transformations ii
DESCRIPTION
Transformations II. CS5600 Computer Graphics by Rich Riesenfeld 4 March 2002. Lecture Set 6. What About Elementary Inverses?. Scale Shear Rotation Translation. Scale Inverse. Shear Inverse. Shear Inverse. Rotation Inverse. Rotation Inverse. Rotation Inverse. Translation Inverse. - PowerPoint PPT PresentationTRANSCRIPT
Transformations II
CS5600 Computer Graphicsby
Rich Riesenfeld4 March 2002Le
ctur
e S
et 6
CS5600 2
What About Elementary Inverses?
• Scale• Shear• Rotation• Translation
CS5600 3
Scale Inverse
1001
100
100 1
100
100 11
CS5600 4
Shear Inverse
1
1
1
1
00
101
101
00
101
101
bb
aa
CS5600 5
Shear Inverse
101
101
101
101
1
1
bb
aa
CS5600 6
Rotation Inverse
cossin-sincos
cossinsin-cos
)cos(-)sin(-)sin(--)cos(-
cossinsin-cos
CS5600 7
Rotation Inverse
)cos(sin)sincossin(cos
)sincossin(cos)sin(cos22
22
cossin-sincos
cossinsin-cos
1001
CS5600 8
Rotation Inverse
cossin-sincos
cossinsin-cos 1
CS5600 9
Translation Inverse
10010
01
10010
01
10010
01
10010
01
0
0
0
)(
00
xdxd
xdxd
CS5600 10
Translation Inverse
10010
01
10010
01
00
1
xdxd
CS5600 11
Shear in x then in y)1,0(
)0,0(
)1,0(
)0,0(
)1,0(
)0,0()0,1( )0,1(
),(1 bab
),( 1a
),( 1 aba
),( 11 baba
),( 11 a
)1,0(
)0,0(
)1,1( b
)1,1(
),1( b
CS5600 12
Shear in y then in x)1,0(
)0,0(
)1,0(
)0,0(
)1,0(
)0,0()0,1( )0,1(
),(1 b
),( 1a
),( 1 aba
),( 11 abba
),( 11 a
)1,0(
)0,0(
)1,1( b
)1,1(
)0,1(
),1( b
CS5600 13
Results Are Different
y then x: x then y:
CS5600 14
Want the RHR to Work
jik
ikj
kji
kj
i
CS5600 15
3D Positive Rotations
x
z
y
CS5600 16
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 17
Example
x
y
1
2
34
CS5600 18
Example
is the transformation that
takes a point in coordinate system
j and converts it to a point in
coordinate system i
M ji
p j)(
p i)(
CS5600 19
Example
•
•
•
pMpji
ji)()(
pMpkj
kj)()(
MMM kjki ji
CS5600 20
Example
•
•
•
)2,4(12 TM
)3,2()2,2(32 TSM
)8.1,7.6()45(43 TRM
CS5600 21
Recall the Following
ABAB 111)(
CS5600 22
Since
•
•
•
)2,4(21 TM
)21,
21()3,2(23 STM
MM ijji
1
)45()8.1,7.6(34 RTM
CS5600 23
Change of Coordinate System
• Describe the old coordinate system in terms of the new one.
x’y
x
y’
CS5600 24
Change of Coordinate System
Move to the new coordinate system and describe the one old.
x
y
x
y Old is a negative rotation of the new.
CS5600 25
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 26
Many Kinds of Perspective Used
• Mechanical Engineering
• Cartography
• Art
CS5600 27
Perspective in Art
• Naïve (wrong)• Egyptian• Cubist (unrealistic)• Esher• Miro• Matisse
CS5600 28
Egyptian Frontalism
• Head profile
• Body front
• Eyes full
• Rigid style
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
CS5600 30
Perspective in Cubism
Braque, Georges
Woman with a Guitar
Sorgues, autumn
1913
CS5600 31
Perspective in Cubism
Pablo Picaso, Madre con niño muerto (1937) 32
Pablo Picaso Cabeza de mujer
llorando con
pañuelo33
CS5600 34
Perspective (Mural) Games
M C Esher, Another World II (1947)
CS5600 35
Perspective
M.C. Escher, Ascending
and Descending
(1960)
CS5600 36
M. C. Esher
M.C. Escher, Ascending
and Descending
(1960)
CS5600 37
M. C. Esher
• Perspective is “local”
• Perspective consistency is not “transitive”
• Nonplanar (hyperbolic) projection
CS5600 38
Nonplanar Projection
M C Esher, Heaven and
Hell
CS5600 39
Nonplanar Projection
M C Esher, Heaven and
Hell
CS5600 40
David McAllister
The March of Progress,
(1995)
CS5600 41
Joan MiroThe Tilled
Field
Flat Perspective:
What cues are missing?
Henri Matisse, La Lecon de
Musique
Flat Perspective:What cues are
missing?
42
Henri Matisse, Danse II (1910) 43
CS5600 44
CS5600 45
Norway is at High Latitude
CS5600 46
Isometric View
CS5600 47
Engineering Drawing
A
A
Section AA
Engineering Drawing: Exploded
View
Understanding 3D Assembly in a
2D Medium
48
CS5600 49
“True” Perspective in 2Dy
x
(x,y)
p
h
CS5600 50
“True” Perspective in 2D
pxpyh
pxy
ph
CS5600 51
“True” Perspective in 2D
11 110
010001
px
p
yx
yx
CS5600 52
“True” Perspective in 2D
px
pypx
px
pxpy
pxpx
ppx
yx
px
yx
11
This is right answer for screen projection
CS5600 53
Geometry is Same for Eye at Origin
y
x
(x,y)
p
h
Screen Plane
CS5600 54
What Happens to Special Points?
00
10
101010001 pp
p
What is this point??
CS5600 55
Look at a Limit
axis-on 0
0100
1
1
xn
nn
n
CS5600 56
Where does Eye Point Go?
• It gets sent to on x-axis
• Where does on x-axis go?
CS5600 57
What happens to ?
0100
1
001
10010001
11
pp
pp
It comes back to virtual eye point!
CS5600 58
What Does This Mean?
x
p
y
CS5600 59
The “Pencil of Lines” Becomes Parallel
x
y
CS5600 60
Parallel Lines Become a “Pencil of Lines” !
x
y
CS5600 61
What Does This Mean?
x
p
y
CS5600 62
“True” Perspective in 2Dy
p
CS5600 63
“True” Perspective in 2Dy
p
CS5600 64
Viewing Frustum
CS5600 65
What happens for large p?”
1100
010001
110010001
1
yx
yx
p
CS5600 66
Projection Becomes Orthogonal
x
(x,y)
p
h=y
The End of
Transformations II
67
Lect
ure
Set
6