λ = What About Elementary Inverses? Transformations II Scale Inverse Shear Inverse λ = λ 0 1 CS Scale Shear CS5600 Computer Graphics by
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1 Lecture Set 6 Transformations II CS56 Comuter Grahics b Rich Riesenfeld March 23 Scale Shear What About Elementar Inverses? Rotation Translation CS56 2 Scale Inverse Shear Inverse λ λ λ λ CS56 3 b a b a CS56 4 CS56
2 CS56 2 CS56 5 Shear Inverse b b a a CS56 6 Rotation Inverse - - (- (- -(- (- - CS56 7 Rotation Inverse + + ( ( ( ( CS56 8 Rotation Inverse - -
3 Translation Inverse Translation Inverse d ( d + d d CS56 9 d d CS56 (, Shear in then in (, (, ( a, ( + a, (, Shear in then in (, (, ( a, ( + a, (, (, (, (, (, (, (, (, (, + b ( a, ab ( + a + ab, + b (, + b ( a, + ab ( + a, + b + ab (, (, (, b (, (+ ab, b CS56 (, (, (, (, b (, (, (, + b CS56 2 (, CS56 3
4 Results Are Different Want the RHR to Work then : then : CS56 3 i j k j k i k i j i j k CS56 4 3D Positive Rotations Transformations as a Change in Coordinate Sstem z Useful in man situations Use most natural coordination sstem locall Tie things together in a global sstem CS56 5 CS56 6 CS56 4
5 Eamle Eamle 3 4 M i j is the transformation that takes a oint j in coordinate sstem ( ( i j and converts it to a oint in 2 coordinate sstem i CS56 7 CS56 8 Eamle Eamle ( i M i j ( j M 2 T (4, 2 ( j M j k ( k M 2 3 S(2,2 T(2,3 M i k M i j M j k M 3 4 T R( 45 (6.7,.8 CS56 9 CS56 2 CS56 5
6 Recall the Following Since M i j M j i ( AB B A M T M 2 ( 4, T ( 2, 3 S(, 2 2 M 4 3 T( 6.7,.8 R( + 45 CS56 2 CS56 22 Change of Coordinate Sstem Describe the old coordinate sstem in terms of the new one. Change of Coordinate Sstem Move to the new coordinate sstem and describe the one old. Old is a negative rotation of the new. CS56 23 CS56 24 CS56 6
7 What is Persective? Man Kinds of Persective Used A mechanism for ortraing 3D in 2D True Persective corresonds to rojection onto a lane True Persective corresonds to an ideal camera image Mechanical Engineering Cartograh Art CS56 25 CS56 26 Persective in Art Egtian Frontalism Naïve (wrong Egtian Cubist (unrealistic Esher Miro Matisse Head rofile Bod front Ees full Rigid stle CS56 27 CS56 28 CS56 7
8 Uccello's ( hand drawing was the first etant comle geometrical form rendered according to the laws of linear ersective Persective in Cubism Braque, Georges Persective Stud of a Chalice, Drawing, Gabinetto dei Disegni, Uffizi, Florence, ca 43 Woman with a Guitar Sorgues, autumn CS56 3 Persective in Cubism Pablo Picaso, Madre con niño muerto (937 CS CS56 8
9 Persective (Mural Games Pablo Picaso Cabeza de mujer llorando con añuelo M C Esher, Another World II ( CS56 34 Persective M. C. Esher M.C. Escher, Ascending and Descending (96 M.C. Escher, Ascending and Descending (96 CS56 35 CS56 36 CS56 9
10 M. C. Esher Nonlanar (Herbolic Projection Persective is local Persective consistenc is not transitive Nonlanar (herbolic rojection M C Esher, Heaven and Hell CS56 37 CS56 38 Nonlanar (Herbolic Projection David McAllister M C Esher, Heaven and Hell The March of Progress, (995 CS56 39 CS56 4 CS56
11 Joan Miro: Flat Persective The Tilled Field Flat Persective: What cues are misg? What cues are misg? Henri Matisse, La Lecon de Musique CS Atlas Projection Henri Matisse, Danse II (9 43 CS56 44 CS56
12 Norwa is at High Latitude Isometric View There is considerable size distortion CS56 45 CS56 46 Isometric View Engineering Drawing A Section AA A CS56 47 CS56 48 CS56 2
13 Engineering Drawing: Eloded View True Persective in 2D (, Understanding 3D Assembl in a 2D Medium 49 CS56 5 h True Persective in 2D h h + + True Persective in 2D CS56 5 CS56 52 CS56 3
14 Geometr is Same for Ee at Origin Screen Plane (, What Haens to Secial Points? h CS56 53 What is this oint?? CS56 54 Let s Look at a Limit Where does Ee Point Go? Observe, We see that n n n lim n n + on -ais CS56 55 It gets sent to Where does + on -ais on -ais go? CS56 56 CS56 4
15 What haens to +? What Does This Mean? It comes back to virtual ee oint! CS56 57 CS56 58 The Pencil of Lines Becomes Parallel Parallel Lines Become a Pencil of Lines! CS56 59 CS56 6 CS56 5
16 What Does This Mean? True Persective in 2D + CS56 6 CS56 62 True Persective in 2D Viewing Frustum CS56 63 CS56 64 CS56 6
17 What haens for large? Projection Becomes Orthogonal: Right Thing Haens lim CS56 65 (, h CS56 66 Lecture Set 6 The End of Transformations II 67 CS56 7
What is Perspective?
Fall 25 M ss =M screen * M ersective * M view What is Persective? A mechanism for ortraing 3D in 2D True Persective corresons to rojection onto a lane True Persective corresons to an ieal camera image
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