Projection (Part 2): DerivationCreated by Dr. Slim BECHIKHfor SPSU course - CS4363 Computer Graphics and MultimediaFall 2014
Parallel Projection
normalization find 4x4 matrix to transform user!specified
view volume to canonical view volume (cube)
glOrtho(left, right, bottom, top,near, far)
Canonical
View Volume
User!specified
View Volume
Parallel Projection: Ortho
Parallel projection: 2 parts
1. Translation: centers view volume at origin
Parallel Projection: Ortho
2. Scaling: reduces user!selected cuboid to canonical
cube (dimension 2, centered at origin)
Parallel Projection: Ortho
Translation lines up midpoints: E.g.midpoint of x = (right + left)/2
Thus translation factors:
!(right + left)/2, !(top + bottom)/2, !(far+near)/2
Translation matrix:
!!!!!
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1000
2/)(100
2/)(010
2/)(001
nearfar
bottomtop
leftright
Parallel Projection: Ortho
Scaling factor: ratio of ortho view volume to cube dimensions
Scaling factors: 2/(right ! left), 2/(top ! bottom), 2/(far ! near)
Scaling Matrix M2:
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1000
02
00
002
0
0002
nearfar
bottomtop
leftright
Parallel Projection: Ortho
Concatenating Translation x Scaling, we get Ortho Projection matrix
X
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1000
02
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002
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0002
nearfar
bottomtop
leftright
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1000
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2/)(010
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nearfar
bottomtop
leftright
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*
+
,,,,,,,,
-
.
('
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((
((
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1000
200
02
0
002
nearfar
nearfar
farnear
bottomtop
bottomtop
bottomtop
leftright
leftright
leftright
P = ST =
Final Ortho Projection
Set z =0
Equivalent to the homogeneous coordinate
transformation
Hence, general orthogonal projection in 4D is
))))
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+
,,,,
-
.
1000
0000
0010
0001
Morth =
P = MorthST
Perspective Projection
Projection – map the object from 3D space to
2D screen
x
y
z
Perspective()Frustrum( )
Perspective Projection: Classical
(0,0,0)
-N
Projection plane
Eye (COP)
(x,y,z)
(x’,y’,z’)
-z
- z
y
Based on similar triangles:
y’ N y -z
Ny’ = y x
-z
=
VRP
COP
Object in 3 space
Projectors
Projected image
Near Plane(VOP)
+ z
Perspective Projection: Classical
So (x*,y*) projection of point, (x,y,z) unto near plane N is
given as:
Numerical example:
Q. Where on the viewplane does P = (1, 0.5, !1.5) lie for a
near plane at N = 1?
/ 0 !!"
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((1
z
Ny
z
Nxyx ,**,
/ 0 )333.0,666.0(5.1
15.0,
5.1
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((1
z
Ny
z
Nxyx
VRP
COP
Object in 3 space
Projectors
Projected image
Pseudodepth
Classical perspective projection projects (x,y) coordinates to
(x*, y*), drops z coordinates
But we need z to find closest object (depth testing)!!!
VRP
COP
Object in 3 space
Projectors
Projected image
(0,0,0)
z
Map to same (x*,y*)Compare their z values?
Perspective Transformation
Perspective transformationmaps actual z distance of
perspective view volume to range [ –1 to 1] (Pseudodepth)
for canonical view volume
-Near
-Far
-1 1
Canonical view volume
Actual view volume
Pseudodepth
Actual depthWe want perspectiveTransformation and NOT classical projection!!
Set scaling zPseudodepth = az + bNext solve for a and b
Perspective Transformation
We want to transform viewing frustum
volume into canonical view volume
(-1, -1, 1)
(1, 1, -1)
Canonical View Volume
x
y
z
Perspective Transformation using
Pseudodepth
Choose a, b so as z varies from Near to Far, pseudodepth
varies from –1 to 1 (canonical cube)
Boundary conditions
z* = !1 when z = !N
z* = 1 when z = !F
/ 0 !!"
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((1
z
baz
z
Ny
z
Nxzyx ,,**,*,
-Near -Far
Canonical view volume
Actual view volume
Pseudodepth
Actual depth
1 -1
Z*
Z
Transformation of z: Solve for a and b
Solving:
Use boundary conditions
z* = !1 when z = !N………(1)
z* = 1 when z = !F………..(2)
Set up simultaneous equations
z
bazz
('
1*
)1........(1 baNNN
baN'(1(
'(1(
)2........(1 baFFF
baF'(1
'(1
Transformation of z: Solve for a and b
Multiply both sides of (1) by !1
Add eqns (2) and (3)
Now put (4) back into (3)
)1........(baNN '(1(
)2........(baFF '(1
)3........(baNN (1
aFaNNF (1'
)4.........()(
NF
NF
FN
NFa
('(
1('
1
Transformation of z: Solve for a and b
Put solution for a back into eqn (3)
So
bNF
NFNN
!
