Transformations II

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# Transformations II - PowerPoint PPT Presentation

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

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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?

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Want to rotate about arbitrary axis a

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3

Now in the

(y-z)-plane

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Rotate in the

(y-z)-plane

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Now a-axis aligned

with z-axis

6

Rotate again in the (y-z)-plane

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Now to original position of a

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• 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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In matrix terms,

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so,

Similarly,

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Recall,

Consequently, for

because,

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It follows directly that,

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Constructively, we have shown,

This will be useful later

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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
• Mechanical Engineering
• Cartography
• Art

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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

(x,y)

h

p

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“True” Perspective in 2D

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“True” Perspective in 2D

This is right answer for screen projection

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“True” Perspective in 2D

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

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Egyptian Frontalism
• Body front
• Eyes full
• Rigid style

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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

GeorgesBraque

Woman with a Guitar (1913)

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Perspective in Cubism

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Pablo Picaso

68

Perspective (Mural) Games

M C Esher, Another World II (1947)

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Perspective

Ascending and Descending (1960)

M C Escher

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M. C. Escher

M C Escher, Ascending and Descending (1960)

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

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Nonplanar (Hyperbolic) Projection

M C Esher, Heaven and Hell

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Nonplanar (Hyperbolic) Projection

M C Esher, Heaven and Hell

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David McAllister

The March of Progress, (1995)

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Joan Miro: Flat Perspective

The Tilled Field

What cues are missing?

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Flat Perspective: What cues are missing?

Henri Matisse, La Lecon de Musique

78

Next 2 Images Contain Nudity !

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Atlas Projection

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Norway is at High Latitude

There is considerable size distortion

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Isometric View

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Engineering Drawing: 2 Planes

AA

AA

SectionAA

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Engineering Drawing: Exploded View

Understanding 3D Assembly in a 2D Medium

86

“True” Perspective in 2D

(x,y)

h

p

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“True” Perspective in 2D

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“True” Perspective in 2D

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Geometry is Same for Eye at Origin

Screen Plane

(x,y)

h

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What Happens to Special Points?

What is this point??

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Let’s Look at Limit

Observe,

We see that

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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 ?

It comes back to virtual eye point!

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What Does This Mean?

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What Does This Mean?

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What Does This Mean?

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“True” Perspective in 2D

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“True” Perspective in 2D

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ViewingFrustum

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What happens for large p?”

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