The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
Perspective projection (outline)
1
The graphics pipeline: part I
2
Windowing transforms
3
Perspective transform
4
Camera transformation
5
Wrap-up
Graphics, 1st period 2006/2007
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
Projecting from arbitrary camera positions
Camera transformation
Orthographic projection and the canonical view volume
Windowing transform
The graphics pipeline: part I
Given an arbitrary camera
position, we want to display
the objects in the model in an
image
The projection should be
perspective projection.
Graphics, 1st period 2006/2007
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
Projecting from arbitrary camera positions
Camera transformation
Orthographic projection and the canonical view volume
Windowing transform
Camera transformation
We simplify by moving the
camera viewpoint to the
origin, such that we look into
the direction of the negative
z-axis.
−Z
Graphics, 1st period 2006/2007
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
Projecting from arbitrary camera positions
Camera transformation
Orthographic projection and the canonical view volume
Windowing transform
Orthographic projection
Orthographic projection is a
lot simpler than perspective
projection, so we transform the
clipped view frustum to an
axis-parallel box
−Z
Graphics, 1st period 2006/2007
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
Projecting from arbitrary camera positions
Camera transformation
Orthographic projection and the canonical view volume
Windowing transform
The canonical view volume
To simplify our calculations,
we transform to the canonical
view volume
(1, 1)
(−1, −1)
Graphics, 1st period 2006/2007
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
Projecting from arbitrary camera positions
Camera transformation
Orthographic projection and the canonical view volume
Windowing transform
Windowing transform
We apply a windowing
transform to display the square
[−1, 1] × [−1, 1] onto an
m × n image.
Graphics, 1st period 2006/2007
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
Projecting from arbitrary camera positions
Camera transformation
Orthographic projection and the canonical view volume
Windowing transform
The graphics pipeline: part I
Every step in the sequence can
be represented by a matrix
operation, so the whole process
can be applied by performing a
single matrix operation!
(Well: almost.)
Graphics, 1st period 2006/2007
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
The canonical view volume
The orthographic view volume
The orthographic projection matrix
The canonical view volume
The canonical view volume is a
2 × 2 × 2 box, centered at the
origin.
Vertices in the canonical view
volume are orthographically
projected onto an m × n
image.
Graphics, 1st period 2006/2007
(1, 1, −1)
y
The clipped view frustum is
transformed to this box (and
the objects within the view
frustum undergo the same
transformation).
−z
x
(1, −1, −1)
(−1, −1, 1)
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
The canonical view volume
The orthographic view volume
The orthographic projection matrix
Mapping the canonical view volume
(1, 1)
We need to map the square
[−1, 1]2 onto a rectangle
[− 12 , m − 21 ] × [− 12 , n − 12 ].
(−1, −1)
The following matrix takes
care of that:
m

1
0 m
2
2 − 2
0 n n − 1
2
2
2
0 0
1
n
m
Graphics, 1st period 2006/2007
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
The canonical view volume
The orthographic view volume
The orthographic projection matrix
The orthographic view volume
(r, t, f )
The orthographic view volume
is an axis-aligned box
[l, r] × [b, t] × [n, f ].
(l, b, n)
Transforming it to the
canonical view volume is done
by

2
r−l
 0

 0
0
0
2
t−b
0
0
0
0
2
n−f
0

0
1 0 0
0 1 0
0

0 0 0 1
0 0 0
1
(1, 1, −1)
y

− l+r
2

− b+t
2 
n+f 
− 2
1
−z
x
(1, −1, −1)
(−1, −1, 1)
Graphics, 1st period 2006/2007
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
The canonical view volume
The orthographic view volume
The orthographic projection matrix
The orthographic projection matrix
We can combine the matrices into one:
M
 om=
2
0
0

