graphics pipeline : geometric operations

University of Calgary GraphicsJungle Project CPSC 587 2005

CPSC 587/687

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Graphics Pipeline : Geometric Operations


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•• Tools for creating and manipulating a “camera” Tools for creating and manipulating a “camera” that produces pictures of a 3D scenethat produces pictures of a 3D scene

•• Viewing transformationsViewing transformations and and projectionsprojections

•• Perform Perform cullingculling or backor back--face eliminationface elimination

Viewing transformation

3D Rendering Pipeline3D Rendering Pipeline

Object definitionObject definition

Compose sceneCompose sceneDefine view referenceDefine view referenceDefine lightingDefine lighting

CullCullClip to 3D view volumeClip to 3D view volume

Hidden surface removalHidden surface removalRasterizationRasterizationShadingShading

Local coordinate space

World coordinate space

View space

3D screen space

Display space

Modeling transformation

View transformation

Projection

Projection divide


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Yw

Zw

Xw

Compose sceneCompose sceneDefine view referenceDefine view reference


View space


View transformation

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

11w

w

w

viewv

v

v

zyx

Tzyx

( )vvv zyx

( )www zyx


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View plane eye

VU

-N

RTTview = ⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⋅−⋅−⋅−

=

1000100010001

NeVecVeVecUeVec

T

( )( )000−= eyeeVec

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

1000000

zyx

zyx

zyx

NNNVVVUUU

R

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

11w

w

w

viewv

v

v

zyx

Tzyx


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Viewing transformation


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Viewing Transforms

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

1000000

zyx

zyx

zyx

NNNVVVUUU

RLemma:Where does thismatrix come from:

U

V

Nt

ge

e = Eye Positiong = gaze directiont = view up vector

N= ||g||g

U= ||t x N||t x N

V= N x U

N


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Special Orthogonal Matrics

See 453 slides


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Taxonomy of ProjectionsTaxonomy of Projections

Parallel Perspective


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⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

==

1/100010000100001

d

TT persproj

obs: d is N in the picture

Perspective ProjectionsPerspective Projections

dzwzZyYxX

v

v

v

v

====


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⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

==

1000000000100001

ortproj TT

Parallel Projections (orthographic)Parallel Projections (orthographic)


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Yw

Zw

Xw

Compose sceneCompose sceneDefine view Define view referencereference


View space


View transformation( )vvv zyx

( )www zyx

ProjectionProjection

3D screen space( )WZYX

Projection DivideProjection Divide

( )sss zyx

Perspective DividePerspective Divide


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PerspectivePerspective

wZz

wYy

wXx

s

s

s

=

=

=

dzwzZyYxX

v

v

v

v

====

where

Parallel Parallel (orthographic)(orthographic)

0===

s

vs

vs

zyyxx



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See 453 slides for ramifications of divide by W



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View VolumeView Volume


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Xw

eye

VU

View planewindow

eye

Yw

Zw

View volume

We specify it!

-NWhat appears on the screen: projection of part of the scene contained within the view volume


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View planewindow

eye

View volume

dhzx v

v −=

h

d

dhzx v

v =

dhzy v

vol =

dhzy v

v −=

dzv =

fzv =f


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( )

( ) ⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

−−

−⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

000

10000100001

10000100000

000

dffd

dffh

dh

d

Tpers

Scaling (d/h in x and y)Truncated pyramid into a

Regular pyramid

Regular pyramid into a box

Perspective MatrixPerspective Matrix


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( )

( ) ⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−−

−=

000

100

100

000

dffd

dff

hd

hd

Tpers


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wZz

wYy

wXx

s

s

s

=

=

=

( ) ( )v

v

v

v

zwdf

dfdf

fzZ

yhdY

xhdX

=−

−−

=

=

=where

( )

( ) ⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−−

−=

000

100

100

000

dffd

dff

hd

hd

Tpers


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OpenGL: Viewing Process, Graphics PipelineOpenGL: Viewing Process, Graphics Pipeline

Scales and shifts each vertex so that all of them that lie inside the view volume will lie inside a standard cube

Provides the CTCombines two effects: Modeling transformations applied to objects +Transformation that orients and positions the camera in space


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OpenGL: Positioning and Aiming the CameraOpenGL: Positioning and Aiming the Camera

View plane

eye

VU

-N

glMatrixMode (GL_MODELVIEW)glLoadIdentity();

glLookAtglLookAt (eye.x, eye.y, eye.z, look.x, look.y, look.z, up.x, up.y, up.(eye.x, eye.y, eye.z, look.x, look.y, look.z, up.x, up.y, up.z);z);


