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

CS56002 What About Elementary Inverses? Scale Shear Rotation Translation

CS56003 Scale Inverse

CS56004 Shear Inverse

CS56005 Shear Inverse

CS56006 Rotation Inverse

CS56007 Rotation Inverse

CS56008 Rotation Inverse

CS56009 Translation Inverse

CS Translation Inverse

CS Shear in x then in y

CS Shear in y then in x

CS Results Are Different y then x:x then y:

CS Want the RHR to Work

CS D Positive Rotations

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

CS Example

CS Example is the transformation that takes a point in coordinate system j and converts it to a point in coordinate system i

CS Example

CS Example

CS Recall the Following

CS Since

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

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

CS 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

CS Many Kinds of Perspective Used Mechanical Engineering Cartography Art

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

CS Egyptian Frontalism Head profile Body front Eyes full Rigid style

Uccello's ( ) 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 ) 29

CS Perspective in Cubism Braque, Georges Woman with a Guitar Sorgues, autumn 1913

CS Perspective in Cubism

Pablo Picaso, Madre con niño muerto (1937) 32

Pablo Picaso Cabeza de mujer llorando con pañuelo 33

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

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

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

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

CS Nonplanar Projection M C Esher, Heaven and Hell

CS Nonplanar Projection M C Esher, Heaven and Hell

CS David McAllister The March of Progress, (1995)

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

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

Henri Matisse, Danse II (1910) 43

CS560044

CS Norway is at High Latitude

CS Isometric View

CS Engineering Drawing A A Section AA

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

CS “True” Perspective in 2D (x,y) p h

CS “True” Perspective in 2D

CS “True” Perspective in 2D

CS “True” Perspective in 2D This is right answer for screen projection

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

CS What Happens to Special Points? What is this point??

CS Look at a Limit

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

CS What happens to? It comes back to virtual eye point!

CS What Does This Mean? p

CS The “Pencil of Lines” Becomes Parallel

CS Parallel Lines Become a “Pencil of Lines” !

CS What Does This Mean? p

CS “True” Perspective in 2D p

CS “True” Perspective in 2D p

CS Viewing Frustum

CS What happens for large p?”

CS Projection Becomes Orthogonal (x,y) p h=y

The End of Transformations II 67 Lecture Set 6