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1、hearn-50265;ISBN:0-13-015390-7bookAugust4,200314:49APPENDIXAMathematicsforComputerGraphicsvarietyofmathematicalconceptsandtechniquesareemployedincomputer-graphicsalgorithms.Here,weprovideabriefrefer-enceforthemethodsfromanalyticgeometry,linearalgebra,
2、vectorAanalysis,tensoranalysis,complexnumbers,quaternions,calculus,numericalanalysis,andotherareasthatarereferredtointhedis-cussionsthroughoutthisbook.A-1COORDINATEREFERENCEFRAMESBothCartesianandnon-Cartesianreferenceframesareoftenusefulincomputer-gra
3、phicsapplications.WetypicallyspecifycoordinatesinagraphicsprogramusingaCartesianreferencesystem,buttheinitialspecificationofascenecouldbegiveninanon-Cartesianframeofreference.Spherical,cylindrical,orothersymmetriesoftencanbeexploitedtosimplifyexpressio
4、nsinvolvingobjectdescriptionsormanipulations.Two-DimensionalCartesianScreenCoordinatesForthedevice-independentcommandswithinagraphicspackage,screen-coordinatepositionsarereferencedwithinthefirstquadrantofatwo-dimensionalCartesianframeinstandardposition
5、,asshowninFig.A-1(a).Thecoordinateori-ginforthisreferenceframeisatthelower-leftscreencorner.Scanlines,however,arenumberedfrom0atthetopofthescreen,sothatscreenpositionsarerepre-sentedinternallywithrespecttotheupper-leftcornerofthescreen.Therefore,devic
6、e-dependentcommands,suchasthoseforinteractiveinputanddisplay-windowmanipulations,oftenreferencescreencoordinatesusingtheinvertedCartesianframeshowninFig.A-1(b).Horizontalcoordinatevaluesinthetwosystemsarethesame,andaninvertedyvalueisconvertedtoayvalue
7、measuredfromthebottomofthescreenwiththecalculationy=ymax−yinvert(A-1)787hearn-50265;ISBN:0-13-015390-7bookAugust4,200314:49788APPENDIXAMathematicsforComputerGraphicsxymaxyyinvertFIGUREA-1Cartesianscreen-coordinatepositionsarereferencedwithrespecttoyma
8、xthelower-leftscreencorner(a)ortheupper-leftscreenxcorner(b).(a)(b)ruyaxisyPruxxaxisFIGUREA-2Apolar-coordinateFIGUREA-3Relationshipreferenceframe,formedwithbetweenpolarandCartesianconcentriccirclesandradiallines.coordinates.Insomeapplicationpa
