AngelCG17 projection matrices - Computer Science | … Computer Graphics I, Fall 2008 4 Pipeline...

191.427 Computer Graphics I, Fall 2008

Projection Matrices


Objectives

•Derive projection matrices used forstandard OpenGL projections

•Introduce oblique projections

•Introduce projection normalization


Normalization

•Rather than derive different projectionmatrix for each type of projection

•can convert all projections toorthogonal projections with defaultview volume

•==> can use standard transformationsin pipeline

•==> efficient clipping


Pipeline View

modelviewtransformation

projectiontransformation

perspective division

clipping projection

nonsingular4D → 3D

against default cube 3D → 2D


Notes

•Stay in 4-d homogeneous coordinates throughboth modelview and projectiontransformations Both nonsingular Default to identity matrices (orthogonal view)

•Normalization lets us clip against simple cuberegardless of type of projection

•Delay final projection until end Important for hidden-surface removal to retain depth

information as long as possible


OrthogonalNormalization

glOrtho(left,right,bottom,top,near,far)

normalization ⇒ find transformation to convertspecified clipping volume to default


Orthogonal Matrix

•Two steps Move center to origin

T(-(left+right)/2, -(bottom+top)/2,(near+far)/2)) Scale to sides of length 2

S(2/(left-right),2/(top-bottom),2/(near-far))

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nearfar

nearfar

farnear

bottomtop

bottomtop

bottomtop

leftright

leftright

leftright

P = ST =


Final Projection

•Set z = 0•Equivalent to homogeneous coordinatetransformation

•Hence, general orthogonal projection in 4Dis

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Morth =

P = MorthST


Oblique Projections

•OpenGL projection functions cannot producegeneral parallel projections such as

•However if look at example of cubeappears that cube sheared

•==> Oblique Projection =Shear + Orthogonal Projection


General Shear

top view side view


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H(θ,φ) =

P = Morth H(θ,φ)

P = Morth STH(θ,φ)

Shear Matrix

xy shear (z values unchanged)

Projection matrix

General case:


Equivalency


Effect on Clipping

•Projection matrix P = STH transformsoriginal clipping volume to defaultclipping volume

top view

DOP DOP

near plane

far plane

object

clippingvolume

z = -1

z = 1

x = -1x = 1

distorted object(projects correctly)


Simple Perspective

Consider simple perspective: COP at origin,near clipping plane at z = -1, and 90 degreefield of view determined by planes

x = ±z, y = ±z


Perspective Matrices

Simple projection matrix inhomogeneous coordinates

Note: this matrix is independent of farclipping plane

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M =


Generalization

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after perspective division, point (x, y, z, 1) goes to

x’’ = x/zy’’ = y/zZ’’ = -(α + β/z)which projects orthogonally to desired point regardless of α and β

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N =


Picking α and β

If pick

α =

β =

nearfar

farnear

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farnear2

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near plane mapped to z = -1far plane mapped to z =1and sides mapped to x = ± 1, y = ± 1

==> new clipping volume = default clipping volume


NormalizationTransformation

original clipping volume original object new clipping

volume

distorted objectprojects correctly


Normalization andHidden-Surface Removal

•Selection of form of perspective matrices mayappear somewhat arbitrary

•But chosen so that if z1 > z2 in original clippingvolume ==> for transformed points z1’ > z2’

•==> hidden surface removal works if first applynormalization transformation

•However, formula z’’ = -(α + β/z) ==> distances distortedby normalization

•==> can cause numerical problems•especially if near distance is small


OpenGL Perspective

•glFrustum allows for non-symmetricviewing frustum (althoughgluPerspective does not)


OpenGL Perspective Matrix

•Normalization in glFrustum requires

•initial shear to form right viewingpyramid

•followed by scaling to get normalizedperspective volume

•Finally, perspective matrix results inneeding only final orthogonaltransformation P = NSH

our previously defined perspective matrix

shear and scale


Why do it this way?

•Normalization allows for singlepipeline

•for both perspective and orthogonalviewing

•Stay in 4-d homogeneous coordinatesas long as possible

•to retain 3-d information needed forhidden-surface removal and shading

•Simplify clipping

AngelCG17 projection matrices - Computer Science | … Computer Graphics I, Fall 2008 4 Pipeline...

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