Computer Graphics Viewing
Shmuel Wimer
Bar Ilan Univ., School of Engineering
May 2010
1
Clipping Window
ywmax
World Coordinates
The clipping window is
mapped into a viewport.
ywmin
xwmin
xwmax
Viewing world has its own
coordinates, which may be
a non-uniform scaling of
world coordinates.
Viewport
yvmax
yvmin
Viewport Coordinates
May 2010
xvmin
xvmax
2
2D viewing transformation pipeline
Modeling
Coordinates
Construct WorldCoordinate Scene
From ModelingCoordinate
Transformations
World
Coordinates
Convert WorldCoordinates to
ViewingCoordinates
Viewing Coordinates
Transform ViewingCoordinates to
NormalizedCoordinates
May 2010
Normalized
Coordinates
Device
Map NormalizedCoordinates
Coordinates to
Device-Coordinates
3
Normalization and Viewport Transformations
• First approach:
– Normalization and window-to-viewport transformations are
combined into one operation.
– Viewport range can be in [0,1] x [0,1].
– Clipping takes place in [0,1] x [0,1].
– Viewport is then mapped to display device.
• Second approach:
– Normalization and clipping take place before viewport
transformation.
– Viewport coordinates are specified in screen coordinates.
May 2010
4
1
Clipping Window
Normalized Viewport
ywmax
yvmax
 xw, yw
 xv, yv 
yvmin
ywmin
xwmin
xwmax
0
xvmin
1
xvmax
Maintain relative size and position between clipping window and viewport.
xv  xvmin
xw  xwmin

xvmax  xvmin xwmax  xwmin
May 2010
yv  yvmin
yw  ywmin

yvmax  yvmin ywmax  ywmin
5
Solving for  xv, yv  obtains:
xv  sx xw  t x , yv  s y yw  t y , where
Scaling factors:
xvmax  xvmin
sx 
,
xwmax  xwmin
yvmax  yvmin
sy 
ywmax  ywmin
Translation factors:
xwmax xvmin  xwmin xvmax
tx 
,
xwmax  xwmin
ywmax yvmin  ywmin yvmax
ty 
ywmax  ywmin
This can also be obtained by composing transformations:
M window,

norm_viewport
 sx
T  xvmin , yvmin   S  sx , s y   T   xwmin ,  ywmin    0
 0
May 2010
0
sy
0
tx 
t y 
1 
6
World clipping window can first be mapped to normalized square between -1
and +1, where clipping algorithm takes place, and then transform the scene
into viewport given in display coordinates.
M window,
norm_square
2

 xw  xw
min
 max


0


0


 xvmax  xvmin  2

M norm_square,  
0
viewport

0
May 2010
xwmax  xwmin 

xwmax  xwmin 

yw  ywmin 
 max
ywmax  ywmin 

1


0
2
ywmax  ywmin
0
0
 yvmax  yvmin 
0
2
 xvmax  xvmin 
 yvmax  yvmin 
1
2

2

7
Clipping Algorithms
p9
Before Clipping
p4
After Clipping
p10
p2
p1
p6
p3 p5
p7
p2
p1
p8
p6
p 5
p 8
p 7
Parametric equations of line segment from  x0 , y0  to  xend , yend 
x  x0  u  xend  x0  , y  y0  u  yend  y0  , 0  u  1.
Used to determine the parts contained in clipping window.
May 2010
8
Cohen-Sutherland Line Clipping Algorithm
• Intersection calculations are expensive. Find first lines
completely inside or certainly outside clipping window.
Apply intersection only to undecided lines.
• Perform cheaper tests before proceeding to expensive
intersection calculations.
May 2010
9
Cohen-Sutherland Line Clipping Algorithm
• Assign code to every endpoint of line segment.
– Borderlines of clipping window divide the plane into two halves.
– A point can be characterized by a 4-bit code according to its
location in half planes.
– Location bit is 0 if the point is in the positive half plane, 1
otherwise.
– Code assignment involves comparisons or subtractions.
• Completely inside / certainly outside tests involve only
logic operations of bits.
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10
Top bit
Bottom bit
Right bit
xwmax  x
x  xwmin
1
0
0
1
Left bit
1
0
0
0
1
0
1
0
ywmax  y
0
0
0
1
0
0
0
0
0
0
1
0
y  ywmin
0
1
0
1
0
1
0
0
0
1
1
0
Endpoint codes are 0000 for both iff line is completely inside.
If endpoint codes has 1 in same bit, line is certainly outside.
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11
Lines that cannot be decided are intersected with window
border lines.
Each test clips the line and the remaining is tested again
for full inclusion or certain exclusion, until remaining is
either empty or fully contained.
Endpoints of lines are examined against left, right, bottom
and top borders (can be any order).
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12
p2
1
0
0
0
0
0
1
1
p 2
0
0
0
0
0
0
p 2
1
0
p3
p1
p 3
0
May 2010
1
0
1
p4
0
1
0
0
p1
13
Liang-Barsky Line Clipping Algorithm
Treat undecided lines in Cohen-Sutherland more efficiently.
Define clipping window by intersections of four half-planes.
xwmin

