清华大学：《计算机图形学基础》课程教学资源（授课教案）参数曲线曲面、Bezier曲线、Bezier曲面

Bezier Curves and Surfaces. Parametric curves and surface: SomeConcepts Bézier Cuvres: Concept and properties Bezier surfaces: Rectangular andTriangular Conversion of Rectangular and TriangularBezier surfaces

Bézier Curves and Surfaces • Parametric curves and surface: Some Concepts • Bézier Cuvres: Concept and properties • Bézier surfaces: Rectangular and Triangular • Conversion of Rectangular and Triangular Bézier surfaces

Parametric curves and surface. In Graphics, we usually design a scene andthen generate realistic image by usingrendering equation.It's necessary to introduce geometricmodeling for scene designHowtorepresent3Dshapes (models)incomputer?

Parametric curves and surface • In Graphics, we usually design a scene and then generate realistic image by using rendering equation. • It’s necessary to introduce geometric modeling for scene design. • How to represent 3D shapes (models) in computer?

History of Geometric Modeling·Surface Modeling（曲面造型）： In 1962, Pierre Bézier, an engineer ofFrenchRenault Car company, propose a new kind ofcurve representation, and finally developed asystem UNISURF for car surface design in 1972

History of Geometric Modeling • Surface Modeling : – In 1962, Pierre Bézier, an engineer of French Renault Car company, propose a new kind of curve representation, and finally developed a system UNISURF for car surface design in 1972

BeziercurvelPierre Bezier<EullGlossarysDefinition:ABézier curve is a curved line orpath defined bymathematical equations.Itwas named after Pierre Bezier, a French mathematician and engineer who developed thismethod of computer drawing inthe late 1g6os while working forthe car manufacturerRenault.Mostgraphics softwareincludes a pentoolfordrawingpaths withBeziercurves.(Continuedbelow...)The most basic Bezier curve is made up of two end points and control handles attached toeachnode.Thecontrol handles definethe shape.ofthe curveon eithersideofthecommonnode.Drawing Bezier curves may seem baffling at first; it's something that requires somestudy and practice to grasp the geometry involved, But once mastered, Bezier curves are awonderful waytodraw!Pierre Bezier was bornSeptember 1, 1910 and died November 25,19g9 at the age of 89.In1985hewasrecognizedbyACMSIGGRAPHwithaStevenA.Coons'awardforhislifetimecontributiontocomputergraphics andinteractivetechniques.A Bezier curve with three nodes.The center node is selected and the control handles arevisible

• Pierre Etienne Bezier was born on September 1, 1910 in Paris. Son and grandson of engineers • He entered the Ecole Superieure d'Electricite and earnt a second degree in electrical engineering in 1931. In 1977, 46 years later, he received his DSc degree in mathematics from the University of Paris. • In 1933, aged 23, Bezier entered Renault and worked for this company for 42 years. • Bezier's academic career began in 1968 when he became Professor of Production Engineering at the Conservatoire National des Arts et Metiers. He held this position until 1979

·Solid Modeling（实体造型）：- In1973, Ian Braid of Cambridge Universitydeveloped a solid modeling system fordesigning engineering parts- Ian Braid presented in his dissertation'Designing with volumes", this work beingdemonstrated with the BUILD-1 system

• Solid Modeling : – In1973, Ian Braid of Cambridge University developed a solid modeling system for designing engineering parts. – Ian Braid presented in his dissertation “Designing with volumes”, this work being demonstrated with the BUILD-1 system

SolidModeling.orgHomeIanBraid,AlanGrayerandCharlesLangWelcomeThe2008Pierre B朗ier AwardRecipientsSymposiaThis group of three people made many fundamental contributions to practical solidCuirentmodelling, and their work has had a profound influence on today's commercial solidPastmodelling systems. They commenced working together in the CAD Group at theComputerLaboratory,Cambridge University,TheGroup was setupbyCharles Langunder Prof MauriceWilkesdirectionin1g65toundertakeresearchontools forPeoplebuilding mechanical CAD/CAM systemswithemphasis on software systemBszierAwardcomponents,computer graphics and computationalgeometry.Initial experiments inWho'sWhosolid modelling were made in 1969.Also in 196g Ian Braid joined the Groupwhere,JOin SMTAunder Charles Lang's supervision,he developed the BUiLD boundaryrepresentationmodeller,the most advanced such system of its day.Whereas other systems usedfaceting to avoid the problems of calculatingintersectionsbetweennon-planarBusinesssurfaces,theBuILDteamtackledsuchproblemshead-on.IanwasawardedhisPhDExecutive Boardin 1973.Alan Grayerjoined thegroup in1971 and,also underCharlesFrcceduressupervision, developed algorithms for the automatic machining of prismatic partsSponsoremodelled InBUILD,These weremachined onamodel making machine,builtbytheGroup in1971ifollowing aninspirational visit toB朗ieratRenault inParis,Alanwasawarded his PhD in197completalynewsolidmodeller,tnenCeVBIOOeOBUILD 2,whichwasa significalitmade a clear separation of geometrytacvanand algorithms. This made it possible toand topologyinbothits datastructuresimplementgeneralisedsystematicallyextend thegeometric coverage and thefunctionalitythe modeller with operations such asblending.SubsequentlyotherPhDthessupervisedbyIanandbasedontheBUILDmodellersincludedDimensionsandTolerances (Hillyard1g78),FeatureRecognition(Kyprianou1980),AutomaticMeshGeneration(Widenweber1g82)andandapSurface IntersectionsThesetheses were some of the earliest(Solomon1986)完成

How to represent a curve?: There are three major types of objectrepresentation:- Explicit representation: the explicit form of acurve in 2D gives the value of one variable.the dependent variable, in terms of the other.the independent variable. In x, y space, wemay writey= f(x)y=mx+h- For the line, we usually write

How to represent a curve? • There are three major types of object representation: – Explicit representation: the explicit form of a curve in 2D gives the value of one variable, the dependent variable, in terms of the other, the independent variable. In x, y space, we may write – For the line, we usually write y f x  ( ) y mx h  

- Implicit representation: In two dimensions, animplicit curve can be represented by the equationf(x,y)= 0- For the lineax +by +c = 0-Forthecircle2—2=0x+

– Implicit representation: In two dimensions, an implicit curve can be represented by the equation – For the line – For the circle 2 2 2 x y r    0 f x y ( , ) 0  ax by c    0

- Parametric form: The parametric form of a curveexpresses the value of each spatial variable forpoints on the curve in terms of an independentvariable t , the parameter.- In 3D, we have three explicit functionsx = x(t)y = y(t)z = z(t)

– Parametric form: The parametric form of a curve expresses the value of each spatial variable for points on the curve in terms of an independent variable , the parameter. – In 3D, we have three explicit functions t ( ) ( ) ( ) x x t y y t z z t   

- One of the advantages of the parametric form isthat it is the same in two and three dimensions. Inthe former case, we simply drop the equation forZ.（容易推广到高维）- A useful representation of the parametric form isto visualize the locus of pointsS0200(点的轨迹)AP(O)-P(t+A)P(t) =[x(t), y(t), z(t)}being drawn as t varies

– One of the advantages of the parametric form is that it is the same in two and three dimensions. In the former case, we simply drop the equation for z. – A useful representation of the parametric form is to visualize the locus of points ( ) being drawn as t varies. ( ) [ ( ), ( ), ( )]T P t x t y t z t 