程序代写代做代考 cse3431-lecture8
cse3431-lecture8
Derivation of the perspective
transformation
• It is basically a mapping of planes
• Normalized view volume is a left
handed system
• However, there is an easier
derivation
Text
Interpretation of the Perspective
transformation
Warps the view volume and
the objects in it
• Eye becomes a point
at infinity, and the projection
rays become parallel lines
(orthographic projection)
• We also want to keep z
Two step derivation
• Start with the Canonical Perspective Projection matrix
• Adjust CPP to scale z so that after the application of CPP
z = -near maps to z’=-near and z=-far maps to z’=-far)
• We now have the standard orthographic view volume
• Use the previously derived orthographic projection matrix MO
MP
MO
First: What are the view volume
points in viewing coordinates?
-n -f
t
tf
y
-z
From similar triangles:
t
�n
=
tf
�f
! tf = t
f
n
Similarly y = b, and for x = r and x = l
So for example: black point: (r, t,�n) ! Red point: (rf/n, tf/n,�f)
b
bf
First step: What does the CPP
produce?
• P’x = n Px/ Pz
• P’y = n Py/Pz
• P’z = -n
MP?
The four front( near) corners map as follows:
(left,bottom,-near) –> (left,bottom,-near)
(left,top,-near) –> (left,top,-near)
(right,bottom,-near) –> (right,bottom,-near)
(right,top,-near) –> (right,top,-near)
The four back (far) corners map as follows:
(l*f/n,b*f/n, -f) –> (l, b,-n) instead of (l,b,-f)
(l*f/n,t*f/n,-f) –> (l, t, -n) instead of (l,t,-f)
(r*f/n,b*f/n, -f) –> (r,b,-n) instead of (r,b,-f)
(r*f/n,t*f/n,-f) –> (r,t,-n) instead of (r,t,-f)
So we just have to adjust the matrix for z
First step:Adjust the matrix to
keep and scale Z
• We want to keep z and
• We want P’z = -n for Pz= -n and P’z = -f for Pz =-f after
the perspective division
• Where do we do the changes?
2
66
4
1 0 0 0
0 1 0 0
0 0 1 0
0 0 �1
n
0
3
77
5
2
66
4
n 0 0 0
0 n 0 0
0 0 n 0
0 0 �1 0
3
77
5
First step: With some intuition…
Reminder: n,f are positive
Sanity check
• (l*f/n,b*f/n,-f) –> (l,b,-f)
• (l,b,-n) –> (l,b,-n)
MP
2
66
4
Px
Py
Pz
1
3
77
5 =
2
66
4
Px
Py
Pz
n+f
n
+ f
�Pz
n
3
77
5 �!
Homogenize with h=�Pz/n
2
666
4
�Pxn
Pz
�Pyn
Pz
�(n+ f)� fn
Pz
1
3
777
5
Therefore
MP =
2
66
4
1 0 0 0
0 1 0 0
0 0 n+f
n
f
0 0 �1
n
0
3
77
5 or MP =
2
66
4
n 0 0 0
0 n 0 0
0 0 n+ f fn
0 0 �1 0
3
77
5 .
First step: MP
• Creates the previous orthographic view volume
• Which can then be transformed to NDCS with MO
MP
MO
Second Step: Combine with
Orthographic Projection Matrix
Now all we need to do is an orthographic
transformation
• We can use matrix Mo that transforms an orthographic
view volume to a normalized one (NDCS)
MO =
2
66
4
2
r�l 0 0 �
r+l
r�l
0 2
t�b 0 �
t+b
t�b
0 0 �2
f�n �
f+n
f�n
0 0 0 1
3
77
5
Second Step: Combine with
Orthographic Projection Matrix
MP
MO
Mprom = MOMP
OpenGL Perspective Matrix
Old form
• Still widely used
MOpenGL =
�
⇧⇧⇧
⇤
2|n|
r�l 0
r+l
r�l 0
0 2|n|
t�b
b+t
t�b 0
0 0 |n|+|f ||n|�|f |
2|f ||n|
|n|�|f |
0 0 �1 0
⇥
⌃⌃⌃
⌅
.
Projections in OpenGL
(mimicking the old way)
New way
projMat = frustum(-left, right, bottom, top, near, far);
projMat = ortho(-left, right, bottom, top, near, far) ;
projMat = Perspective(fov, aspect, near far) ;
near plane at z = -near
far plane at z = -far
Matrix Order
Normally projection has to apply to all
objects (i.e. the entire scene) thus it must
pre-multiply the modelview matrix
• M = MprojMmodelview or
• M = MprojMviewMmodel
Important
Projection parameters are given in CAMERA
Coordinate system (Viewing).
So if camera is at z = 50, is aligned with the
world CS, and you give near = 10 where is
the near plane with respect to the world?
Important
Projection parameters are given in CAMERA
Coordinate system (Viewing).
So if the camera is at z = 50, is aligned with
the world CS, and you give |near| = 10 where
is the near plane with respect to the world?
• Transformed by inverse(Mvcs)
• i.e. (0,0,40)
Perspective Division in Pipeline
The perspective division is done
automatically
• Typically the vertex shaders writes CCS in gl_Position.
Modeling
transformation
Viewing
transformation
Projection
transformation
Perspective
division
Viewport
transformation
OCS WCS VCS CCS
NDCSDCS
(e.g. pixels)
lookAt()translate()…
ModelView Matrix
Perspective Division in Pipeline
However, we can do it ourselves if we want.
The vertex shader has total freedom on how
to deal with projections
