Projection (Part 2) : Derivation Created by Dr. Slim BECHIKH for - - PowerPoint PPT Presentation

Projection (Part 2): Derivation Created by Dr. Slim BECHIKH for SPSU course - CS4363 Computer Graphics and Multimedia Fall 2014

ParallelProjection

normalization find4x4matrixtotransformuserspecified

viewvolume tocanonicalviewvolume(cube)

glOrtho(left, right, bottom, top,near, far) Canonical ViewVolume Userspecified ViewVolume

ParallelProjection:Ortho

Parallelprojection:2parts

1.

Translation: centersviewvolumeatorigin

ParallelProjection:Ortho

2.

Scaling: reducesuserselectedcuboidtocanonical cube(dimension2,centeredatorigin)

ParallelProjection:Ortho

• Translationlinesupmidpoints:E.g. midpointofx=(right+left)/2
• Thustranslationfactors:

(right+left)/2,(top+bottom)/2,(far+near)/2

Translationmatrix:

• 1

2 / ) ( 1 2 / ) ( 1 2 / ) ( 1 near far bottom top left right

ParallelProjection:Ortho

• Scalingfactor:ratiooforthoviewvolumetocubedimensions
• Scalingfactors:2/(right left),2/(top bottom),2/(far near)
• ScalingMatrixM2:
• 1

2 2 2 near far bottom top left right

ParallelProjection:Ortho

Concatenating Translation x Scaling, we get Ortho Projection matrix

X

• 1

2 2 2 near far bottom top left right

• 1

2 / ) ( 1 2 / ) ( 1 2 / ) ( 1 near far bottom top left right

• 1

2 2 2 near far near far far near bottom top bottom top bottom top left right left right left right

P = ST =

FinalOrthoProjection

Setz =0 Equivalenttothehomogeneouscoordinate

transformation

Hence,generalorthogonalprojectionin4Dis

• 1

1 1

Morth = P = MorthST

PerspectiveProjection

Projection– maptheobjectfrom3Dspaceto

2Dscreen

x y z Perspective() Frustrum( )

PerspectiveProjection: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 N y’ = y x

• z

=

VRP COP Object in 3 space Projectors Projected image

Near Plane (VOP) + z

PerspectiveProjection:Classical

So(x*,y*)projectionofpoint,(x,y,z)untonearplaneNis

givenas:

Numericalexample:

Q.Whereontheviewplane doesP=(1,0.5,1.5)liefora nearplaneatN=1?

• z

N y z N x y x , * *,

• )

333 . , 666 . ( 5 . 1 1 5 . , 5 . 1 1 1 , * *,

• z

N y z N x y x

VRP COP Object in 3 space Projectors Projected image

Pseudodepth

Classicalperspectiveprojectionprojects(x,y)coordinatesto

(x*,y*),dropszcoordinates

Butweneedztofindclosestobject(depthtesting)!!!

VRP COP Object in 3 space Projectors Projected image

(0,0,0) z

Map to same (x*,y*) Compare their z values?

PerspectiveTransformation

Perspectivetransformation mapsactualzdistanceof

perspectiveviewvolumetorange[–1to1](Pseudodepth) forcanonicalviewvolume

• Near
• Far
• 1

1 Canonical view volume Actual view volume

Pseudodepth Actual depth

We want perspective Transformation and NOT classical projection!! Set scaling z Pseudodepth = az + b Next solve for a and b

PerspectiveTransformation

Wewanttotransformviewingfrustum

volumeintocanonicalviewvolume

(-1, -1, 1) (1, 1, -1) Canonical View Volume x y z

PerspectiveTransformationusing Pseudodepth

Choosea,bsoaszvariesfromNeartoFar,pseudodepth

variesfrom–1to 1(canonicalcube)

• Boundaryconditions
• z*=1whenz=N
• z*=1whenz=F
• z

b az z N y z N x z y x , , * *, *,

• Near
• Far

Canonical view volume Actual view volume

Pseudodepth Actual depth

1

• 1

Z* Z

Transformationofz:Solveforaandb

Solving: Useboundaryconditions

z*=1whenz=N………(1) z*=1whenz=F………..(2)

Setupsimultaneousequations

z b az z

• *

) 1 ........( 1 b aN N N b aN

• )

2 ........( 1 b aF F F b aF

Transformationofz:Solveforaandb

Multiplybothsidesof(1)by1 Addeqns (2)and(3) Nowput(4)backinto(3)

) 1 ........( b aN N

• )

2 ........( b aF F

• )

3 ........( b aN N

• aF

aN N F

• )

4 .........( ) ( N F N F F N N F a

Transformationofz:Solveforaandb

Putsolutionforabackintoeqn(3) So

b N F N F N N

• )

( ) 3 ........( b aN N

• N

F N F N N b

• )

( N F NF N F N NF N NF N F N F N N F N b

• 2

) ( ) (

2 2

N F N F a

• )

( N F FN b

• 2

Whatdoesthismean?