xwmax
 xend , yend 
ywmax


 x0 , y0 
ywmin

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14
Parametric presentation:
x  x0  u  xend  x0  , y  y0  u  yend  y0  , 0  u  1.
A point on the line is cotained in the clipping window iff:
xwmin  x0  u  xend  x0   xwmax ,
ywmin  y0  u  yend  y0   ywmax .
It can be expressed by: upk  qk , k  1, 2,3, 4, where
p1  x0  xend , q1  x0  xwmin ;
p2  xend  x0 , q2  xwmax  x0 .
p3  y0  yend , q3  y0  ywmin ;
p4  yend  y0 , q4  ywmax  y0 .
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15
In the inequality upk  qk if pk  0 (pk  0), the
traversal from  x0 , y0  to  xend , yend  by increasing
u from   to + proceeds the line from the  ( )
half-plane to  ( ) one (with respect to the k -th
border).
Intersection of  ,   extension with k -th border
occurs at u  qk pk .
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16
We calculate and update u0 and uend progressively for
k  1, 2,3, 4 borders (left, right, bottom, top).
If pk  0 u0 is calculated since progression is from 
to  half planes. Similarly, if pk  0 uend is calculated.
u0 is the maximum among 0 and all qk pk . uend is the
minimum among 1 and all qk pk . The feasibility
condition uend  u0 is progressively checked. The line
is completely outside if uend  u0 .
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17
Notice that qk pk doesn't need actual division since
comparison of q p with q p can be done by
comparison of qp with qp, and the quotient q p
can be stored as a pair  q, p 
Only if uend  u0 , given by  q0 , p0  and  qend , pend  ,
the actual ends of the clipped portion are calculated.
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18
This is more efficient than Cohen-Sutherland Alg,
which computes intersection with clipping window
borders for each undecided line, as a part of the
feasibility tests.
May 2010
19
Nicholl-Lee-Nicholl Line Clipping Algorithm
• Creates more regions around clipping window to avoid
multiple line intersection calculations.
• Performs fewer comparisons and divisions than
Cohen-Sutherland and Liang-Barsky, but cannot be
extended to 3D, while they can.
• For complete inclusion in clipping window or certain
exclusion we’ll use Cohen-Sutherland.
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20
P0
P0
P0
Examine first where the starting point P0 is located.
Only three regionsare considered. Location in any
of the other six regions can be handled by symmety
transformation.
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21
The location of Pend in each
T
R
P0
B
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L
region defines what edge the
line  P0 , Pend  is intersecting.
22
LT
Detecting whether Pend is in
L
P0
L
LR
L
any of regions L is immediate.
LB
Else, Pend is detected for being positioned in any of
LB, LR or LT, case where  P0 , Pend  is clipped with
left border and bottom, right or top border, resp.
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23
Pend
The slope of  P0 , Pend  is
LT
compared to  P0 , Pcorner 
L
L
P0
LR
L
LB
for each corner to find the
region of Pend .
Pend
Once one of LT, LR or LB regions is found,
intersection point with appropriate border is calculated.
 P0 , Pend  is entierely clipped if Pend is positioned
outside the regions.
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24
P0
P0
T
TR
L
L
LB
T
T
TR
L
LB
LR
TB
There are two cases, depending on whether P0 is
closer to left or top borders.
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25
P0
P0
T
TR
L
L
LB
T
T
TR
L
LB
LR
TB
Notice that simple inclusion test of Pend in
clipping rectangle is not enough since there
are both T and L labels for the regions inside.
Testing of the angular regions is a must.
May 2010
26
Sutherland-Hodgman Polygon Clipping
1
1’’
1’’
1’
3’’
Clipping
1’
3’’
3
Clipping Window
3’
2’
2’’
3’
2’
2’’
2
Efficient algorithm for clipping convex polygons.
Edges are clipped against every border line of clipping window. Edges
are processed successively.
Allows pipelining of edge clipping of polygons, as well as pipelining of
different polygons.
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27
The four possible outputs generated by the left clipper, depending on
the relative position of pair of edge endpoints.
out
in
in
output: v1 v 2
output: v 2
v1
in
output:
v1
v2
v1
v1
v2
v2
in
v1
out
out
output: none
v1
out
v1
v2
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28
2
2’
2”
3
1’
3’
1
Input
Left Clipper
Right Clipper
Bottom Clipper
Top Clipper
[1,2]:
(in-in)>{2}
[2,3]:
(in-out)>{2’}
[2,2’]:(in-in)>{2’}
[3,1]:
(out-in)>{3’,1}
[2’,3’]:(in-in)>{3’}
[2’,3’]:(in-out)>{2”}
[3’,1]:(in-in)>{1}
[3’,1]:(out-out)>{}
[1,2]:(in-in)>{2}
[1,2]:(out-in)>{1’,2}
[2”,1’]:(in-in)>1’}
[2,2’]:(in-in)>{2’}
