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.

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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.

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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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Pictorial Representation

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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)

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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:

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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!

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General Shear

side view

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top view

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(,)

Equivalency

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Effect on Clipping

• Projection matrix P = STH transforms the original clipping volume to the default clipping volume.

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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.

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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.

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