Originalpointzinoriginalviewvolume,transformed

intoz*incanonicalviewvolume

where

• Near
• Far

Canonical view volume Actual view volume 1

• 1

Original vertex z value Transformed vertex z* value

z b az z

• *

N F N F a

• )

(

N F FN b

• 2

HomogenousCoordinates

Wanttoexpressprojectiontransformas4x4matrix Previously,homogeneouscoordinatesof

P=(Px,Py,Pz)=>(Px,Py,Pz,1)

Introducearbitraryscalingfactor,w,sothat

P=(wPx,wPy,wPz,w)(Note:wisnonzero)

Forexample,thepointP=(2,4,6)canbeexpressedas (2,4,6,1)

• r(4,8,12,2)wherew=2
• r(6,12,18,3)wherew=3,or….

Toconvertfromhomogeneousbacktoordinarycoordinates,

firstdivideallfourtermsbyw anddiscard4th term

PerspectiveProjectionMatrix

RecallPerspectiveTransform Wehave: Inmatrixform:

• 1

) ( 1 z b az z N y z N x wz b az w wNy wNx w wz wy wx b a N N

• z

b az z N y z N x z y x , , * *, *,

z N x x

• *

z N y y

• *

z b az z

• *

Perspective Transform Matrix Original vertex Transformed Vertex Transformed Vertex after dividing by 4th term

PerspectiveProjectionMatrix

Inperspectivetransformmatrix,alreadysolvedfora

andb:

So,wehavetransformmatrixtotransformz

values

• 1

) ( 1 z b az z N y z N x wP b aP w wNP wNP w wP wP wP b a N N

z z y x z y x

N F N F a

• )

(

N F FN b

• 2

PerspectiveProjection

• Notdoneyet!!Cannowtransformz!
• Alsoneedtotransformthex=(left,right)andy=(bottom,top)

rangesofviewingfrustumto[1,1]

• SimilartoglOrtho,weneedtotranslateandscalepreviousmatrix

alongxandytogetfinalprojectiontransformmatrix

• wetranslateby
• –(right+left)/2inx
• (top+bottom)/2iny
• Scaleby:
• 2/(right– left)inx
• 2/(top– bottom)iny

1

• 1

x y left right bottom top

PerspectiveProjection

TranslatealongxandytolineupcenterwithoriginofCVV

• –(right+left)/2inx
• (top+bottom)/2iny

Multiplybytranslationmatrix:

1

• 1

x y left right bottom top

• 1

1 2 / ) ( 1 2 / ) ( 1 bottom top left right

Line up centers Along x and y

PerspectiveProjection

• TobringviewvolumesizedowntosizeofofCVV,scaleby
• 2/(right– left)inx
• 2/(top– bottom)iny

Multiplybyscalematrix:

1

• 1

x y left right bottom top

Scale size down along x and y

• 1

1 2 2 bottom top left right

PerspectiveProjectionMatrix

glFrustum(left, right, bottom, top, N, F) N = near plane, F = far plane

• 1

2 ) ( 2 min max 2 N F FN N F N F bottom top bottom top bottom top N left right left right x x N

• 1

1 1 2 / ) ( 1 2 / ) ( 1 1 1 2 2 b a N N bottom top left right bottom top left right

Scale

Final Perspective Transform Matrix

Translate

Previous Perspective Transform Matrix

PerspectiveTransformation

Afterperspectivetransformation,viewing

frustumvolumeistransformedintocanonical viewvolume

(-1, -1, 1) (1, 1, -1) Canonical View Volume x y z

GeometricNatureofPerspective Transform

a)

Linesthrougheyemapintolinesparalleltozaxisaftertransform

b) Linesperpendiculartozaxismaptolinesperptozaxisaftertransform

NormalizationTransformation

• riginal clipping

volume

• riginal object

new clipping volume distorted object projects correctly

Implementation

Setmodelview andprojectionmatricesinapplicationprogram Passmatricestoshader

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

Andthecorrespondingshader

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

InteractiveComputerGraphics(6th edition),Angeland

Shreiner

ComputerGraphicsusingOpenGL(3rd edition),HillandKelley