[1’,2]:(in-in)>{2}
[2,2’]:(in-in)>{2’}
[2’,2”]:(in-in)>{2”}
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29
• The four clippers can work in parallel.
– Once a pair of endpoints it output by the first clipper,
the second clipper can start working.
– The more edges in a polygon, the more effective
parallelism is.
• Processing of a new polygon can start once first
clipper finished processing.
– No need to wait for polygon completion.
May 2010
30
Convex Polygon Intersection
P
P∩Q
Q
Theorem: The intersection of an L-vertex convex polygon and an M-vertex
convex polygon is a convex polygon of L+M vertices at most.
Proof: P∩Q is an intersection of L+M interior half planes determined by the
two polygons.
Intersection of convex polygons can answer the question of whether two
sets of points can be separated by a line.
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31
Theorem: The intersection of an L-vertex convex polygon and an M-vertex
convex polygon can be found in θ(L+M) time.
Proof: Polygons are given in cyclic order of vertices. We start from the
leftmost vertex of both polygons and progress along the border of both
in a left-to-right fashion, defining O(L+M) slabs.
Inside a slab each polygon forms a trapezoid. The intersection of two
trapezoids can be calculated in constant time.
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32
Another solution: The non empty intersection of two convex polygons
forms a sequence of “sickles” enclosing the intersection. The border of a
sickle comprises internal and external sequence of vertices originating
from P and Q, which are alternating in every sickle.
sickle
R=P∩Q
P
Q
Let P: (p1,p2,…,pL) and Q: (q1,q2,…,qM) be counterclockwise cyclically
ordered. The algorithm advances such that ∂P and ∂Q are “chasing” one
another, adjusting their speeds so that they meet at every intersection.
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33
Let pi and q j be the vertices where the traversal is. pi 1 pi
and q j 1q j are the current edges, defining the half planes
h  pi  and h  q j  , containing P and Q, respectively.
Clearly R  h  pi  h  q j  .
There are four possible situations of
qj
pi and q j with respect to R. For each
an advancing rule is in order. The
pi
idea is to progress along boundary of
the "lagging" polygon.
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34
Advance is from pi since there's no
qj
(a)
chance for a future intersection point
in the edge pi 1 pi . Edge q j 1q j may
pi
have a future intersection with the
boundary of P.
Edge pi 1 pi may still be intersected
pi
(b)
qj
later by Q, while edge q j 1q j have
already exahusted its intersections.
Hence advance is from q j .
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35
qj
Edge q j 1q j may still be intersected
pi
(c)
later by P, while edge pi 1 pi have
already exahusted its intersections.
Hence advance is from pi .
Arbitrary choice. Advancing from
(d)
pi
May 2010
qj
q j yields case (a), while advancing
from pi yields case (b).
36
void CONVEX_POLYGON_INTERSECT (polygon P, polygon Q) {
i  1; j  1; k  1; // initialization
while ( k  2  L  M  ) {
if ( pi 1 pi and q j 1q j intersect) { report intersection }
ADVANCE; // apply one of (a), (b), (c) or (d) cases
 k;
}
if (no intersection reported) {
if ( pi  Q ) { P  Q }
elseif ( q j  P ) { Q  P }
else { Q
P }
}
}
May 2010
37
May 2010
38
Correcness of algorithm : If pi and q j belong to the
same sickle, the intersection point terminating the sickle
must be found since algorithm never ADVANCE along
the polygon whose current edge may contain a sought
intersection point.
This in turn guarantees that once an intersection point
of a sickle is found, all the others will be constructed
successively.
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39
To complete the proof we must show that an
intersection point must be found if P Q  .
Let the current edge pi 1 pi on P, contain an
intersection point v with edge qr 1qr on Q, so
v  h  pi  .
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40
pi
qr
qr 1
v
pi 1
Cs
Cr
qs
farthest vertex
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41
Let q j 1q j be current edge and q j  Cr . Sequence of
cases (a), (c), (d) and (b) (possibly empty) occurs, while
pi 1 pi stays stationary,
pi
v qr 1
until v is found.
qr
case (b)
pi 1
Cs
case (a)
case (d)
Cr
case (c)
qs
farthest vertex
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42
Let q j  Cs , l be a line determined by q j 1q j and
l  l supporting P at pm , the first
support reached when traversing
l
l
from pi along P.
qj
q j 1
A sequence of cases (a)
starts at pi and will stay
pm
so until crossing l ,
where case (c)
holds up to pm (may be
pi 1
qr
qr 1
empty). q j 1q j stays stationary.
May 2010
pi
43
 
