Absolutelynosharingorcopyingofanycodeforboth

theassignmentsandtheproject!Offenderswillbegiven

afailuregradeandthecasewillbereportedtothe

department.

■Youarewelcometoturnoffyourmobilephonebefore

attendinglectures.

7

Thiscourseconcentrateson

sevenmainissues:

■NURBScurvesandsurfaces(includingBezier,B-

splinecurvesandsurfaces)

■Triangularsurfaces

■Gordon-Coonssurfaces

■Subdivisionsurfacesofarbitrarytopology

■The2ndgenerationwaveletsformulti-resolution

modeling

■Solidmodeling

■Newtechnologyforgeometricmodeling

ContentsofThisCourse

1.Introduction

2.AMathematicBasics

□Affinemaps

□Divideddifference

□Functionspace

□Geometricbasicsfromcurvesandsurfaces

3.AInterpolatoryPolynomialSplines

□Hermiteinterpolation

9

ContentsofThisCourse

।(Continued)

□Quadricpolynomialsplinecurves

□Cubicpolynomialsplinecurves

□Solvingalinearsystemofequationswitha

tridiagonalcoefficientmatrix

□Cubicparametricsplinecurves

4.*BezierCurvesandSurfaces

□Beziercurvesdefinedbyedgevectors

□Bernstein-Beziercurves

□PropertiesofBernstein-Beziercurves

10

ContentsofThisCourse

(Continued)

□DeCasteljaualgorithm

□DiscretegenerationofBeziercurves

□DegreeelevationofBeziercurves

□DegreereductionofBeziercurves

□Beziersplinecurves

□Bezierinterpolationcurves

□MatrixformulaofBeziercurves

□RationalBeziercurves

□Product&innerproductofBeziercurves

□Beziersurfaces

11

ContentsofThisCourse

(Continued)

5.*B-splineCurvesandSurfaces

□B-splinebasisfunctionsandtheir

properties

□B-splinecurves

-Opencurvesandknotvectors

・UniformB-splinecurves

・EndpointinterpolatingB-splinecurves

QosedB-splinecurves

12

ContentsofThisCourse

(Continued)

□Chaikinalgorithm

□DeBooralgorithm

□InsertingknotsinB-splinecurves

Boehmalgorithm

Oisoalgorithm

GeneralknotinsertionforB-splinecurves

□DegreeelevationofB-splinecurves

・Marsdenidentityandrecursivedegree

elevation

Prautzschalgorithm

13

ContentsofThisCourse

(Continued)

・ArbitrarilyhighdegreeelevationforB-spline

curves

□DegreereductionofB-splinecurves

□B-splinesurfaces

□InterpolatingB-splinecurvesand

surfaces

□MatrixformulasofB-splinecurvesand

surfaces

14

ContentsofThisCourse

(Continued)

・MatrixformulaofuniformBLsplinecurves

・Matrixformulaofnon-uniformB_splines

□InnerproductofB-splinecurves

■GeneralizedMarsdenidentity

・B-splinecurveproduct

・InnerproductofB-splinebasisfunctions

・InnerproductofB-splinecurves

6.*NURBSCurvesandSurfaces

□NURBScurves

□RepresentingconicsusingNURBS15

ContentsofThisCourse

(Continued)

□Parameterizationofcurves

□NURBSsurfaces

□RepresentingquadricsusingNURBS

surfaces

□InterpolatingNURBScurvesandsurfaces

7.BlossomingPrinciple

□LookingatdeCasteljaualgorithmfroma

blossomingpointofview

□Knotinsertionfromablossomingpointof

view

16

ContentsofThisCourse

(Continued)

GeneratingdeBoorpointsbasedonthe

blossomingprinciple

□DegreeraisingofB-splinecurvesby

blossoming

8.*TriangularSurfaces

□Barycentriccoordinates

□TriangularBeziersurfaces

□ContinuityconditionsfortriangularBezier

patches

□RationalTriangularsurfaces

17

ContentsofThisCourse

(Continued)

9.*Gordon-CoonsSurfaces

□Coonssurfaces

□Gordon-Coonssurfacesonrectangles

□Gordon-Coonssurfacesontriangles

10.*SubdivisionSurfacesofArbitrary

Topology

□Catmull-Oarksurfaces

□Doo-Sabinsurfaces

□Continuityofuniformsubdivisionsurfaces

□Non-uniformsubdivisionsurfaces

18

ContentsofThisCourse

(Continued)

