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