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OpenGL: Setting the Camera OpenGL: Setting the Camera

(Parallel Projection)(Parallel Projection)

glMatrixMode (GL_PROJECTION)glLoadIdentity();

glOrthoglOrtho (left, right, bottom, top, near, far);(left, right, bottom, top, near, far);


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OpenGL: Setting the Camera OpenGL: Setting the Camera (Perspective Projection)(Perspective Projection)

glMatrixMode (GL_PROJECTION)glLoadIdentity();

glFrustumglFrustum (left, right, bottom, top, near, far);(left, right, bottom, top, near, far);

gluPerspectivegluPerspective ((viewAngleviewAngle, aspect, near, far);, aspect, near, far);

or

more intuitive

3D Rendering Pipeline3D Rendering Pipeline

1. INTRODUCTION

2. IMAGING

3. 3D RENDERING OVERVIEW

4. MODELING (1)shapes and transformations

5. RENDERING (1, 2, 3, 4)

1. Graphics Pipeline : Geometric Operations

6. MODELING (2)Representation and Modeling

7. ANIMATION

8. TOPICS

Object definitionObject definition

Compose sceneCompose sceneDefine view referenceDefine view referenceDefine lightingDefine lighting

CullCullClip to 3D view volumeClip to 3D view volume

Hidden surface removalHidden surface removalRasterizationRasterizationShadingShading

Local coordinate space


View space

3D screen space

Display space


View transformation

Projection

Projection divide

Yw

Zw

Xw

Compose sceneCompose sceneDefine view referenceDefine view reference


View space


View transformation

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

11w

w

w

viewv

v

v

zyx

Tzyx

( )vvv zyx

( )www zyx

( ) 0>− FmdoteyeP

Culling (backCulling (back--face removal)face removal)View plane

eye

VU

-N

F is F is back faceback face ifif

Dot ProductDot Product

Angle between Two Vectors

∑=

=⋅=n

iiiwvwvd

1

Sign and Perpendicularity

( )∧∧

⋅= cbθcos

“perpendicular”orthogonalnormal

Normal to a PlaneNormal to a Plane

Finding the Normal VectorsFinding the Normal Vectors

Flat Face:1V

2V 3Vm

( ) ( )2321 VVVVm −×−=

Two Problems:

1V

2V

3V

Vectors nearly parallel, the cross product will be very small,Numerical inaccuracies may result

Polygon is not perfect planar

Finding the Normal VectorsFinding the Normal Vectors

Robust Method (Newell): 1V

2V 3Vm

( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( )∑

∑

∑

−

=

−

=

−

=

+−=

+−=

+−=

1

0

1

0

1

0

N

iinextiinextiz

N

iinextiinextiy

N

iinextiinextix

yyxxm

xxzzm

zzyym

Vertex and Face TableVertex and Face Table

Shared vertices

Still no topology information

Each face lists vertex references

( )111 ,, zyx

( )222 ,, zyx

( )333 ,, zyx

( )444 ,, zyx

( )555 ,, zyx

F1F2

F3

VERTEX TABLE

V1

V2

V3

V4

V5

( )111 ,, zyx

( )222 ,, zyx

( )333 ,, zyx

( )444 ,, zyx

( )555 ,, zyx

FACE TABLE

V1 V2 V3

V3 V2 V4

V3 V4 V5

F1

F2

F2

Mesh

pt

norm

face

Point3

x, y, zVector3

dx, dy, dz

VertexID

vertIndexnormIndex

numVerts

numNormals

numFacesnVerts

Face

vert

( ) 0>− FmdoteyeP

Culling (backCulling (back--face removal)face removal)View plane

eye

VU

-N

F is F is back faceback face ifif

void Mesh :: draw ()

{

for (int f = 0; f < numFaces; f++)

{

glBegin (GL_POLYGON)

for (int v = 0; v < face[f].nVerts; v++)

{

int iv = face[f].vert[v].vertIndex;

glVertex3f(pt[iv].x, pt[iv].y, pt[iv].z);

}

glEnd();

}

}

void Mesh :: draw ()

{

for (int f = 0; f < numFaces; f++)

{

If ( isBackFace ( f, eye ) ) continue;glBegin (GL_POLYGON)

for (int v = 0; v < face[f].nVerts; v++)

{

int iv = face[f].vert[v].vertIndex;

glVertex3f(pt[iv].x, pt[iv].y, pt[iv].z);

}

glEnd();

}

}

graphics pipeline : geometric operations

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