student name server spring 2005utah school of computing1 what is “ perspective ?” a mechanism...
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Spring 2005 Utah School of Computing 1Student Name ServerStudent Name Server
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
Spring 2005 Utah School of Computing 2Student Name ServerStudent Name Server
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 2Dy
x
(x,y)
p
h
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“True” Perspective in 2D
pxpyh
pxy
ph
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“True” Perspective in 2D
px
py
px
px
px
py
px
px
p
pxpx
y
x
y
x
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This is right answer for screen projection
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“True” Perspective in 2D
1 1
1
1 0 0
0 1 0
0 1 1px
x
xp p
pyx p
x
p
pp
x x
y y
x
y
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Perspective in Art
• Naïve (wrong)
• Egyptian
• Cubist (unrealistic)
• Esher
• Miro
• Matisse
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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)
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Perspective in Cubism
Woman with a Guitar (1913) G
eorg
es B
raqu
e
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Perspective (Mural) Games
M C Esher, Another World II
(1947)
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PerspectiveAscending and Descending (1960)
M C
Escher
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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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Norway is at High Latitude
There is considerable size distortion
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Engineering Drawing: 2 Planes
AA
AA
Section AA
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“True” Perspective in 2Dy
x
(x,y)
p
h
Spring 2005 Utah School of Computing 29Student Name ServerStudent Name Server
“True” Perspective in 2D
pxpyh
pxy
ph
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“True” Perspective in 2D
1
1
1
1 0 0
0 1 0
0 1 1 xp p
pyx p
pxx p
x pp
px
x p
py
x p
x x
y y
x
y
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Geometry is Same for Eye at Originy
x
(x,y)
p
h
Screen Plane
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What Happens to Special Points?
What is this point??
1
1 0 0
0 1 0
0 1 1 0
0 0
p
p p
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Let’s Look at Limit
1
1lim 0 0
01
n
nn
n
We see that
Observe,
on -axis0
nx
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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 ?
1 1
1 11 0 0
0 1 0
0 1 10
0 0 00
p p
p p
It comes back to virtual eye point!
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What Does This Mean?
x
y
p
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What Does This Mean?y
p
x
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The “Pencil of Lines” Becomes Parallel
y
x
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Parallel Lines Become “Pencil of Lines” !
x
y
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Parallel Lines Become “Pencil of Lines” !
x
y
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What Does This Mean?
x
y
p
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“True” Perspective in 2Dy
p
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“True” Perspective in 2Dy
p
p
p
p
p
p
p
p
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What happens for large p?”
1 0 01 0 0
0 1 0 0 1 0
0 1 0 1
1
1 01 1
lim 0
p
p
x x
y y
p