□Convergenceandcontinuityofnon-uniform

subdivisionsurfaces

11.*The2ndGenerationWaveletsfor

Multi-resolutionmodeling

□B-splinewaveletsforMulti-resolutionmodeling

・EndpointinterpolatingB-splinewavelets

・ArbitraryNon-uniformB-splinewavelets

-B-splinewaveletswithconstraints

□Subdivision-basedSurfacewavelets

・LoopSubdivisionWavelets

・Catmull-QarkSubdivisionWavelets

・也-subdivision-basedBi-orthogonalWavelets

19

ContentsofThisCourse

(Continued)

12.AScatteredDataInterpolation

13.*IntersectionsofCurvesandSurfaces

14.*SolidModeling

15.ParameterizationModelingforShape

DesignandFeature-basedModeling

16.*NewTechnologyforGeometric

Modeling

□HierarchicalB-splines

□Physics-basedmodeling20

ContentsofThisCourse

(Continued)

□Modelingtractalizedscenes(mountains,

flowersetc.)

□Particlesystemformodelingfires,clouds,

water,forestsetc.

21

1.Introduction

SomeApplicationsofCAGD

■Representationoflargedatasets

■Visualizingproducts

■Automaticallyproducingsectional

drawings

■Modelingsurfacesarisingin

constructionofcars,ships&airplanes

■Designingpipesystems,e.g.in

chemicalplants

22

SomeApplicationsofCAGD

(continued)

■Drawingmarinechartsandcityandrelief

mapsincartography

■Productionandqualitycontrol,e.g.in

thesewingmachine,textileandshoe

industries

■Planningandcontrollingsurgery

■Creatingimagesinadvertising,television

andfilmindustries

23

SomeApplicationsofCAGD

(continued)

■Constructingvirtualenvironments

■Describingrobotpathsandcontrolling

theirmovements

■Controllingmillingmachinesusedin

manufacturing

24

CM

卑

CurvemodelingwithconstrainedB-splinewavelets

保特征点的多分辨率曲线造型

K

K

29

(b3)(c3)

曲线的多分辨率分段无缝表示

30

4细分曲面

FT1

co

3

o

左图是采用经典B样条曲面小波分片多分辨率表示的结果，右图是采用带约束

的B样条曲面小波分片多分辨率表示的结果,其中约束施加在接合线处。对比

两图，可以看到带约束的B样条曲面小波消除了经典B样条曲面小波引入的缝

隙。

41

《带约束的样条曲面小波

(a)原始高分辨率曲面(b)不带约束小波分解(c)带约束小波分解

图(蓟是用张量积B样条曲面表示的一个函数曲面,同时包含低频和高频特征；

(b)莫采用经典B样条曲面小波平滑的结果，图(c)是采用带约束的非张量积B样

条曲面小波平滑的结果,其位置约束如图中的血个标记所示。与经典B样条曲

面小波相比，带约束的山羊条小波在分解变换时不改变约束点处的曲面位置。

42

细分曲面

控制多边形/细分一次/极限曲面

(a)普通细分曲面(b)支持尖锐特征的细分曲面

43

_细分曲面

控制多边形/细分一次/极限曲面

(a)普通细分曲面(b)支持尖锐特征的细分曲面

44

细分曲而

(a)普通细分曲面(b)支持尖锐特征的细分曲面

45

%支持尖锐特征的细分曲面小波(1)

46

支持尖锐特征的细分曲面小波(2)

分解fandisk:6///

47

支持尖锐特征的细分曲面小波(2)

分解fandisk:/5//

48

支持尖锐特征的细分曲面小波(2)

分解fandisk://4/

.

49

支持尖锐特征的细分曲面小波(2)

分解fandisk:///0

50

支持尖锐特征的细分曲面小波(3)

51

支持尖锐特征的细分曲面小波(3)

分解mannequin:/4//

52

支持尖锐特征的细分曲面小波(3)

分解mannequin://3/

53

支持尖锐特征的细分曲面小波(3)

分解mannequin:///0

54

%支持尖锐特征的细分曲面小波(4)

分解te叩ot:5//

痔

55

%支持尖锐特征的细分曲面小波(4)

56

%支持尖锐特征的细分曲面小波(4)

分解te叩ot://3

♦

57

%支持尖锐特征的细分曲面小波(5)

58

%支持尖锐特征的细分曲面小波(5)

59

%支持尖锐特征的细分曲面小波(5)

60

■几何压缩(1)-

(a)venus(1,239B)(e)venus(2,477B)(i)venus(5,583B)

61

几何压缩(2)

(b)feline(93IB)(f)feline(2,135B)(j)feline(4.955B)

62

几何压缩(3)

(c)horse(1,328B)(g)horse(2,832B)(k)horse(6414B)

63

,几何压缩⑷

(d)bunny(1.395B)(h)bunny(3358B)(1)bunny(8,524B)

64

(b)feline(c)bu