Transformations IIWhat About Elementary Inverses?Scale InverseShear InverseSlide 5Rotation InverseSlide 7Slide 8Translation InverseSlide 10Shear in x then in yShear in y then in xResults Are DifferentWant the RHR to Work3D Positive RotationsTransformations as a Change in Coordinate SystemExampleSlide 18Slide 19Slide 20Recall the FollowingSinceChange of Coordinate SystemSlide 24What is “Perspective?”Many Kinds of Perspective UsedPerspective in ArtEgyptian FrontalismPowerPoint PresentationPerspective in CubismSlide 31Slide 32Slide 33Perspective (Mural) GamesPerspectiveM. C. EsherSlide 37Nonplanar ProjectionSlide 39David McAllisterJoan MiroSlide 42Slide 43Slide 44Norway is at High LatitudeIsometric ViewEngineering DrawingSlide 48“True” Perspective in 2DSlide 50Slide 51Slide 52Geometry is Same for Eye at OriginWhat Happens to Special Points?Look at a LimitWhere does Eye Point Go?What happens to ?What Does This Mean?The “Pencil of Lines” Becomes ParallelParallel Lines Become a “Pencil of Lines” !Slide 61Slide 62Slide 63Viewing FrustumWhat happens for large p?”Projection Becomes OrthogonalThe End of Transformations IITransformations IICS5600 Computer GraphicsbyRich Riesenfeld4 March 2002Lecture Set 6CS5600 2What About Elementary Inverses?•Scale•Shear•Rotation•TranslationCS5600 3Scale Inverse1001100100110010011CS5600 4Shear Inverse11110010110100101101bbaaCS5600 5Shear Inverse10110110110111bbaaCS5600 6Rotation Inversecossin-sincoscossinsin-cos)cos(-)sin(-)sin(--)cos(-cossinsin-cosCS5600 7Rotation Inverse)cos(sin)sincossin(cos)sincossin(cos)sin(cos2222cossin-sincoscossinsin-cos1001CS5600 8Rotation Inversecossin-sincoscossinsin-cos1CS5600 9Translation Inverse1001001100100110010011001001000)(00xdxdxdxdCS5600 10Translation Inverse10010011001001001xdxdCS5600 11Shear in x then in y)1,0()0,0()1,0()0,0()1,0()0,0()0,1()0,1(),(1bab),(1a),(1aba   ),(11baba ),(11a)1,0()0,0()1,1( b)1,1(),1( bCS5600 12Shear in y then in x)1,0()0,0()1,0()0,0()1,0()0,0()0,1()0,1(),(1b),(1a),(1aba ),(11abba ),(11a)1,0()0,0()1,1( b)1,1()0,1(),1( bCS5600 13Results Are Differenty then x: x then y:CS5600 14Want the RHR to WorkjikikjkjikjiCS5600 153D Positive RotationsxzyCS5600 16Transformations as a Change in Coordinate System •Useful in many situations•Use most natural coordination system locally•Tie things together in a global systemCS5600 17Examplexy1234CS5600 18Example is the transformation that takes a point in coordinate system j and converts it to a point in coordinate system iMji pj)(pi)(CS5600 19Example• • • pMpjiji)()(pMpkjkj)()(MMMkjkijiCS5600 20Example• • • )2,4(12TM )3,2()2,2(32TSM-)8.1,7.6()45(43TRM-CS5600 21Recall the FollowingABAB111)(CS5600 22Since • • • )2,4(21TM)21,21()3,2(23STM-MMijji1)45()8.1,7.6(34-RTMCS5600 23Change of Coordinate System•Describe the old coordinate system in terms of the new one.x’yxy’CS5600 24Change of Coordinate SystemMove to the new coordinate system and describe the one old. xyxyOld is a negative rotation of the new.CS5600 25What is “Perspective?”•A mechanism for portraying 3D in 2D•“True Perspective” corresponds to projection onto a plane•“True Perspective” corresponds to an ideal camera imageCS5600 26Many Kinds of Perspective Used•Mechanical Engineering•Cartography•ArtCS5600 27Perspective in Art•Naïve (wrong)•Egyptian•Cubist (unrealistic)•Esher•Miro•MatisseCS5600 28Egyptian Frontalism•Head profile•Body front•Eyes full•Rigid styleUccello's (1392-1475) handdrawing was the first extant complex geometrical form rendered accor-ding to the laws of linear perspective (Perspective Study of a Chalice, Drawing, Gabinetto dei Disegni, Uffizi, Florence, ca 1430 1440)29CS5600 30Perspective in Cubism Braque, Georges Woman with a Guitar Sorgues, autumn 1913CS5600 31Perspective in CubismPablo Picaso, Madre con niño muerto (1937) 32Pablo Picaso Cabeza de mujer llorando con pañuelo33CS5600 34Perspective (Mural) GamesM C Esher, Another World II (1947)CS5600 35PerspectiveM.C. Escher, Ascending and Descending (1960)CS5600 36M. C. EsherM.C. Escher, Ascending and Descending (1960)CS5600 37M. C. Esher•Perspective is “local”•Perspective consistency is not “transitive”•Nonplanar (hyperbolic) projectionCS5600 38Nonplanar ProjectionM C Esher, Heaven and HellCS5600 39Nonplanar ProjectionM C Esher, Heaven and HellCS5600 40David McAllisterThe March of Progress, (1995)CS5600 41Joan MiroThe Tilled FieldFlat Perspective:What cues are missing?Henri Matisse, La Lecon de MusiqueFlat Perspective:What cues are missing?42Henri Matisse, Danse II (1910)43CS5600 44CS5600 45Norway is at High LatitudeCS5600 46Isometric ViewCS5600 47Engineering DrawingAASection AAEngineering Drawing: Exploded ViewUnderstanding 3D Assembly in a 2D Medium48CS5600 49“True” Perspective in 2Dyx(x,y)   phCS5600 50“True” Perspective in 2DpxpyhpxyphCS5600 51“True” Perspective in