If pm  h q j
ADVANCE
continues marching
l
along P with cases (d)
qj
l
q j 1
and (b) until first
 
point pt  h q j
pm
is found.
pi 1
qr
qr 1
pi
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44
If an edge like pi 1 pi exists, L  M steps must reach
it since the boundary of at least one polygon must be
consumed.
Additional L  M ADVANCE steps suffice to
obtain all the intersection points in cyclic order,
yielding 2  L  M  total steps of ADVANCE.
If no intersection was found then the relations
P  Q, Q  P or P Q   can be resolved
in O  L  M  time.
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45
3D Viewing Concepts
World Coordinate System
May 2010
Viewing Coordinate System
46
2D Reminder
Choose viewing position, direction and orientation of the
camera in the world.
A clipping window is defined by the size of the aperture
and the lens.
Viewing by computing offers many more options which
camera cannot, e.g., parallel or perspective projections,
hiding parts of the scene, viewing behind obstacles, etc.
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47
Clipping window: Selects what we want to see.
Viewport: Indicates where it is to be viewed on the output
device (still in world coordinates).
Display window: Setting into screen coordinates.
In 3D the clipping is displayed on the view plane, but
clipping of the scene takes place in the space by a clipping
volume.
3D transformation pipeline is similar to 2D with addition of
projection transformation.
May 2010
48
3D Viewing Transformation Pipeline
Modeling
Coordinates
World
Coordinates
Construct WorldCoordinate Scene
From ModelingCoordinate
Transformations
Viewing
Coordinates
Convert WorldCoordinates to
ViewingCoordinates
Projection
Transformation
Projection Coordinates
Transform ProjectionCoordinates to
NormalizedCoordinates
May 2010
Normalized
Coordinates
Device
Map NormalizedCoordinates
Coordinates to
Device-Coordinates
49
Model is given in model (self) coordinates.
Conversion to world coordinates takes place.
Viewing coordinate system which defines the position and
orientation of the projection plane (film plane in camera) is
selected, to which scene is converted.
2D clipping window (lens of camera) is defined on the
projection plane (film plane) and a 3D clipping, called view
volume, is established.
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50
The shape and size of view volume is defined by the
dimensions of clipping window, the type of projection and
the limiting positions along the viewing direction.
Objects are mapped to normalized coordinated and all
parts of the scene out of the view volume are clipped off.
The clipping is applied after all device independent
transformation are completed, so efficient transformation
concatenation is possible.
Few other tasks such as hidden surface removal and
surface rendering take place along the pipeline.
May 2010
51
World to Viewing 3D Transformation
yworld
yview
view point
v
p0
xworld
zworld
May 2010
u
view-up vector
n
zview
xview
52
yview
n : viewing directionon
u  v : viewing plane
view point
v
u   ux , u y , uz 
p0
v   vx , v y , vz 
view-up vector
u
n
zview
n   nx , n y , nz 
xview
p 0   x0 , y0 , z0 
world to viewing
transformation
M WC,VC
u x
v
 x
 nx

