10 Canonical Screen Space

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Universidade Federal da Paraíba
Centro de Informática
Canonical and Screen 
Space
Lecture 10
1107190 - Introdução à Computação Gráfica – Turma 01
Prof. Christian Azambuja Pagot
CI / UFPB
Universidade Federal da Paraíba
Centro de Informática
How the space looks like after 
homogenization?
● After projection and division by w:
y
z
y
z Geometry was 
distorted by the 
homogenization.
However, is the 
new space a 
canonical 
space?
y
z
y
z
Universidade Federal da Paraíba
Centro de Informática
Canonical Space
● The concept of a canonical space involves a space that 
contains our scene and that is limited by unit 
coordinates:
z
y x
(1, 1, 1)
(-1, -1, -1)
How to fit a scene 
into this space?
Universidade Federal da Paraíba
Centro de Informática
Generating Canonical Spaces
● The canonical space should contain our scene. 
● Therefore, we have to define a bounding volume around our 
scene somehow in camera space. 
● That bounding volume will become the the canonical space.
● We will have to include additional transformations to both 
projection transforms:
– Orthogonal.
– Perspective.
Universidade Federal da Paraíba
Centro de Informática
Canonical Space for Orthogonal 
Projections
● As seen before, the orthogonal projection 
matrix is just the identity. 
● To generate a canonical space, we will bound 
our scene (a portion of camera space) with an 
axis-aligned bounding box.
● The whole camera space is translated and 
stretched such that the axis-aligned bounding 
box becomes a canonical cube centered at 
the origin.
Universidade Federal da Paraíba
Centro de Informática
Canonical Space for Orthogonal 
Projections
z
y
x
Camera
Space
Bounding box
z
y x
(1, 1, 1)
(-1, -1, -1)
Canonical
Space
l = left
b = bottom
n = near
r = right
t = top
f = far
(r, t, f)
(l, b, n)
Universidade Federal da Paraíba
Centro de Informática
Canonical Space for Orthogonal 
Projections
● 1st: Translate the bounding 
box to the origin:
● 2nd: Scale the bounding 
box:
d x=−
r+l
2
, d y=−
t+b
2
, d z=−
n+ f
2
sx=
2
r−l
, sx=
2
t−b
, s x=
2
n− f
p '=Mcanp Mcan=[
2
r−l
0 0 − r +l
r−l
0 2
t−b
0 − t+b
t−b
0 0 2
n− f
− n+ f
n− f
0 0 0 1
], onde
z
y
x
Camera
Space
Bounding box
l = left
b = bottom
n = near
r = right
t = top
f = far
(r, t, f)
(l, b, n)
Universidade Federal da Paraíba
Centro de Informática
Canonical Space for Perspective 
Projections
● It would be convenient to have a perspective 
transform capable to deliver vertices in 
clipping space such that, after 
homogenization, we could have everything 
within the canonical space.
Universidade Federal da Paraíba
Centro de Informática
Canonical Space for Perspective 
Projections
z
yp
p'
d
z
y
y'
c = camera position
p(z,y) = a point in cam. space.
p'(d,y') = projection of p onto the vp.
c
Mproj=[ 1 0 0 00 1 0 00 0 1 d0 0 −1
d
0]
Our previous Perspective 
Projection Setup
● Does not automatically 
generate a canonical 
volume after /W !
Universidade Federal da Paraíba
Centro de Informática
Canonical Space for Perspective 
Projections
z
y
n
f
y
n = distance of the near plane.
f = distance of the far plane.
c
The New Setup
● Scene is bounded along the 
z axis by a near and a far 
plane.
● The new projection matrix 
M
proj
 is shown below:
Mproj=[
1 0 0 0
0 1 0 0
0 0 −n+ f
n
f
0 0 −1
n
0 ]
q
p
Mproj=[ xyz1 ]=[
x
y
z n+ f
n
−f
z
n
]→[
nx
z
ny
z
n+ f− fn
z
1
]
Universidade Federal da Paraíba
Centro de Informática
Canonical Space for Perspective 
Projections
z
y
q
n
f
n = distance of the near plane.
f = distance of the far plane.
p
After Homogenization
● The view frustum becomes 
a rectangle (not yet in 
canonic space).
● Now we can use the same 
procedure used for the 
orthogonal projection in 
order to obtain the 
canonical volume. The 
transformed points must be 
multiplied by the matrix 
bellow: 
Mcan=[
2
r−l
0 0 − r+l
r−l
0 2
t−b
0 − t+b
t−b
0 0 2
n− f
− n+ f
n− f
0 0 0 1
]
Universidade Federal da Paraíba
Centro de Informática
From Canonical to Screen Space
z
y x
p1 = (1, 1, 1)
p2 = (-1, -1, -1)
x
y
Width = W pixels
Height = H pixels
Universidade Federal da Paraíba
Centro de Informática
From Canonical to Screen Space
[ xyz1 ][
1 0 0 0
0 −1 0 0
0 0 1 0
0 0 0 1][
w
2
0 0 0
0 h
2
0 0
0 0 1 0
0 0 0 1
][1 0 0
w−1
2
0 1 0 h−1
2
0 0 1 0
0 0 0 1
][ x 'y 'z '1 ]=
Point p in canonical 
space
Invert Y axis 
direction
Scale along X 
and Y acis.
Transtale.Point p' in 
Screen space.
y
c
x
c
(-1, -1)
(1, 1)
Canonical
Space
p
y
s
x
sScreen
Space
w = width
in pixels.
h = heigth 
in pixels.p'
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