机械CADChapter 5Basic Graphics Concepts.docx

机械CADChapter 5Basic Graphics Concepts.docx

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机械CADChapter5BasicGraphicsConcepts

Chapter5BasicGraphicsConcepts

5.1Introduction

WhenweuseCADsystemstobuildamodel,geometrytransformationsareoftenperformedtochangetheposition,orientation,andsizeofthemodelingobjects.Geometrytransformationsareoftenperformedinhomogenouscoordinates,andmatricesaresuitabletorepresentthesetransformations.

Visualizationisimportantbothduringandaftermodelgenerations,Inordertodisplaya3Dobject,techniqueslikeperspectiveprojections,hiddenlineandsurfaceremovalareused.Variousshadingmodelsareusedtogivetheobjectarealisticimage.Thischaptergivesanintroductiontothesebasictechniquesusedinmodelgenerationsandvisualization.

5.2GeometryTransformation

5.2.13DTransformations

5.2.1.1Translation

ApointP（x,y,z）maybetranslatedintotoanewpositionP*（x*,y*,z*）byaddingtranslationamountsd（dx,dy,dz）tothecoordinatesofthepoint,i.e.

P*=P+d（5.1）

whereP*isthefinalpositionvector,Pistheinitialpositionvector,anddisshowninFig.5.1.

Inscalarform,Eq.（5.1）canbewrittenas

（5.2）

5.2.1.2Scaling

Thepointcanbescaledbysxalongxaxis,sya1ongyaxisandszalongzaxis,whichmaybemathematicallyexpressedas

P*=[S]P（5.3）

where[S]isadiagonalmatrix,

（5.4）

Inscalarform,Eq.（5.3）maybewrittenas

（5.5）

5.2.1.3Reflection

Reflection（ormirror）operationsareoftenusedtogeneratesymmetricmodels.Reflectionoperationsrelativetothemodelcoordinatesystemcanbeexpressedas

P*=[M]P（5.6）

Where

（5.7）

wherem11,m22,m33=

1.

Forexample,thetransformationmatrixtoreflectPrelativetoxyplaneis

（5.8）

5.2.1.4Rotation

AtypicalrotationoperationisdemonstratedinFig.5.2,wherepointPisrotatedanangel

aboutzaxis,andthenewpositionisP*.

Fromthefigure,itcanbeobtainedthat

.Utilizingthetrigonometricfunctions,wegetthematrixformfortherotationaboutzaxis

P*=[R]P（5.9）

Where

（5.10）

Therotationsaboutxandyaxiscanbederivedinthesimilarway.

5.2.2HomogenousRepresentation

Thetranslationtransformationinpreviouschapterisrepresentedbymatrixaddition,whilescalingandrotationbymatrixmultiplications.Itisconvenienttorepresentalloperationsbymatrixmultiplication,suchas

P*=[T]P（5.11）

where[T]isthetransformationmatrixwewanttodeduce,PandP*arethecoordinatesofpointsbeforeandafterthetransformations.Thiscanbedonewithhomogenouscoordinates,andthetransformationmatrix[T]becomesa4

4matrix.

Foranypoint（x,y,z）inCartesiancoordinates,wedefinethefollowingrepresentation:

①Fromthe3Dcoordinatetothehomogenouscoordinate:

（x,y,z）=>（.x,y,z,1）

②Promthehomogenouscoordinatetothe3Dcoordinate:

（x,y,z,W）=>（x/W,y/W,z/W）（W

）.

③Hormogenizing:

（x,y,z,W））=>（x/W,y/W,z/W,1）.

Forexample,（4,0,0,2）and（2,0,0,1）representthesamecoordinates（2,0,0）.

Wenowusethehomogenouscoordinatestorepresentthegeometrytransformationdiscussedinpreviouschapter.

GiveP=[xyz1]T,thefollowingoperationsmaybedefined.

（1）Translation

P*=T（dx,dy,dz）P（5.12）

Where

（5.13）

（2）Scaling

P*=S（sx,sy,sz）P（5.14）

Where

（5.15）

（3）Rotaiton

（5.16）

Whereforrotationaboutxaxis

（5.17a）

forrotationaboutyaxis

（5.17b）

forrotationaboutzaxis

（5.17c）

（4）Compositetransformation

Inpractice,aseriesoftransformationsmaybeappliedtoageometrymodel,resultinginacompositetransformation.Compositetransformationmatrixisequaltotheproductofthesequenceofthegiventransformationmatrices.Theorderofmultiplicationofthematricesisimportantbecausematrixmultiplicationisnotcommutative.

Mostofthetransformationsthatwenormallydealwithcanbeobtainedasacompositetransformationofbasictransformations.Ingeneralwehave

（5.18）

where[Ti]aretransformationmatricesand[T]istheproductofthem.

5.3MappingofGeometryModels

Thegeometrytransformationsdiscussedsofarassumethatthecoordinatesystemstaysunchangedandtheobjectistransformedwithrespecttotheoriginofthecoordinatesystem.Analternativebutequivalentwayofthinkingistotakethegeometrytransformationasachangeofcoordinatesystem.Thegeometrytransformationscanberegardedasmappingsofthegeometrymodelbetweendifferentcoordinatesystems,andtherelatedmatricesarecalledmappingmatrices.

Thisviewisusefulduringthemodelgeneration,asitisoftenconvenienttobuildsomepartofthemodelinthelocal（working）coordinatesystems.Thesoftwaremapsthesecoordinatestoglobalcoordinates.

