Transformations II

Presentation Transcript

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?

Utah School of Computing

Want to rotate about arbitrary axis a

3

First rotate about z by

Now in the

(y-z)-plane

Then rotate about x by

Rotate in the

(y-z)-plane

Now perform rotation about

Now a-axis aligned

with z-axis

6

Then rotate about x by

Rotate again in the (y-z)-plane

Then rotate about z by

Now to original position of a

We effected a rotation by about arbitrary axis a

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We effected a rotation by about arbitrary axis a

10

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

In matrix terms,

so,

Similarly,

Recall,

Consequently, for

because,

It follows directly that,

Constructively, we have shown,

This will be useful later

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

Differert Perspectives Used

Mechanical Engineering
Cartography
Art

Perspective in Art

“Naïve” (wrong)
Egyptian
Cubist (unrealistic)
Esher

Impossible (exploits local property)
Hyperpolic (non-planar)
etc

“True” Perspective in 2D

(x,y)

h

p

This is right answer for screen projection

Perspective in Art

Naïve (wrong)
Egyptian
Cubist (unrealistic)
Esher
Miro
Matisse

Egyptian Frontalism

Head profile
Body front
Eyes full
Rigid style

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)

Perspective in Cubism

Pablo Picaso

Madre con niño muerto (1937)

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Pablo Picaso,Cabeza de mujer llorando con pañuelo

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Perspective (Mural) Games

M C Esher, Another World II (1947)

Perspective

Ascending and Descending (1960)

M C Escher

M. C. Escher

M C Escher, Ascending and Descending (1960)

M C Escher

Perspective is “local”
Perspective consistency is not “transitive”
Nonplanar (hyperbolic) projection

Nonplanar (Hyperbolic) Projection

M C Esher, Heaven and Hell

Nonplanar (Hyperbolic) Projection

M C Esher, Heaven and Hell

David McAllister

The March of Progress, (1995)

Joan Miro: Flat Perspective

The Tilled Field

What cues are missing?

Flat Perspective: What cues are missing?

Henri Matisse, La Lecon de Musique

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Next 2 Images Contain Nudity !

Henri Matisse, Danse (1909)

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Henri Matisse, Danse II (1910)

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Atlas Projection

Norway is at High Latitude

There is considerable size distortion

Isometric View

Engineering Drawing: 2 Planes

AA

SectionAA

Engineering Drawing: Exploded View

Understanding 3D Assembly in a 2D Medium

86

(x,y)

h

p

Geometry is Same for Eye at Origin

Screen Plane

(x,y)

h

What Happens to Special Points?

What is this point??

Let’s Look at Limit

Observe,

We see that

Where does Eye Point Go?

It gets sent to on x-axis
Where does on x-axis go?

What happens to ?

It comes back to virtual eye point!

What Does This Mean?

The “Pencil of Lines” Becomes Parallel

Parallel Lines Become “Pencil of Lines” !

ViewingFrustum

What happens for large p?”

Projection Becomes Orthogonal: “Right Thing Happens”

(x,y)

h=y

The End Transformations II

Lecture Set 7