3d viewing perspective projections single point perspective cop on x-axis cop (-1/p 0 0 1) vp x (1/p...
TRANSCRIPT
![Page 1: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/1.jpg)
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3D ViewingPerspective Projections
Single Point Perspective
1111
1
1
1000
0100
0010
001
1
***
pxz
pxy
pxx
zyx
pxzyx
p
zyx
COP on X-axis
COP (-1/p 0 0 1) VPx (1/p 0 0 1)
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3D ViewingPerspective Projections
Two Point Perspective
1000
0100
010
001
q
p
PPP qppq
![Page 4: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/4.jpg)
3D ViewingPerspective Projections
Three Point Perspective
1000
100
010
001
P P qp
r
q
p
PP rpqr
![Page 5: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/5.jpg)
3D ViewingPerspective Projections
![Page 6: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/6.jpg)
3D ViewingVanishing Points
Two ways• Intersection of transformed lines• Transformation of points at infinity
X
Z
Y
Y
X
VPx
VPz
![Page 7: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/7.jpg)
3D Viewing
Orthographic
Plane Geometric Projections
Parallel Perspective
Axonometric Oblique
Trimetric Dimetric Isometric
Cavalier Cabinet
Single Point
Two Point
Three Point
![Page 8: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/8.jpg)
3D ViewingImplementation Issues
More from Interface point of view
Y
Z
X
Eye
N
V
U
World Coordinate System (WCS)
Viewing Coordinate System (VCS)
![Page 9: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/9.jpg)
3D ViewingView Coordinate System (VCS)
Viewing coordinate system• Position and orientation of the view plane• Extent of the view plane (window)• Position of the eye
View Plane• View Reference Point (VRP): the origin of VCS specified as (rx , ry, rz) in WCS: center of the scene• Normal to the view plane (nx , ny, nz )
![Page 10: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/10.jpg)
3D Viewing
View Plane• Normal Direction (View Plane Normal VPN) n (nx ,ny ,nz)
User may provide normalized vectore.g.
nx = sin cos ny = sin sin
nz = cos
Z
Y
r
X
View Coordinate System (VCS)
![Page 11: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/11.jpg)
3D Viewing
View Plane• Direction vv is a unit vector intuitively corresponding to “up” vector“up” vector is specified by the user in WCS
n
upup’
v
up’ = up – (up.n)n
v = up’ / |up’|
• Direction u
u = n x v ( Left Handed)
View Coordinate System (VCS)
![Page 12: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/12.jpg)
3D Viewing
Window and Eye
• Window : left, right, bottom,top (wl,wr,wb,wt) generally is centered at VRP (origin)
• Eye : e = (eu,ev,en) Typically e = (0,0,-E)
View Coordinate System (VCS)
u
e
wt
wb
wr
wl
n v
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3D ViewingTransformation from WCS to VCS
rba
rv
ubayx
M ) (
) () (
(x, y)
X
Y
O
O’
u
v
r
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3D ViewingTransformation from WCS to VCS
Point object is represented as • (a,b,c) in VCS• (x,y,z) in WCS
zyx
zyx
zyx
nnn
vvv
uuu
n
v
u
M
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3D Viewing
TMrp
Mrpcba
rMcbazyxp
)(
)(
1
Transformation from WCS to VCS
Conversion from one coordinate system to another
Therefore a=(p-r).u, b=(p-r).v, c=(p-r).n
![Page 16: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/16.jpg)
3D Viewing
1???
0
0
0T
wv
MA
In Homogenous Coordinates
(a,b,c,1) = (x,y,z,1) Awv
Transformation from WCS to VCS
![Page 17: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/17.jpg)
3D Viewing
ntranslatio
TTT rMpMMrpcba )(
In Homogenous Coordinates
r’= -rMT = (-r.u,-r.v,-r.n) = (rx’,ry’,rz’)puvn=pxyzAwv
1'''
0
0
0
1'''
0
0
0
zyx
zzz
yyy
xxx
zyx
T
wv
rrr
nvu
nvu
nvu
rrr
MA
Transformation from WCS to VCS
![Page 18: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/18.jpg)
3D ViewingTransformation from VCS to View Plane
et=0
p*
pt=1
t=t’
unv
e
p (pu,pv,pn)
p*(u*,v*)
Parametrically r(t) = e(1-t)+p.t
![Page 19: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/19.jpg)
3D ViewingTransformation from VCS to View Plane
On u-v plane,r(t)n = 0
nn
nvvn
nn
nuun
nn
n
nn
pepepe
v
pepepe
u
pee
t
tpte
*
*
'
'' )()1(0
![Page 20: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/20.jpg)
3D Viewing
1000
100
0010
0001
1
n
p eM
When eye is on n-axis eu=ev=0u*=pu/(en-pn), v*=pv/(en-pn)
Matrix form (n*=0) Perspective Transformation
1000
000
0010
0001
1
ne
Transformation from VCS to View Plane
![Page 21: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/21.jpg)
3D Viewing
Using Perspective Transformation Mp
Transformation from VCS to View Plane
)depth pseudo(
)1,,,(),,(
*
*
*
****
nn
n
nn
v
nn
u
pnvu
pep
n
pep
v
pep
u
Mpppnvup
![Page 22: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/22.jpg)
3D Viewing
1000
01
0010
0001
11
nvnu
s eeeeM
p*=(pu,pv,pn,1)MsMp
q : in WCS maps to p*=qAwvMsMp
Transformation from VCS to View Plane
If eye is off n-axis we have another matrix
![Page 23: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/23.jpg)
3D ViewingView Volume
Back Plane n=B
Front Plane n=F
View Plane, n=0Eye
![Page 24: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/24.jpg)
3D ViewingView Volume
v v
n n
wt
wb
wt
wbF B
F/(1-F/en) B/(1-B/en)
![Page 25: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/25.jpg)
3D ViewingVolume Normalization Transformation
Vt
Vb
Vl Vr
0
1v
u
![Page 26: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/26.jpg)
3D ViewingVolume Normalization Transformation
F/(1-F/en) B/(1-B/en)
no
0
nt
1
For n
)()(
)(
))((
11
1
2 BFeBeF
FBe
nBeFe
eFF
eBB
eFF
n
nn
n
n
onn
nn
n
o
t
Scaling sn Translation rn
![Page 27: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/27.jpg)
3D Viewing
1
000
000
000
nvu
n
v
n
rrr
s
s
s
N )(/)))(((
)/()()/()(
2 FBeFeBes
wwvvswwvvs
nnnn
btbtv
rlrlu
Volume Normalization Transformation
where
)(/)(
)/()()/()(
BFeBeFr
wwwvwvrwwwvwvr
nnn
btbttbv
rlrllru
Total Transformation: AwvMsMpN
![Page 28: 3D Viewing Perspective Projections Single Point Perspective COP on X-axis COP (-1/p 0 0 1) VP x (1/p 0 0 1)](https://reader035.vdocument.in/reader035/viewer/2022062421/56649e4d5503460f94b43d21/html5/thumbnails/28.jpg)