0
uy
uz
vy
vz
ny
nz
0
0
rotation
May 2010
0  1
0  0

0 0
 
1  0
0 0  x0 
1 0  y0 
0 1  z0 

0 0 1 
translation
53
Projection Transformations
Next step in 3D viewing pipeline is projection of object to
viewing plane
Parallel Projection
Coordinate are transferred
to viewing plane along
parallel lines.
View Plane
Preserves relative size of object’s portions.
Projection can be perpendicular or oblique to viewing plane.
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Perspective Projection
Projection lines converge
in a point behind viewing
plane.
Doesn’t preserve relative size but looks more realistic.
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55
Orthogonal (orthographic) projections
Projection lines are parallel to normal.
Plane View
Front
Elevation
View
Side
Elevation
View
Used in engineering and architecture. Length and angles can be
measured directly from drawings.
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Clipping Window and View Volume
Orthogonal Projection
View Volume
View
Plane
Far
Clipping
Plane
Near
Clipping
Plane
yview
Clipping window
xview
zview
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57
Normalizing Orthogonal Projection
Orthogonal Projection View Volume
ynorm
 xwmax , ywmax , zwfar 
znorm
1,1,1
xnorm
 xwmin , ywmin , znear 
yview
zview
xview
 1, 1, 1
Normalized View Volume
Display coordinate system is usually left-handed.
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58
M ortho,norm 
2

 xw  xw
min
 max

0



0



0
0
0
2
ywmax  ywmin
0
0
2
znear  zfar
0
0
xwmax  xwmin 

xwmax  xwmin 

yw  ywmin 
 max
ywmax  ywmin 
znear  zfar 

znear  zfar 

1
The complete transformation from world coordinated
to normalized orthogonal-projection coordinates is
obtained by M ortho,norm  M WC,VC .
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59
Oblique Parallel Projections
Projection is defined by a viewing vector Vp .
x
p
, y p , zvp 
Projection lines are
L
parallel to vector Vp .
Vp
 x, y , z 
vp
 x, y, z 
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60
xp  x
zvp  z
yp  y
zvp  z


V px
x
p
, y p , zvp 
V pz
V py
L
Not orthogonal
as viewed
V pz
Vp
 x, y , z 
vp
 x, y, z 
x p  x   zvp  z V px V pz  , y p  y   zvp  z V py V pz 
The above is called shear transformation, where the
displacement in x and y linearly increases with z.
May 2010
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View Volume of Parallel Oblique Projection
View Plane
Clipping
Window
View
Volume
(Side)
Near
Plane
View
Volume
(Top)
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Vp
Vp
Far
Plane
62
M oblique

