CS559: Computer Graphics
Lecture 9: Rasterization
Li Zhang
Spring 2008
Today
•
•
•
•
Finish Projection
Rasterization
Antialising
Hidden Surface Removal
• Reading:
– Shirley ch 7,4,8
3D Geometry Pipeline
Model Space
(Object Space)
Eye Space
(View Space)
World Space
Rotation
Translation
Resizing
Rotation
Translation
1
0

0

0
0
0
1
0
0
1
0 1/ d
Projective,
Scale
Translation
0  x   ( d / z ) x 
0  y  (d / z ) y 

0  z   d 
  

0  1   1 
Clipping and the viewing frustum
The center of projection and the portion of the projection
plane that map to the final image form an infinite pyramid.
The sides of the pyramid are clipping planes.
Frequently, additional clipping planes are inserted to
restrict the range of depths. These clipping planes are
called the near and far or the hither and yon clipping
planes.
1
0

0

0
0
0
1
0
0
1
0 1/ d
0  x   ( d / z ) x 
0  y  (d / z ) y 

0  z   d 
  

0  1   1 
All of the clipping planes bound the viewing frustum.
View Volume Transformation
• We want a matrix that will take points in our
perspective view volume and transform them
into the orthographic view volume
– This matrix will go in our pipeline before an
orthographic projection matrix
(r,t,f)
(r,t,n)
(l,b,n)
(r,t,n)
(l,b,n)
(r,t,f)
Mapping Lines
y
-z
• We want to map all the lines
through the center of projection to
parallel lines
– This converts the perspective case to
the orthographic case, we can use all
our existing methods
• The relative intersection points of
lines with the near clip plane
should not change
• The matrix that does this looks like
the matrix for our simple
perspective case
n
f
How to get this mapping?
y
1
0

0

0
-z
0  x   ( n / z ) x 
1 0 0  y  (n / z ) y 

0 1 0  z   n 
  

0 1 / n 0  1   1 
0
1
0
M 
0

0
0
0
0
1
0
0
A
0 1/ n
x
 x 

 y 

y

M  
 n  ( An  B)
  

1
1
  

n
n
0
0 
B

0
 x   (n / f ) x 
 y   (n / f ) y 

M  
 f  (n / f )( Af  B)
  

1
1 

 f
f
f
A  1
n
Bf
Properties of this mapping
y
1
0

0

0
0
1
0
0
1
0
M 
0

0
0  x   ( n / z ) x 
0 0  y  (n / z ) y 

1 0  z   n 
  

1 / n 0  1   1 
0
0
0 
1
0
0 
0 (n  f ) / n  f 

0
1/ n
0 
-z
0
nx



 x'
 x 
z

 y '
 y 
ny

  M  
z

 z '
z 
 
   f  n  fn 
1
1  
z


1
n
f
If z = n, M(x,y,z,1) = [x,y,z,1]
near plane is unchanged
If x/z = c1, y/z = c2, then x’=n*c1, y’=n*c2
bending converging rays to parallel rays
If z1 < z2, z1’ < z2’
z ordering is preserved
In pyramid  In cube
General Perspective
1
0
MViewOrthoView  
0

0
0  n
1
0
0  0

0 n  f  n  f  0
 
0
1n
0  0
0
0
0
0
n
0
0 n f
0
1
0 
0 
 nf 

0 
To Canonical view volume
• Perspective projection transforms a pyramid
volume to an orthographic view volume
(r,t,f)
(r,t,f)
(r,t,n)
(r,t,n)
(l,b,n)
(l,b,n)
MViewOrthoView
2





r  l 
0
0
0
2 t  b 
0
0
2 n  f 
0
0
0
0
0 1
0 0

0  0

1  0
 r  l  2 
1 0  t  b  2 
0 1  n  f  2

0 0
1

Scale, translate
0 0
(-1,1,-1)
(1,-1,1)
Canonical view volume [-1,1]^3
Orthographic View to Canonical Matrix
 xcanonical  2
y
 
 canonical   
 zcanonical  

 
1

 
r  l 
2





r  l 
0
0
0
2 t  b 
0
0
2 n  f 
0
0
0
0
0
0
0
2 t  b 
0
0
2 n  f 
0
0
0
0
0 1
0 0
0  0

1  0
 r  l  2   xorthoView
1 0  t  b  2   yorthoView
0 1  n  f  2  zorthoView


0 0
1
1


0 0
 r  l  r  l    xorthoView
 t  b  t  b    yorthoView
 n  f  n  f   zorthoView


1
1


xcanonical  M orthoViewcanonicalxorthoView
Complete Perspective Projection
• After applying the perspective matrix, we map
the orthographic view volume to the canonical
view volume:
M view canonical  M orthoViewCanonicalM vieworthoView
 2
 r  l 