0
0
0
0
n
2
m
2
n
2
0
0
1
0
 2
− 21
r−l
 0
− 21 

0  0
1
0
0
2
t−b
0
0
0
0
2
n−f
0

0
1 0 0
0 1 0
0

0 0 0 1
0 0 0
1

− l+r
2

− b+t
2 
n+f 
− 2
1
Given a point p in the orthographic view volume, we map it to a
pixel [i, j] as follows:


xp

 yp 



 
zcanonical  = Mo  zp 
1
1
i
j


Graphics, 1st period 2006/2007
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
Transforming the view frustum
Perspective transform matrix
Homogeneous coordinates
Transforming the view frustum
We have to transform the view
frustum into the orthographic view
volume. The transformation needs to
Map lines through the origin to
lines parallel to the z axis
Map points on the viewing
plane to themselves.
Map points on the far plane to
(other) points on the far plane.
Preserve the near-to-far order of
points on a line.
Graphics, 1st period 2006/2007
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
Transforming the view frustum
Perspective transform matrix
Homogeneous coordinates
Transforming the view frustum
(cf. book, fig. 7.12)
Graphics, 1st period 2006/2007
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
Transforming the view frustum
Perspective transform matrix
Homogeneous coordinates
Transforming the view frustum
(cf. book, fig. 7.10)
Graphics, 1st period 2006/2007
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
Transforming the view frustum
Perspective transform matrix
Homogeneous coordinates
Perspective transform matrix
The following matrix does the trick:

1
0
Mp = 
0
0
0
1
0
0
0
0
n+f
n
1
n

0
0 

−f 
0
  

x
x
y  

 =  n+fy

We have Mp 
 z  z

−
f
n
z
1
n
Hmmmm. . . is that what we
want. . . ?
Graphics, 1st period 2006/2007
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
Transforming the view frustum
Perspective transform matrix
Homogeneous coordinates
Homogeneous coordinates
We have seen homogeneous coordinates before, but so far, the
fourth coordinate of a 3D point has alway been 1.
In general, however, the homogeneous representation of a point
(x, y, z) in 3D is (hx, hy, hz, h). Choosing h = 1 has just been a
convenience.
So we
 


 have:
nx
x
x
z
ny
y  


y
z
 


Mp 
 z  = z n+f − f  homogenize
−−−−−−−−→ n + f −
n
z
1
1
n
Graphics, 1st period 2006/2007



fn 
z
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
Transforming the view frustum
Perspective transform matrix
Homogeneous coordinates
Transforming the view frustum
(cf. book, fig. 7.9)
Graphics, 1st period 2006/2007
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
Transforming the view frustum
Perspective transform matrix
Homogeneous coordinates
Homogeneous coordinates
Note: moving and scaling still work properly
For example: translation

1
0

0
0
0
1
0
0



  
x + tx
hx
hx + htx
0 tx

  


0 ty 
 hy  =  hy + hty  homogenize  y + ty 






hz
hz + htz −−−−−−−−→ z + tz 
1 tz
1
0 1
h
h
Graphics, 1st period 2006/2007
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
Aligning coordinate systems
Transformation matrix
Aligning coordinate systems
Given a camera coordinate system
with origin e and orthonormal base
(u, v, w), we need to tranform
objects in world space coordinates
into objects in camera coordinates.
This can alternatively be considered
as aligning the canonical coordinate
system with the camera coordinate
system.
Graphics, 1st period 2006/2007
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
Aligning coordinate systems
Transformation matrix
Camera transformation
The required transformation is taken
care
 of by Mv =  

xu
 xv

xw
0
yu
yv
yw
0
zu
zv
zw
0
0
1 0 0 −xe
0 1 0 −ye 
0


0 0 0 1 −ze 
1
0 0 0
1
Graphics, 1st period 2006/2007
Lecture 6: Orthographic and perspective projection
The graphics pipeline: part I
Windowing transforms
Perspective transform
Camera transformation
Wrap-up
Wrap-up
If we combine all steps, we get:
compute Mo
compute Mv
compute Mp
M = Mo Mp Mv
for each line segment (ai , bi ) do
p = Mai
q = Mbi
drawline(xp /hp , yp /hp , xq /hq , yq /hq )
Graphics, 1st period 2006/2007
Lecture 6: Orthographic and perspective projection