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Transformations II. CS5600 Computer Graphics Rich Riesenfeld Spring 2005. Lecture Set 7. Arbitrary 3D Rotation. What is its inverse? What is its transpose? Can we constructively elucidate this relationship?. Want to rotate about arbitrary axis a. 3.

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transformations ii

Transformations II

CS5600 Computer Graphics

Rich Riesenfeld

Spring 2005

Lecture Set 7

arbitrary 3d rotation
Arbitrary 3D Rotation
  • What is its inverse?
  • What is its transpose?
  • Can we constructively elucidate this relationship?

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want to rotate about arbitrary axis a
Want to rotate about arbitrary axis a

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3

first rotate about z by
First rotate about z by

Now in the

(y-z)-plane

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then rotate about x by
Then rotate about x by

Rotate in the

(y-z)-plane

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now perform rotation about
Now perform rotation about

Now a-axis aligned

with z-axis

6

then rotate about x by1
Then rotate about x by

Rotate again in the (y-z)-plane

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then rotate about z by
Then rotate about z by

Now to original position of a

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rotation about arbitrary axis a
Rotation about arbitrary axis a
  • Rotation about a-axiscan be effected by a composition of 5elementary rotations
  • We show arbitrary rotation as succession of 5rotations about principal axes

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slide12

In matrix terms,

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slide13

so,

Similarly,

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recall
Recall,

Consequently, for

because,

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it follows directly that
It follows directly that,

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slide17

Constructively, we have shown,

This will be useful later

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what is perspective
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

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differert perspectives used
Differert Perspectives Used
  • Mechanical Engineering
  • Cartography
  • Art

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perspective in art
Perspective in Art
  • “Naïve” (wrong)
  • Egyptian
  • Cubist (unrealistic)
  • Esher
    • Impossible (exploits local property)
    • Hyperpolic (non-planar)
    • etc

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true perspective in 2d
“True” Perspective in 2D

(x,y)

h

p

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true perspective in 2d1
“True” Perspective in 2D

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true perspective in 2d2
“True” Perspective in 2D

This is right answer for screen projection

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true perspective in 2d3
“True” Perspective in 2D

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perspective in art1
Perspective in Art
  • Naïve (wrong)
  • Egyptian
  • Cubist (unrealistic)
  • Esher
  • Miro
  • Matisse

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egyptian frontalism
Egyptian Frontalism
  • Head profile
  • Body front
  • Eyes full
  • Rigid style

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slide27

Uccello's (1392-1475) hand drawing was the first extant complex geometrical form rendered according to the laws of linear perspective

Perspective Study of a Chalice, Drawing, Gabinetto dei Disegni, Uffizi, Florence, ca 1430)

65

perspective in cubism
Perspective in Cubism

GeorgesBraque

Woman with a Guitar (1913)

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perspective in cubism1
Perspective in Cubism

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slide30

Pablo Picaso

Madre con niño muerto (1937)

68

perspective mural games
Perspective (Mural) Games

M C Esher, Another World II (1947)

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perspective
Perspective

Ascending and Descending (1960)

M C Escher

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m c escher
M. C. Escher

M C Escher, Ascending and Descending (1960)

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m c escher1
M C Escher
  • Perspective is “local”
  • Perspective consistency is not “transitive”
  • Nonplanar (hyperbolic) projection

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nonplanar hyperbolic projection
Nonplanar (Hyperbolic) Projection

M C Esher, Heaven and Hell

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nonplanar hyperbolic projection1
Nonplanar (Hyperbolic) Projection

M C Esher, Heaven and Hell

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david mcallister
David McAllister

The March of Progress, (1995)

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joan miro flat perspective
Joan Miro: Flat Perspective

The Tilled Field

What cues are missing?

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slide40

Flat Perspective: What cues are missing?

Henri Matisse, La Lecon de Musique

78

next 2 images contain nudity
Next 2 Images Contain Nudity !

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atlas projection
Atlas Projection

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norway is at high latitude
Norway is at High Latitude

There is considerable size distortion

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isometric view
Isometric View

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engineering drawing 2 planes
Engineering Drawing: 2 Planes

AA

AA

SectionAA

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slide48

Engineering Drawing: Exploded View

Understanding 3D Assembly in a 2D Medium

86

true perspective in 2d4
“True” Perspective in 2D

(x,y)

h

p

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true perspective in 2d5
“True” Perspective in 2D

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true perspective in 2d6
“True” Perspective in 2D

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geometry is same for eye at origin
Geometry is Same for Eye at Origin

Screen Plane

(x,y)

h

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what happens to special points
What Happens to Special Points?

What is this point??

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let s look at limit
Let’s Look at Limit

Observe,

We see that

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where does eye point go
Where does Eye Point Go?
  • It gets sent to on x-axis
  • Where does on x-axis go?

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what happens to
What happens to ?

It comes back to virtual eye point!

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what does this mean
What Does This Mean?

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what does this mean1
What Does This Mean?

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what does this mean2
What Does This Mean?

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true perspective in 2d7
“True” Perspective in 2D

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true perspective in 2d8
“True” Perspective in 2D

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viewing frustum
ViewingFrustum

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what happens for large p
What happens for large p?”

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