 0


 0

 0
0
2
t  b 
0
0
0
0
2
n  f 
0
 r  l  
r  l   n
 t  b   
0

t  b   0
 n  f   

n  f   0
1

0
0
n
0
0
n  f 
0
1
0 
0 
 nf 

0 
3D Geometry Pipeline
Rotation
Translation
Rotation
Translation
Resizing
Model Space
(Object Space)
M world view
Eye Space
(View Space)
World Space
M viewcanonical : Projective, Scale, Translation
Canonical View Space
Screen Space (2D)
Image Space (pixels)
Raster Space
3D Geometry Pipeline
Rotation
Translation
Rotation
Translation
Resizing
Model Space
(Object Space)
M world view
World Space
Eye Space
(View Space)
M world viewM viewcanonical
Canonical View Space
Screen Space (2D)
Image Space (pixels)
Raster Space
3D Geometry Pipeline
Rotation
Translation
Rotation
Translation
Resizing
Model Space
(Object Space)
M world view
World Space
Eye Space
(View Space)
M viewcanonicalM world view Order matters!!!
Canonical View Space
Screen Space (2D)
Image Space (pixels)
Raster Space
Canonical  Window Transform
(xmax, ymax)=(nx-0.5,ny-0.5)
ny-1
(1,1)
(-1,-1)
nx-1
(xmin,ymin)=(-0.5,-0.5)
 x pixel  xmax
y  
 pixel   
 z pixel  

 
1

 
 xmin  2
0
0
xmax  xmin  2   xcanonical 
 ymax  ymin  2  ycanonical 
0
 ymax  ymin  2
0
0
1
0
0
0
0
1
0
M canonical  pixel
  zcanonical 


