computer graphics (cs 543) 5 (part projection (part...
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Computer Graphics (CS 543) Lecture 5 (part 2): Projection (Part 2):
Derivation
Prof Emmanuel Agu
Computer Science Dept.Worcester Polytechnic Institute (WPI)
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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‐specifiedView Volume
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Parallel Projection: Ortho
Parallel projection: 2 parts1. Translation: centers view volume at origin
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Parallel Projection: Ortho
2. Scaling: reduces user‐selected cuboid to canonical cube (dimension 2, centered at origin)
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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:
10002/)(1002/)(010
2/)(001
nearfarbottomtop
leftright
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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:
1000
0200
0020
0002
nearfar
bottomtop
leftright
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Parallel Projection: Ortho
Concatenating Translation x Scaling, we get Ortho Projection matrix
X
1000
0200
0020
0002
nearfar
bottomtop
leftright
10002/)(1002/)(010
2/)(001
nearfarbottomtop
leftright
1000
200
020
002
nearfarnearfar
farnear
bottomtopbottomtop
bottomtop
leftrightleftright
leftright
P = ST =
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Final Ortho Projection
Set z =0 Equivalent to the homogeneous coordinate transformation
Hence, general orthogonal projection in 4D is
1000000000100001
Morth =
P = MorthST
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Perspective Projection
Projection – map the object from 3D space to 2D screen
x
y
z
Perspective()Frustrum( )
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Perspective Projection: Classical
(0,0,0)
-N
Projection plane
Eye (COP)
(x,y,z)
(x’,y’,z’)
-z
- z
yBased on similar triangles:
y’ N y -z
Ny’ = y x
-z
=
VRP
COP
Object in 3 space
Projectors
Projected image
Near Plane(VOP)
+ z
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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?
zNy
zNxyx ,**,
)333.0,666.0(5.1
15.0,5.1
11,**,
zNy
zNxyx
VRP
COP
Object in 3 space
Projectors
Projected image
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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?
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Perspective Transformation
Perspective transformationmaps actual z distance of perspective view volume to range [ –1 to 1] (Pseudodepth) for canonical view volume
NearFar
1 -1
Canonical view volume
Actual view volume
Pseudodepth
Actual depth We want perspectiveTransformation and NOT classical projection!!
Set scaling zPseudodepth = az + bNext solve for a and b
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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
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Perspective Transformation using Pseudodepth
Choose a, b so as z varies from Near to Far, pseudodepthvaries from –1 to 1 (canonical cube)
Boundary conditions z* = ‐1 when z = ‐N z* = 1 when z = ‐F
zbaz
zNy
zN
xzyx ,,**,*,
Near Far
Canonical view volume
Actual view volume
Pseudodepth
Actual depth
-1 1
Z*
Z
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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
zbazz
*
)1........(1 baNNN
baN
)2........(1 baFFF
baF
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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
)2........(baFF
)3........(baNN
aFaNNF
)4.........()(NFNF
FNNFa
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Transformation of z: Solve for a and b
Put solution for a back into eqn (3)
So
bNF
NFNN
)(
)3........(baNN
NFNFNNb
)(
NFNF
NFNNFNNF
NFNFNNFNb
2)()( 22
NFNFa
)(
NFFNb
2
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What does this mean?
Original point z in original view volume, transformed into z* in canonical view volume
whereNear Far
Canonical view volume
Actual view volume
-1 1
Original vertex z value
Transformed vertex z* value
zbazz
*
NFNFa
)(
NFFNb
2
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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
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Perspective Projection Matrix
Recall Perspective Transform
We have:
In matrix form:
1
)(0100
00000000
zbazz
Nyz
Nx
wzbazw
wNywNx
wwzwywx
baN
N
zbaz
zNy
zN
xzyx ,,**,*,
zNxx
*z
Nyy
*zbazz
*
Perspective Transform Matrix
Original vertex
TransformedVertex
Transformed Vertex after dividing by 4th term
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Perspective Projection Matrix
In perspective transform matrix, already solved for a and b:
So, we have transform matrix to transform z values
1
)(0100
00000000
zbazz
Nyz
Nx
wPbaPw
wNPwNP
wwPwPwP
baN
N
z
z
y
x
z
y
x
NFNFa
)(
NFFNb
2
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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
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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 rightbottom
top
10000100
2/)(0102/)(001
bottomtopleftright
Line up centersAlong x and y
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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 rightbottom
top
Scale size downalong x and y
10000100
0020
0002
bottomtop
leftright
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Perspective Projection Matrix
glFrustum(left, right, bottom, top, N, F) N = near plane, F = far plane
0100
2)(00
020
00minmax
2
NFFN
NFNF
bottomtopbottomtop
bottomtopN
leftrightleftright
xxN
010000
000000
10000100
2/)(0102/)(001
10000100
0020
0002
baN
Nbottomtop
leftright
bottomtop
leftright
Scale
Final PerspectiveTransform Matrix
Translate
Previous PerspectiveTransform Matrix
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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
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Geometric Nature of Perspective Transform
a) Lines through eye map into lines parallel to z axis after transformb) Lines perpendicular to z axis map to lines perp to z axis after transform
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Normalization Transformation
original clippingvolume original object new clipping
volume
distorted objectprojects correctly
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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 shaderglUniformMatrix4fv(matrix_loc, 1, GL_TRUE, model_view);glUniformMatrix4fv(projection_loc, 1, GL_TRUE, projection);
.....}
Build 4x4 projection matrix
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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;
}
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References Interactive Computer Graphics (6th edition), Angel and
Shreiner Computer Graphics using OpenGL (3rd edition), Hill and Kelley