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

What About Elementary Inverses? Scale Shear Rotation Translation CS5600

Scale Inverse CS5600

Shear Inverse CS5600

Shear Inverse CS5600

Rotation Inverse CS5600

Rotation Inverse CS5600

Rotation Inverse CS5600

Translation Inverse CS5600

Translation Inverse CS5600

Shear in x then in y CS5600

Shear in y then in x CS5600

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

Want the RHR to Work CS5600

3D Positive Rotations CS5600

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

Example 4 3 2 1 CS5600

Example is the transformation that takes a point in coordinate system j and converts it to a point in coordinate system i CS5600

Example CS5600

Example CS5600

Recall the Following CS5600

Since CS5600

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

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

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

Many Kinds of Perspective Used Mechanical Engineering Cartography Art CS5600

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

Egyptian Frontalism Head profile Body front Eyes full Rigid style CS5600

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 Woman with a Guitar Sorgues, autumn 1913 Braque, Georges Woman with a Guitar Sorgues, autumn 1913 CS5600

Perspective in Cubism CS5600

Pablo Picaso, Madre con niño muerto (1937) 32

Pablo Picaso Cabeza de mujer llorando con pañuelo 33

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

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

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

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

M C Esher, Heaven and Hell Nonplanar Projection M C Esher, Heaven and Hell CS5600

M C Esher, Heaven and Hell Nonplanar Projection M C Esher, Heaven and Hell CS5600

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

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

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

Henri Matisse, Danse II (1910) Sample speaker note (rfr) Henri Matisse, Danse II (1910) 43 CS5600

CS5600

Norway is at High Latitude CS5600

Isometric View CS5600

Engineering Drawing A A Section AA CS5600 CS5600 Section AA shows: 1. What object looks like when viewed orthogonally to cut-away looking in direction of arrows. 2. Bottom part shows solid material, no holes encountered. 3. Top part of Section AA shows orthogonal view in looking in direction of arrows, BUT as though the cut-away were pivoted to vertical. 4. Depth of "counter bore hole" which cannot be determined from main drawing. Concentric holes only indicate that something like this is going on. 5. This is a interesting way in which engineering drawing communicate 3D info on 2D media. A Section AA CS5600 CS5600

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

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

“True” Perspective in 2D CS5600

“True” Perspective in 2D CS5600

“True” Perspective in 2D This is right answer for screen projection CS5600

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

What Happens to Special Points? What is this point?? CS5600

Look at a Limit CS5600

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

It comes back to virtual eye point! What happens to ? It comes back to virtual eye point! CS5600

What Does This Mean? p CS5600

The “Pencil of Lines” Becomes Parallel CS5600

Parallel Lines Become a “Pencil of Lines” ! CS5600

What Does This Mean? p CS5600

“True” Perspective in 2D CS5600

“True” Perspective in 2D CS5600

Viewing Frustum CS5600

What happens for large p?” CS5600

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

The End of Transformations II Lecture Set 6 67