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)3........(baNN "
NF
NFNNb
!
"#)(
NF
NF
NF
NNFNNF
NF
NFNNFNb
"
! "
!
"#2)()( 22
NF
NFa
!
")(
NF
FNb
"2
What does this mean?
Original point z in original view volume, transformed
into z* in canonical view volume
where
-Near -Far
Canonical view volume
Actual view volume
1 -1
Originalvertex z value
Transformedvertex z* value
z
bazz
!
"*
NF
NFa
!
")(
NF
FNb
"2
Homogenous Coordinates
Want to express projection transform as 4x4 matrix
Previously, homogeneous coordinates of
P = (Px,Py,Pz) => (Px,Py,Pz,1)
Introduce arbitrary scaling factor, w, so that
P = (wPx, wPy, wPz, w) (Note: w is non!zero)
For example, the point P = (2,4,6) can be expressed as
(2,4,6,1)
or (4,8,12,2) where w=2
or (6,12,18,3) where w = 3, or….
To convert from homogeneous back to ordinary coordinates,
first divide all four terms by w and discard 4th term
Perspective Projection Matrix
Recall Perspective Transform
We have:
In matrix form:
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z
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z
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z
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PerspectiveTransform Matrix
Originalvertex
TransformedVertex
Transformed Vertex after dividing by 4th term
Perspective Projection Matrix
In perspective transform matrix, already solved for a
and b:
So, we have transform matrix to transform z values
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w
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wP
wP
ba
N
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z
z
y
x
z
y
x
NF
NFa
!
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NF
FNb
"2
Perspective Projection
Not done yet!! Can now transform z!
Also need to transform the x = (left, right) and y = (bottom, top)
ranges of viewing frustum to [!1, 1]
Similar to glOrtho, we need to translate and scale previous matrix
along x and y to get final projection transform matrix
we translate by
–(right + left)/2 in x
!(top + bottom)/2 in y
Scale by:
2/(right – left) in x
2/(top – bottom) in y
1 -1
x
y
left rightbottom
top
Perspective Projection
Translate along x and y to line up center with origin of CVV –(right + left)/2 in x
!(top + bottom)/2 in y
Multiply by translation matrix:
1 -1
x
y
left right
bottom
top
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!
!
1000
0100
2/)(010
2/)(001
bottomtop
leftright
Line up centersAlong x and y
Perspective Projection
To bring view volume size down to size of of CVV, scale by
2/(right – left) in x
2/(top – bottom) in y
Multiply by scale matrix:
1 -1
x
y
left right
bottom
top
Scale size downalong x and y
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(
)
1000
0100
002
0
0002
bottomtop
leftright
Perspective Projection Matrix
glFrustum(left, right, bottom, top, N, F) N = near plane, F = far plane
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0100
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00minmax
2
NF
FN
NF
NF
bottomtop
bottomtop
bottomtop
N
leftright
leftright
xx
N
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0100
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000
1000
0100
2/)(010
2/)(001
1000
0100
002
0
0002
ba
N
N
bottomtop
leftright
bottomtop
leftright
Scale
Final PerspectiveTransform Matrix
Translate
PreviousPerspectiveTransformMatrix
Perspective Transformation
After perspective transformation, viewing
frustum volume is transformed into canonical
view volume
(-1, -1, 1)
(1, 1, -1)
Canonical View Volume
x
y
z
Geometric Nature of Perspective
Transform
a) Lines through eye map into lines parallel to z axis after transform
b) Lines perpendicular to z axis map to lines perp to z axis after transform
Normalization Transformation
original clippingvolume original object new clipping
volume
distorted object
projects correctly
Implementation
Set modelview and projection matrices in application program
Pass matrices to shader
void display( ){
.....
model_view = LookAt(eye, at, up);
projection = Ortho(left, right, bottom,top, near, far);
// pass model_view and projection matrices to shader
glUniformMatrix4fv(matrix_loc, 1, GL_TRUE, model_view);
glUniformMatrix4fv(projection_loc, 1, GL_TRUE, projection);
.....
}
Build 4x4 projection matrix
Implementation
And the corresponding shader
in vec4 vPosition;
in vec4 vColor;
Out vec4 color;
uniform mat4 model_view;
Uniform mat4 projection;
void main( )
{
gl_Position = projection*model_view*vPosition;
color = vColor;
}
References
Interactive Computer Graphics (6th edition), Angel and
Shreiner
Computer Graphics using OpenGL (3rd edition), Hill and Kelley