Viewing System
한신대학교류승택
3D Transformation
3D Transformation
ModelingCoordinates
(Xm, Ym, Zm)Modeling
Transformation
ModelingTransformation
WorldCoordinates(Xw, Yw, Zw)
ViewingTransformation
ViewingTransformation
ViewingCoordinates(Xv, Yv, Zv)
ProjectionTransformation
ProjectionTransformation
ProjectionCoordinates
(Xp, Yp)Workstation
Transformation
WorkstationTransformation
DeviceCoordinates
(Xd, Yd)
Viewing Transformation (1/5)
Pview = R T
Translation (T)
Rotation (R)
1000
100
010
001
z
y
x
C
C
C
T
1000
0
0
0
zyx
zyx
zyx
nnn
vvv
uuu
R
Scalar Product
Scalar Product (= dot product) ( 내적 ) The sum of the products of their corresponding components
Using the law of cosine, the angle between two vectors a and b satisfies the equation
Scalar Product Properties
• If a is perpendicular to b, then
zzyyxx bababa ba
cosbaba
ba
ba 1cos
abba 2
aaa
cabac)(ba )()()( bababa kkk
cosbaba
0ba
Scalar ??A quantity that is completely specified by its magnitude and has no direction.
a
b
Scalar Product
Scalar Product Use the dot product to project a vector onto another vector
• V unit vector
• The dot product of V and W the length the projection of W onto V
A property of dot product used in CG• Sign
WV|W||V|
WV|W||W||X|
cos
WV (unit vector)
X
o
o
o
90 if 0
90 if 0
90 if 0
WV
WV
WV
Vector Product
Vector Product (= Cross Product) 외적
c = a x b c is perpendicular to both a and b direction: right-hand property
• Perpendicular to the pane defined by a and b
kjiba )()()( xyyxxzzzxyzzy babababababa
)()()( xyyxxzzzxyzzy babababababa ba
zyx
zyx
bbb
aaa
kji
ba b
a
c
Viewing Transformation (2/5)
N vector
V vector Up vector
U vector
),,( zyx uuuVNU
zyx vvvV
VV ,,
0,1,0Up
NNUpUpV )(
zyx nnnN
NN ,,
V
Up-N
Viewing Transformation (3/5)
Another Way
),,(
),,(
,,
321
321
321
vvvunv
uuuNV
NVu
nnnN
Nn
대문자 V Up 벡터를 말함
Viewing Transformation (4/5)
Projection
Parallel Projection vs xx vs yy 0sz
1
ortv
v
v
z
y
x
w
Z
Y
X
P
1000
0000
0010
0001
ortP
Parallel Perspective
Viewing Transformation (5/5)
Perspective Projection
v
vs
z
x
d
x
v
vs
z
y
d
y
dz
yy
v
vs /
dz
xx
v
vs /
1
persv
v
v
z
y
x
w
Z
Y
X
P
0/100
0100
0010
0001
pers
d
P
wXxs / wYys / wZzs / dzw v /
Conclusion (1/2)
3D Viewing Transformation Parallel Projection
1
ort z
y
x
w
Z
Y
X
RTP
11000
100
010
001
1000
0
0
0
1000
0000
0010
0001
z
y
x
C
C
C
nnn
vvv
uuu
w
Z
Y
X
z
y
x
zyx
zyx
zyx
(Xv, Yv, Zv, 1)
(Xp, Yp, 0, 1)
(Xw, Yw, Zw, 1)
Conclusion (2/2)
3D Viewing Transformation Perspective Projection
1
per z
y
x
w
Z
Y
X
RTP
11000
100
010
001
1000
0
0
0
0/100
0100
0010
0001
z
y
x
C
C
C
nnn
vvv
uuu
dw
Z
Y
X
z
y
x
zyx
zyx
zyx
(Xv, Yv, Zv, 1)
(Xp, Yp, Zp, 1)
(Xw, Yw, Zw, 1)
3D Viewing Process (1/2)
3D Viewing Process
ModelingCoordinates
(Xm, Ym, Zm)Modeling
Transformation
ModelingTransformation
WorldCoordinates(Xw, Yw, Zw)
ViewingTransformation
ViewingTransformation
ViewingCoordinates(Xv, Yv, Zv)
ProjectionTransformation
ProjectionTransformation
ProjectionCoordinates
(Xp, Yp)Workstation
Transformation
WorkstationTransformation
DeviceCoordinates
(Xd, Yd)
3D Viewing Process (2/2)
3D Viewing Process
ModelingCoordinates
(Xm, Ym, Zm)Modeling
Transformation
ModelingTransformation
WorldCoordinates(Xw, Yw, Zw)
ViewingTransformation
ViewingTransformation
ViewingCoordinates(Xv, Yv, Zv)
ProjectionTransformation
ProjectionTransformation
ProjectionCoordinates
(Xp, Yp)Workstation
Transformation
WorkstationTransformation
DeviceCoordinates
(Xd, Yd)
Normalizing Transformation
Normalizing Transformation ClippingClipping
NormalizingCoordinates(Xn, Yn, Zn)
ClippingCoordinates(Xc, Yc, Zc)