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Projection Matrices
CS4395: Computer Graphics 1
Mohan SridharanBased on slides created by Edward Angel
Objectives
• Derive the projection matrices used for standard OpenGL projections.
• Introduce oblique projections.
• Introduce projection normalization.
CS4395: Computer Graphics 2
Projection Normalization
• Convert all projections to orthogonal projections with the default view volume.
• The same pipeline of operations for all types of projections
• Clip against simple cube regardless of type of projection.• Simplifies clipping because sides of the object are aligned
with coordinate axes.
• Delay final projection until end: important for hidden-surface removal.
CS4395: Computer Graphics 3
Projection Normalization
• Work in 4D using homogeneous coordinates.
• Convert all projections into orthogonal projections:– Distort objects such that the orthogonal projection of distorted
objects is the same as desired projection of original objects.
• Concatenation of normalization matrix (for distortion) and orthogonal projection matrix yields desired homogeneous coordinate matrix.
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Pipeline View
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modelviewtransformation
projectiontransformation
perspective division
clipping projection
nonsingular
4D 3D
against default cube 3D 2D
Stay in 4D coordinates for both the modelview and projection transformations (nonsingular transformations).
Orthogonal Normalization
glOrtho(left,right,bottom,top,near,far)
CS4395: Computer Graphics 7
Normalization find transformation to convert specified clipping volume to default.
Orthogonal Matrix
• Two steps:– Move center to origin:
T(-(left + right)/2, -(bottom + top)/2, (near + far)/2))
– Scale to have sides of length 2:S(2/(right-left), 2/(top-bottom), 2/(near-far))
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1000
200
02
0
002
nearfar
nearfar
farnear
bottomtop
bottomtop
bottomtop
leftright
leftright
leftright
P = ST =
Final Projection
• Set z =0
• Equivalent to the homogeneous coordinate transformation:
• General orthogonal projection in 4D:
CS4395: Computer Graphics 9
1000
0000
0010
0001
Morth =
P = MorthST
Oblique Projections
• The OpenGL projection functions cannot produce general parallel projections such as:
• However the cube it appears to have been sheared.• Oblique Projection = Shear + Orthogonal Projection!
CS4395: Computer Graphics 10
Shear Matrix
• For x-y shear (z values unchanged):
• Projection matrix:• General case: • See Section 5.8.
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1000
0100
0φcot10
0θcot01
H(,) =
P = Morth H(,)
P = Morth STH(,)
Effect on Clipping
• Projection matrix P = STH transforms the original clipping volume to the default clipping volume.
CS4395: Computer Graphics 14
top view
DOPDOP
near plane
far plane
object
clippingvolume
z = -1
z = 1
x = -1x = 1
distorted object(projects correctly)
Simple Perspective
• Consider a simple perspective : COP at origin, the near clipping plane at z = -1, and a 90o field of view determined by planes: x = z, y = z.
CS4395: Computer Graphics 15
Perspective Matrices
• Simple projection matrix in homogeneous coordinates:
• This matrix is independent of the far clipping plane.
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0100
0100
0010
0001
M =
Generalization (Section 5.9)
N =
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0100
βα00
0010
0001• Perspective normalization matrix:
• After perspective division, the point (x, y, z, 1) goes to:
• Orthogonal projection to the desired point regardless of
and .
)("
/"
/"
zz
zyy
zxx
Picking and
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=
=
nearfar
farnear
farnear
farnear2
• If we pick:
• The near plane is mapped to z = -1• The far plane is mapped to z =1• The sides are mapped to x = 1, y = 1• The new clipping volume is the default volume.
Normalization Transformation
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original object new clipping volume
distorted objectprojects correctly
original clipping volume
Normalization and Hidden-Surface Removal
• The perspective projection matrices are chosen such that if z1 > z2 in the original clipping volume then for the transformed points: z1’ > z2’
• Thus hidden surface removal works if the normalization transformation is applied first.
• However, the formula z’’ = -(+ /z) implies that the distances are distorted by the normalization which can cause numerical problems if the near distance is small.
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OpenGL Perspective
• Function glFrustum allows for an un-symmetric viewing frustum (although gluPerspective does not).
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OpenGL Perspective Matrix• The normalization in glFrustum requires an initial shear to
form a right viewing pyramid, followed by a scaling to get the normalized perspective volume. Finally, the perspective matrix needs only a orthogonal transformation:
• See Section 5.9.2.
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P = NSH
previously defined perspective matrix
shear and scale
N =
0100
βα00
0010
0001
Recap: Why do we do it this way?
• Normalization allows for a single pipeline for both perspective and orthogonal viewing.
• Stay in 4D homogeneous coordinates as long as possible to retain three-dimensional information needed for hidden-surface removal and shading.
• Simplifies clipping.
CS4395: Computer Graphics 23