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Rasterization Converts•^
lines and triangles
-^
with floating point vertices
-^
in viewport (screen) coordinates into•^
pixels
-^
with integer coordinates
-^
in viewport (screen) coordinates
pixels centered
at gridvertices, pixels centerednot grid cells
at gridvertices, not grid cells
Line Rasterization •^
Need to rasterize linesbetween any two clippedscreen points, from (
x^0
, y
to (
x^1
, y
•^
Only rasterize lines fromthe origin to a point in thefirst octant
-^
Translate start vertex toorigin
-^
Reflect end vertex into firstoctant
( x
, y 0
) 0
( x^1
, y 1
) ( x
-x 1
,^0 y^1
-y^0
)
( x
-x 0
, 1 y-y^0
) 1
Line Rasterization •^
How to rasterize a linefrom (0,0) to (4,3)
-^
Pixel (0,0) and (4,3) easy
-^
One pixel for each integerx-coordinate
-^
Pixel’s y-coordinateclosest to line
-^
If line equal distancebetween two pixels, pickon arbitrarily butconsistently
0
1
2
3
4
4 3 2 1 0
Midpoint Algorithm •^
Which pixel should beplotted next?– East?– Northeast?
NE E
Midpoint Algorithm •^
Which pixel should beplotted next?– East?– Northeast?
-^
Line equation
y^
mx
b
m
y^1
y
x^1
x
b^
y
0
–^
mx
0
f (
x ,
y ) =
mx
b
y
NE M E
f^ < 0
f^
Midpoint Algorithm •^
Which pixel should beplotted next?– East?– Northeast?
-^
Line equation
y^
mx
b
m
y^1
y
x^1
x
b^
=^
y^0
mx
0
f (
x ,
y ) =
mx
b
y
•^
f (
M
Æ
NE
•^
f (
M
)^
Æ
E
NE M E
Pixel Increments f (
x ,
y )
mx
b
y
M
= P
f (
M
)^
f (
x +1,
y +½)
m
( x
b
y +½)
mx
m
b
y
mx
b
y + m
f (
P
m
f (
E
)^
f (
x +1,
y )
f (
P
m
f (
NE
)^
f (
x +1,
y +1)
f (
P
m
f (0,0)
b
NE M E
P
Midpoint Increments f (
M
)^
f (
P
m
f (
M
) E
f (
x +2,
y +½)
m
( x
b
y +½)
f (
P
m
f (
M
m
f (
M
NE
f (
x +2,
y +1½)
m
( x
b
y +1½)
f (
P
m
f (
M
m
f (1, ½) =
m
b
NE M E
P
NE ENE EE
(^2) M
E M
NE
PolygonRasterization •^
Ignore horizontal lines
0
1
2
3
4
5
6
7
8
9
9 8 7 6 5 4 3 2 1 0
PolygonRasterization •^
Ignore horizontal lines
-^
Sort edges by smallery coordinate
0
1
2
3
4
5
6
7
8
9
9 8 7 6 5 4 3 2 1 0
A
B
C
D
E
F
G
PolygonRasterization •^
For each scanline…
-^
Add edges wherey = ymin
-^
Sorted by x
-^
Then by dx/dy
0
1
2
3
4
5
6
7
8
9
9 8 7 6 5 4 3 2 1 0
A
B
C
D
E
F
G
PolygonRasterization Plotting rules for when
segments lie on pixels
- Plot lefts2. Don’t plot rights3. Plot bottoms4. Don’t plot tops
0
1
2
3
4
5
6
7
8
9
9 8 7 6 5 4 3 2 1 0
A
B
C
D
E
F
G
PolygonRasterization •^
y = 1
-^
Delete y = ymax edges
-^
Update x
-^
Add y = ymin edges
-^
For each pair x
,x 0
, plot 1
from ceil(x
to ceil(x
0
1
2
3
4
5
6
7
8
9
9 8 7 6 5 4 3 2 1 0
A
B
C
D
E
F
G
PolygonRasterization •^
y = 2
-^
Delete y = ymax edges
-^
Update x
-^
Add y = ymin edges
-^
For each pair x
,x 0
, plot 1
from ceil(x
to ceil(x
0
1
2
3
4
5
6
7
8
9
9 8 7 6 5 4 3 2 1 0
A
B
C
D
E
F
G
PolygonRasterization •^
y = 3
-^
Delete y = ymax edges
-^
Update x
-^
Add y = ymin edges
-^
For each pair x
,x 0
, plot 1
from ceil(x
to ceil(x
0
1
2
3
4
5
6
7
8
9
9 8 7 6 5 4 3 2 1 0
A
B
C
D
E
F
G
PolygonRasterization •^
y = 4
-^
Delete y = ymax edges
-^
Update x
-^
Add y = ymin edges
-^
For each pair x
,x 0
, plot 1
from ceil(x
to ceil(x
0
1
2
3
4
5
6
7
8
9
9 8 7 6 5 4 3 2 1 0
A
B
C
D
E
F
G
PolygonRasterization •^
y = 9
-^
Delete y = ymax edges
-^
Update x
-^
Add y = ymin edges
-^
For each pair x
,x 0
, plot 1
from ceil(x
to ceil(x
0
1
2
3
4
5
6
7
8
9
9 8 7 6 5 4 3 2 1 0
A
B
C
D
F
G
E
Gouraud Interpolation •^
Flat shading– Per face normals– Color jumps across edge– Human visual perception
accentuates edges
•^
Smooth shading– Per vertex normals– Colors similar across edge– Edges become harder to
discern
Perception
Gouraud Interpolation •^
Keep track of R, G, B atedge endpoints
-^
Compute dR/dy, dG/dyand dB/dy per edge
-^
Compute dR/dx, dG/dxand dB/dx at eachscanline
-^
Color each pixelR += dR/dxG += dG/dxB += dB/dx
0
1
2
3
4
5
6
7
8
9
9 8 7 6 5 4 3 2 1 0
A
B
C
D
F
G
E