5.3.1Translation

Figure5.3showsthatpointPisdescribedintwocoordinatesystemsxyzandx*y*z*.ThemappingbetweenthesetwocoordinatesystemscanbewrittenasP*=[T]P,where[T]isthesameasT（dx,dy,dz）inEq.（5.13）.

5.3.2Rotation

Figure5.4showstherotationalmappingbetweentwocoordinatesystems.Thesetwocoordinatesystemssharethesameoriginbuttheirorientationsaredifferent.Itisalsoassumedthatthexyandthex*y*planesarecoincident.

Theunitvectorsi*,j*,K*ofthecoordinatesystemx*y*z*canbeexpressedas

（5.19）

Inmatrixform,

（5.20）

5.3.3GeneralMapping

ForageneralmappingasshowninFig.5.5,wehaveP*=[T]P

Where

（5.21）

Inwhich[R]istheorientationvectorsoftheorigintothexyzcoordinatesystem,whiledisthepositionvector,bothexpressedinx*y*z*system.

5.4ProjectionsofGeometryModels

Projectionisusedtoreducethedimensionofamodelfrom3Dto2D.Therearetwotypesofprojections:

parallelprojectionandperspectiveprojection.

5.4.1ParallelProjection

Parallelprojectioncanbedividedintoorthographicandobliqueprojections.Inorthographicprojection,directionofprojectionandprojectionplanenormalareparallel,whileinobliqueprojection,directionofprojectionandprojectionplanenormalarenotparallel.

Thematrixformoforthographicprojectionis:

Pv=[T]P（5.22）

Where

（5.23）

Orthographicprojectionpreservesrelativeproportionsandiscommonlyusedindraftingtoproducedrawingof3Dobjects.

5.4.2PerspectiveProjection

Ifthedistancebetweentheprojectioncenterandtheprojectionplanisfinite,thentheprojectionisperspective.Perspectiveprojectionisrealisticbecauseitresembleshuman’sviewing,andtheobjectbecomessmallerasitismovedawayfromtheviewplane.

Figure5.9showsthatacubeviewedinperspective.Theprojectionplaneisparalleltothexyplaneandperpendiculartothezaxis.Linesthatareparalleltothexoryaxisremainparallel,andxandydistancesbecomeshorteraszbecomesmorenegative.

Toderivethetransformationmatrixinperspectiveprojection,weplacethecentreofprojectionintheoriginofthecoordinatesystemandtheprojectionplanisperpendiculartothezvaxis,seeFig.5.10.Assumethat（xv,yv,zv）and（xp,yp,zp）arecoordinatesbeforeandaftertheperspectiveprojection.Fromthefigure,wegetzp=d,yp=dyv/zv,Similarlyxp=dxv/zv.Thiscanbewritteninhomogenousform,

（5.24）

5.5HiddenSurfaceRemoval

Togivea3Dpictureofthemodel,hiddenlinesandsurfacesmustberemoved.Linesareassumedtobetheedgesofthesurfaceandwerefertothegeneralprocessofmovinghiddensurfacesashiddensurfaceremoval.

Therearevariousalgorithmsforhiddensurfaceremoval.Thesealgorithmscanbecharacterizedastwogroups:

objectprecisionandimageprecision.

Objectprecisionisbasedontheobjectgeometryanddecideswhichpartoftheobjectisvisibleandthendrawsthesesurfaces.Theprecisionisindependentoftheresolutiononscreen.

Imageprecisionisapixelbasedmethod.Thismethodgoesthrougheachpixelonthescreen,anddeterminestheobjectclosesttotheprojectorandthendrawsthepixel.Theresultsdependontheresolution.

5.5.1VisualTechniques

Theoperationsusedinthetypicalimage-precisionandobject-precisionalgorithmsareof-tencostly.Weneedtomaketheseoperationsaseffectivelyaspossible.Thefollowingtechniquesprovidesomegeneralwaystodothat.

5.5.1.1Coherence

①Objectcoherence:

Iftwoobjectsareentirelyseparated,weneedonlytocomparetwoobjects,notbetweencomponentfacesoredgesoftheobjects.

②Facecoherence:

Surfacecomputationcanbemodifiedincrementallytoapplytoadjacentparts.

③Edgecoherence:

Anedgecanchangeitsvisibilityonlywhenitcrossesbehindavisibleedge.

④Scan-linecoherence:

Onescan-lineofanimagetypicallydifferslittlefromthepreviousline.

⑤Areacoherence:

Agroupofadjacentpixelsisoftencoveredbythesamevisibleface.

⑥Depthcoherence:

Oncethedepthatonepointofthesurfaceiscalculated,thedepthofpointsontherestofthesurfacecanoftenbedeterminedbysimpledifferenceequation.

⑦Framecoherence:

Calculationsmadeforonepicturecanbeusedforthenextinsequence.

5.5.1.2PerspectiveTransformation

Theideawiththeperspectivetransformationistotransfertheperspectiveprojectionintotheparallelprojectionofthetransformedobjects,asshowninFig5.11.

ThepyramidinFig.5.11iscutbyafrontclippingplaneandabackclippingplane,bothplanesareperpendiculartothedirectionofprojection.Onlytheobjectsinsidethetruncatedpyramidaredisplayedandthistruncatedpyramidiscalledviewingfrustum.Theviewingfrustumbecomesaboxafterperspectivetransformation.

Theadvantagesofperspectivetransformationarethefollowing:

①Providingz-information,whichcanbeusedinz-bufferalgorithmandtexturemapping.

②Simplifyingclipping,asinthecaseofapyramidbecomingabox.

③Invertible