1


3D Geometry Pipeline
Rotation
Translation
Resizing
Model Space
(Object Space)
Rotation
Translation
M world view
World Space
Eye Space
(View Space)
M viewcanonical
M canonical  pixel
Canonical View Space
Screen Space (2D)
Image Space (pixels)
Raster Space
Polygons are basic shape primitives
• Any 3D shape can be approximated by a polygon using a locally
linear (planar) approximation. To improve the quality of fit, we
need only increase the number of edges
Line Drawing
(x1, y1)
(x0, y0)
y
( x  x0 )  y0
x
x  x1  x0 , y  y1  y0
y
Line Drawing
for x = x0 to x1 do
y=y0+(∆y/∆x)(x-x0)
draw(x,round(y))
(x1, y1)
(x0, y0)
y
( x  x0 )  y0
x
x  x1  x0 , y  y1  y0
y
Line Drawing
y = y0
for x = x0 to x1 do
draw(x,y)
if (some condition) then
y=y+1
(x1, y1)
(x0, y0)
y
( x  x0 )  y0
x
x  x1  x0 , y  y1  y0
y
Implicit Line Drawing
y = y0
for x = x0 to x1 do
draw(x,y)
if (some
condition) then
f(x+1,y+0.5)<0
y=y+1
(x1, y1)
(x0, y0)
y
( x  x0 )  y0
x
x  x1  x0 , y  y1  y0
y
f ( x, y)  x  ( y  y0 )  y  ( x  x0 )
f ( x, y )  0  on the line
f ( x, y )  0  above the line
f ( x, y )  0  below the line
Implicit Line Drawing
y = y0
for x = x0 to x1 do
draw(x,y)
if (some
condition) then
f(x+1,y+0.5)<0
y=y+1
(x1, y1)
(x0, y0)
y
( x  x0 )  y0
x
x  x1  x0 , y  y1  y0
y
f ( x, y)  x  ( y  y0 )  y  ( x  x0 )
Incremental Evaluation
f ( x  1, y )  f ( x, y )  y
f ( x  1, y  1)  f ( x, y )  y  x
Implicit Line Drawing
d=f(x+1,y+0.5)
y = y0
for x = x0 to x1 do
draw(x,y)
if (some
condition) then
d<0
(x1, y1)
(x0, y0)
D=d-∆y+∆x
y=y+1
else
D=d-∆y
y
( x  x0 )  y0
x
x  x1  x0 , y  y1  y0
y
f ( x, y)  x  ( y  y0 )  y  ( x  x0 )
Incremental Evaluation
f ( x  1, y )  f ( x, y )  y
f ( x  1, y  1)  f ( x, y )  y  x
Implicit Line Drawing
d=2f(x+1,y+0.5)
y = y0
for x = x0 to x1 do
draw(x,y)
if (some
condition) then
d<0
(x1, y1)
(x0, y0)
D=d-2∆y+2∆x
y=y+1
else
D=d-2∆y
y
( x  x0 )  y0
x
x  x1  x0 , y  y1  y0
y
f ( x, y)  x  ( y  y0 )  y  ( x  x0 )
Incremental Evaluation
Integer operation
f ( x  1, y )  f ( x, y )  y
f ( x  1, y  1)  f ( x, y )  y  x
Implicit Line Drawing
d=2f(x+1,y+0.5)
y = y0
for x = x0 to x1 do
draw(x,y)
if (some
condition) then
d<0
(x1, y1)
(x0, y0)
D=d-2∆y+2∆x
y=y+1
else
D=d-2∆y
y
( x  x0 )  y0
x
x  x1  x0 , y  y1  y0
y
f ( x, y)  x  ( y  y0 )  y  ( x  x0 )
f ( x  1, y )  f ( x, y )  y
f ( x  1, y  1)  f ( x, y )  y  x
Implicit Line Drawing
d=2f(x+1,y+0.5)
y = y0
for x = x0 to x1 do
draw(x,y)
if (some
condition) then
d<0
D=d-2∆y+2∆x
y=y+1
else
D=d-2∆y
y
( x  x0 )  y0
x
x  x1  x0 , y  y1  y0
y
f ( x, y)  x  ( y  y0 )  y  ( x  x0 )
f ( x  1, y )  f ( x, y )  y
f ( x  1, y  1)  f ( x, y )  y  x
Implicit Line Drawing
y = y0
for x = x0 to x1 do
draw(x,y)
if (some condition) then
y=y+1
(x1, y1)
(x0, y0)
y
( x  x0 )  y0
x
x  x1  x0 , y  y1  y0
y
f ( x, y)  x  ( y  y0 )  y  ( x  x0 )
f ( x  1, y )  f ( x, y )  y
f ( x  1, y  1)  f ( x, y )  y  x
Implicit Line Drawing
y = y0
for x = x0 to x1 do
draw(x,y)
if (some condition) then
y=y+1
y
( x  x0 )  y0
x
x  x1  x0 , y  y1  y0
y
f ( x, y)  x  ( y  y0 )  y  ( x  x0 )
f ( x  1, y )  f ( x, y )  y
f ( x  1, y  1)  f ( x, y )  y  x
Implicit Line Drawing
d=2f(x+1,y+0.5)
y = y0
for x = x0 to x1 do
draw(x,y)
if (some
condition) then
d<0
(x1, y1)
(x0, y0)
D=d-2∆y+2∆x
y=y+1
else
D=d-2∆y
y
( x  x0 )  y0
x
x  x1  x0 , y  y1  y0
y
f ( x, y)  x  ( y  y0 )  y  ( x  x0 )
If ∆y>∆x, switch x and y
f ( x  1, y )  f ( x, y )  y
f ( x  1, y  1)  f ( x, y )  y  x
Drawing Triangle
Drawing Triangle
A2
for all y do
for all x do
if (in triangle) then
draw(x,y) with color c
P
A1
A3
 x3 
 xP 
 x1 
 x2 
 y   t1   y   t2   y   t3   y 
 P
 1
 2
 3
t1  t2  t3  1
t1 
area ( PA2 A3 )
area ( A1 A2 A3 )
t2 
area ( PA3 A1 )
area ( PA1 A2 )
t3 
area ( A1 A2 A3 )
area ( A1 A2 A3 )
A1 : ( x1 , y1 ), c1
A2 : ( x2 , y 2 ), c2
A3 : ( x3 , y3 ), c3
area ( PA2 A3 ) 
area ( PA2 A3 ) 
1
PA2  PA3
2
1
PA2  PA3 sin( A2 PA3 )
2
Drawing Triangle
for all y do
for all x do
compute t1,t2,t3
if t1,t2,t3 in [0,1] then
draw(x,y) with color
t1c1+t2c2+t3c3
 x3 
 xP 
 x1 
 x2 
 y   t1   y   t2   y   t3   y 
 P
 1
 2
 3
t1  t2  t3  1
t1 
area ( PA2 A3 )
area ( A1 A2 A3 )
t2 
area ( PA3 A1 )
area ( A1 A2 A3 )
t3 
area ( PA1 A2 )
area ( A1 A2 A3 )
A2
P
A1
A3
A1 : ( x1 , y1 ), c1
A2 : ( x2 , y 2 ), c2
A3 : ( x3 , y3 ), c3
area ( PA2 A3 ) 
area ( PA2 A3 ) 
1
PA2  PA3
2
1
PA2  PA3 sin( A2 PA3 )
2
Common Edge
for all y do
for all x do
compute t1,t2,t3
if t1,t2,t3 in [0,1] then
draw(x,y) with color
t1c1+t2c2+t3c3
 x3 
 xP 
 x1 
 x2 
 y   t1   y   t2   y   t3   y 
 P
 1
 2
 3
t1  t2  t3  1
t1 
area ( PA2 A3 )
area ( A1 A2 A3 )
t2 
area ( PA3 A1 )
area ( A1 A2 A3 )
t3 
area ( PA1 A2 )
area ( A1 A2 A3 )
A1 : ( x1 , y1 ), c1
A2 : ( x2 , y 2 ), c2
A3 : ( x3 , y3 ), c3
area ( PA2 A3 ) 
area ( PA2 A3 ) 
1
PA2  PA3
2
1
PA2  PA3 sin( A2 PA3 )
2