1


 0


0
0
0 
V px
1 
V py
0
1
0
0
V pz
V pz
V px 
zvp

V pz 
V py 

zvp
V pz 

0 
1 
This is a 3D shear
transformation.
x and y are displaced by
amount proportional to z.
Normalization oblique projection is similar to orthogonal projection.
The composite transformation is obtained by the product of the two.
Moblique,norm  Mortho,norm  Moblique
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Perspective Projections
Closer objects look larger.
p   x, y, z 
x
p
, y p , zvp 
View Plane
Projection
Reference
Point
Projection Point
yview
p prp   x prp , y prp , z prp 
xview
View Plane
zview
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Parametric representation of a point on the line
connecting P   x, y, z  with Pprp   x prp , y prp , z prp  :
x  x   x  x prp  u , y   y   y  y prp  u ,
z   z   z  z prp  u , 0  u  1.
At viewing plane: u   zvp  z 
z
prp
 z .
Substitution for the point on viewing plane:
x p  x  z prp  zvp 
prp
 z   x prp  zvp  z 
y p  y  z prp  zvp
prp
 z   y prp
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z
 z
z
z  z z
vp
prp
prp
 z
 z
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The problem with the above representation is that Z appears in
denominator, so matrix multiplication representation of X and Y
on view plane as a function of Z is not straight forward.
Z is point specific, hence division will be computation killer.
Different representation is in order, so transformations can
be concatenated
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Vanishing Points
Vanishing points occur when the viewing plane intersects with the
axes of viewing coordinate system.
Vanishing
Point
y
x
z
Principle axes for cube
One-Point perspective Projection
Parallel to Z lines of XZ plane and parallel lines to Z in YZ plane will
vanish. Vanishing point on viewing plane correspond to infinity in world.
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Viewing plane is parallel to y-axis,
z-axis vanishing point
intersecting both x-axis and z-axis
x-axis vanishing point
Vanishing points of all three axes occur when viewing plane
intersects all three axes.
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Transformation Matrix
To get rid of z in denominator we define parameter
h  z prp  z and new homogeneous coordinates
 xh ,yh , zh , h  .
x p  xh h , y p  yh h , yielding:
xh  x  z prp  zvp   x prp  zvp  z  ,
yh  y  z prp  zvp   y prp  zvp  z  .
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P   x, y, z,1 , Ph   xh , yh , zh , h  , Ph  Mpers  P,
Matrix Representation:
M pers
 z prp  zvp

0



0

0

0
 x prp
z prp  zvp
 y prp
0
sz
0
1
x prp z prp 
y prp z prp 
tz 

z prp 
zh  zsz  t z
zh  zsz  t z is using a scaling factor sz and translation t z .
sz and t z can be set arbitrarily. We'll set those to satisfy
desired normalization.
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Transformed z coordinate is very useful for deciding
later on the hidden parts of the scene.
Notice that drawing the perspective point requires a
division by h. h depends on z coordinate, hence evey
point must be divided by a different number, which is
very expensive!
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Perspective-Projection View Volume
View
Plane
Rectangular Frustum
View Volume
Clipping Window
Far
Clipping
Plane
Near
Clipping
Plane
yview

xview
Field-of-view Angle
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zview
Projection
Reference Point
72
Symmetric Frustum
View Volume
Parallel Piped
View Volume
Far Plane
Perspective
Mapping
Near Plane
View Plane
Clipping Window
Projection
Reference Point
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Oblique Frustum
View Volume
Parallel Piped
View Volume
Perspective
Mapping
xwmin
xwmax
Projection
Reference Point
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Oblique Perspective-Projection Frustum
We'll take the projection reference point to be the origin of viewing
coordinate system  x prp , y prp , z prp    0, 0, 0  and viewing plane at near
clipping plane. Oblique projection results shear transformation. It
transforms the intersection point of center-line with clipping window
 xwmin  xwmax ywmin  ywmax

,
,
z
near  to  0, 0, z near  .

2
2


M z shear
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1
0

0

0
0 sh zx
1 sh zy
0
1
0
0
0
0 
0

1
  xwmin  xwmax  2 
 0 


 0 
yw

yw
2


min
max


  M z shear  


 Z naer 
Z naer




1
 1 


75
xwmin  xwmax
ywmin  ywmax
which solves to sh zx  
and sh zy  
.
2
2
Substituting  x prp , y prp , z prp    0, 0, 0  in the perspective-projection
matrix and positioning the viewing near z clipping plane at the view
plane simplifies it to
M pers
  znear
 0

 0

 0
0
 znear
0
0
0
0 0 
sz t z 

1 0 
0
The z coordinate scaling factor sz and the translation t z will be
determined by the normalization requirements.
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The complete oblique perspective-projection is obtained by
concatenating the perspective and shear matrices.
M obliquepers  M pers  M z shear
 znear

0


 0

 0
 xwmin  xwmax 
 ywmin  ywmax 
0
 znear
0
sz
0
1
0

2 0
tz 

0
2
For symmetric viewing volume  xwmin   xwmax , ywmin   ywmax  .
Transformation is simplified to:
M symmetricpers  M pers  M z shear
May 2010
  znear
 0

 0

 0
0
 znear
0
0
0
0 0 
sz t z 

1 0 
0
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Normalized Perspective-Projection
Normalization is obtained by multiplying with standard scaling matrix.
M normpers  M xy scale  M obliquepers
 znear sx

