Lecture Set 6 Transformations II CS5600 Computer Graphics by

Lecture Set 6 Transformations II CS5600 Computer Graphics by Rich Riesenfeld 4 March 2002

What About Elementary Inverses? Scale Shear Rotation Translation CS5600 2

Scale Inverse 0 0 1 0 1 0 0 1 0 1 1 1 0 1 0 CS5600 0 1 0 1 3

Shear Inverse 1 a 1 a 1 0 0 1 0 1 0 1 1 0 1 0 1 0 b 1 b 1 0 1 CS5600 4

Shear Inverse 1 0 a 1 1 1 b 0 1 1 1 0 1 b CS5600 a 1 0 1 5

Rotation Inverse cos sin cos sin -sin cos(- ) -sin(- ) cos sin(- ) cos(- ) -sin cos sin -sin cos cos CS5600 6

Rotation Inverse cos sin -sin cos cos -sin 2 2 (cos sin ) (cos sin cos sin ) 1 0 0 1 CS5600 sin cos (cos sin cos sin ) 2 2 (sin cos ) 7

Rotation Inverse cos sin -sin cos 1 cos -sin CS5600 sin cos 8

Translation Inverse 1 0 0 1 0 0 0 d 1 0 x 0 1 1 0 0 0 d x 0 1 1 0 0 ( d d ) 1 0 x 0 1 x CS5600 1 0 0 0 0 1 0 0 1 9

Translation Inverse 1 0 0 0 d 1 0 x 0 1 1 1 0 0 CS5600 0 d 1 0 x 0 1 10

Shear in x then in y (1,1) (0,1) (0,1) (0,0) (1,0) (0,0) (a,1) (1 a, 1) (1,0) (a,1 ab) (1 a ab,1 b) (1,1 b) (0,1) (0,1) (1, b) (0,0) (1 ab, b) CS5600 (0,0) 11

Shear in y then in x (0,1) (1,1) (0,0) (1,0) (0,1) (1 a, 1) (a,1) (0,0) (1,0) (1 a,1 b ab) (a,1 ab) (1,1 b) (0,1) (0,1) (1, b) (0,0) (1, b) CS5600 (0,0) 12

Results Are Different y then x: x then y: CS5600 13

Want the RHR to Work i j k j k i k i j i j CS5600 k 14

3D Positive Rotations z y x CS5600 15

Transformations as a Change in Coordinate System Useful in many situations Use most natural coordination system locally Tie things together in a global system CS5600 16

Example 4 y 3 2 x 1 CS5600 17

Example Mi j is the transformation that takes a point p ( j) in coordinate system j and converts it to a point (i) p in coordinate system i CS5600 18

Example (i ) p M i j p ( j) ( j) (k ) p M j k p M i k M i j M j k CS5600 19

Example M 2 1 T (4, 2) M 2 3 S (2, 2) T (2, 3) M 3 4 R( 45 ) T (6.7,1.8) CS5600 20

Recall the Following ( AB) 1 B CS5600 1 1 A 21

Since M i j 1 M j i M1 2 T ( 4, 2) 1 1 M 3 2 T ( 2, 3) S ( , ) 2 2 M 4 3 T ( 6.7, 1.8) R( 45 ) CS5600 22

Change of Coordinate System Describe the old coordinate system in terms of the new one. x’ y’ y x CS5600 23

Change of Coordinate System y Move to the new coordinate system and describe the one old. y Old is a x x CS5600 negative rotation of the new. 24

What is “Perspective?” A mechanism for portraying 3D in 2D “True Perspective” corresponds to projection onto a plane “True Perspective” corresponds to an ideal camera image CS5600 25

Many Kinds of Perspective Used Mechanical Engineering Cartography Art CS5600 26

Perspective in Art Naïve (wrong) Egyptian Cubist (unrealistic) Esher Miro Matisse CS5600 27

Egyptian Frontalism Head profile Body front Eyes full Rigid style CS5600 28

Uccello'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) 29

Perspective in Cubism Braque, Georges Woman with a Guitar CS5600 Sorgues, autumn 30

Perspective in Cubism CS5600 31

Pablo Picaso, 32 Madre con niño muerto (1937)

Pablo Picaso Cabeza de mujer llorando con pañuelo 33

Perspective (Mural) Games M C Esher, Another World II (1947) CS5600 34

Perspective M.C. Escher, Ascending and Descending (1960) CS5600 35

M. C. Esher M.C. Escher, Ascending and Descending (1960) CS5600 36

M. C. Esher Perspective is “local” Perspective consistency is not “transitive” Nonplanar (hyperbolic) projection CS5600 37

Nonplanar Projection M C Esher, Heaven and Hell CS5600 38

Nonplanar Projection M C Esher, Heaven and Hell CS5600 39

David McAllister The March of Progress, (1995) CS5600 40

Joan Miro The Tilled Field Flat Perspective: What cues are missing? CS5600 41

Flat Perspective: What cues are missing? Henri Matisse, La Lecon de Musique 42

Henri Matisse, Danse II (1910) 43

CS5600 44

Norway is at High Latitude CS5600 45

Isometric View CS5600 46

Engineering Drawing A A Section AA CS5600 47

Engineering Drawing: Exploded View Understanding 3D Assembly in a 2D Medium 48

“True” Perspective in 2D y (x,y) h x p CS5600 49

“True” Perspective in 2D h y p x p py h x p CS5600 50

“True” Perspective in 2D 1 0 1 p 0 1 0 0 0 1 x y 1 x y x 1 p CS5600 51

“True” Perspective in 2D x x y y x x p 1 p p px x p py x p 1 px x p py x p This is right answer for screen projection CS5600 52

Geometry is Same for Eye at Origin y Screen Plane (x,y) h x p CS5600 53

What Happens to Special Points? 1 0 0 1 p 1 0 0 0 1 p p 0 0 1 0 What is this point? CS5600 54

Look at a Limit n 1 0 0 1 n 1 n 0 n 0 on x - axis CS5600 55

Where does Eye Point Go? It gets sent to Where does on x-axis CS5600 on x-axis go? 56

What happens to ? 1 0 1 p 0 1 0 0 0 1 1 0 0 1 0 1 p p 0 1 p 0 It comes back to virtual eye point! CS5600 57

What Does This Mean? y x p CS5600 58

The “Pencil of Lines” Becomes Parallel y x CS5600 59

Parallel Lines Become a “Pencil of Lines” ! y x CS5600 60

What Does This Mean? y x p CS5600 61

“True” Perspective in 2D y p CS5600 62

“True” Perspective in 2D y p CS5600 63

Viewing Frustum CS5600 64

What happens for large p?” 1 0 1 p 0 1 0 0 0 1 x y 1 1 0 0 CS5600 0 0 x 1 0 0 1 y 1 65

Projection Becomes Orthogonal (x,y) h y x p CS5600 66

Lecture Set 6 The End of Transformations II 67