0

 0

 0
0
 znear s y
sx  xwmin  xwmax  2
s y  ywmin  ywmax 
0
sz
0
1
Homogenous coordinates
are obtained as follows:
May 2010
0

2 0
tz 

0
 xh 
 x
y 
 y
 h   M normpers  
 zh 
z
 
 
h
1
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Projection coordinates are:
x p  xh h    znear sx x  sx  xwmin  xwmax  z 2   z ,
y p  yh h    znear s y y  s y  ywmin  ywmax  z 2   z ,
z p  zh h   s z z  t z   z .
We'd like normalization transform to result:
 xwmin , ywmin , znear    1, 1, 1 and  xwmax , ywmax , zfar   1,1,1 .
Substitution in above equations yields:
sx  2  xwmax  xwmin  , s y  2  ywmax  ywmin  ,
sz   znear  zfar   znear  zfar  , t z  2 znear zfar
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 znear  zfar  .
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Back substitution in the normalized perspective transformation yields:
M normpers
 2 znear
 xw  xw
min
 max

0



0



0
0
xwmin  xvmax
xwmax  xwmin
2 znear
ywmax  ywmin
ywmin  ywmax
ywmax  ywmin
0
znear  zfar
znear  zfar
0
1




0


2 znear zfar 


znear  zfar 

0
0
No division by z!
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3D Viewport Transformation
The normalized view volume cube extending from  1, 1, 1 to
1,1,1 is mapped to a screen viewport, extending from  xvmin , yvmin 
to  xvmax , yvmax  . z information is stored for depth calculations. z is
often renormalized to the range from 0 to 1.0, yielding:
M normviewvol,
3D screen
May 2010
 xvmax  xvmin

2


0



0


0

0
0
yvmax  yvmin
2
0
0
0
1
2
0
xvmax  xvmin 

2

yvmax  yvmin 

2

1

2


1

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Settings of Perspective Projection
• Perspective projection point
– Where the viewer (camera, eye) is positioned in the world.
• Positioning viewing plane with respect to viewing
coordinates
– Results vanishing points, one, two or three.
• Clipping window on viewing plane
– Defines the infinite pyramid view volume.
• Near and far clipping planes (parallel to view plane)
– Define the rectangular frustum view volume.
• Scale and translation parameters of perspective matrix
– Define the normalization range.
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3D Clipping
Clipping can take place on normalized cube:
xwmin  1, ywmin  1, zwmin  1, xwmax  1, ywmax  1, zwmax  1.
Similar to 2D, we add two bits to code the far and near planes.
Far bit
Near bit
Top bit
Bottom bit
Right bit
Left bit
Recall that in 3D point is represented in homogeneous form. Hence
P   xh , yh , zh , h  is inside the normalized cube iff
 h  xh  h,  h  yh  h,  h  zh  h if h  0
h  xh  h, h  yh  h, h  zh  h if h  0
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Top
y
Far
Near
Bottom
x
Left
z
Right
011001
011000
011010
001001
001000
001010
101001
101000
101010
010001
010000
010010
000001
000000
000010
100001
100000
100010
010101
010100
010110
000101
000100
000110
100101
100100
100110
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A line is completely accepted if the codes of its ends are both 000000, or
equivalently, if the logical OR between codes is zero.
A line is rejected if the codes of its ends has at least one 1 in same bit, or
equivalently, if the logical AND between codes is nonzero.
Otherwise, the line is tested against each of the planes and the 2D LiangBarsky algorithm can be extended to 3D.

A point P of a line segment P1P2 extending from P1  xh1 , yh1 , zh1 , h1



to P2  xh2 , yh2 , zh2 , h2 is given by P  P1   P2  P1  u, 0  u  1.
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If for instance the codes of the two end points of P1P2 w.r.t the right
clipping plane xmax  1 are different, the intersection point is derived


from x p  xh h   xh1  xh2  xh1 u   h1   h2  h1  u   1.



 

Solving for u yields u   xh1  h1   xh1  h1  xh2  h2  .


Such calculation proceeds for each upk  qk , 1  k  6, while
updating u0 and u1. If at some iteration u1  u0 , line segment is
completely clipped. If upon termination u1  u0 , the end points
of the clipped line segment are obtained.
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