Evening’s Goals

2COEN 290 - Computer Graphics I

Evening’s Goals

Discuss viewing and modeling transformations

Describe matrix stacks and their uses Show basic geometric rasterization and

clipping algorithms

3COEN 290 - Computer Graphics I

Modeling Objects

Recall objects are composed of geometric primitives• each primitive is composed of vertices

Model around the origin• model just means

“determine an object’s vertices”

4COEN 290 - Computer Graphics I

Modeling Transformations

Position objects in world coordinates Move coordinate systems, not objects Affects all objects rendered after

transformation Types

• translation• scale• rotation• shear

5COEN 290 - Computer Graphics I

Translation

Move the origin to a new location

6COEN 290 - Computer Graphics I

1000

100

010

001

),,(z

y

x

zyxt

t

t

tttT

glTranslate[fd]()

glTranslatef( tx, ty, tz );

7COEN 290 - Computer Graphics I

Scale

Stretch, mirror or decimate a coordinate direction

recthreflect/st1rinkreflect/sh01

decimate0shrink10stretch1

ss

ss

Note, there’s a translation applied here to make thingseasier to see

8COEN 290 - Computer Graphics I

glScale[fd]()

glScalef( sx, sy, sz );

1000

000

000

000

),,(z

y

x

zyxs

s

s

sssS

9COEN 290 - Computer Graphics I

Rotation

Rotate coordinate system about an axis in space

Note, there’s a translation applied here to make thingseasier to see

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COEN 290 - Computer Graphics I

glRotate[fd]()

glRotatef( angle, x, y, z );

0

00

xyxz

yzS

zyxu

zyxv

vv

SuuIuuM tt )sin())(cos(

1000

0

0

0

vR M

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COEN 290 - Computer Graphics I

Some Additional Rotation Examples

10000)cos()sin(00)sin()cos(00001

xR

1000

0)cos(0)sin(

0010

0)sin(0)cos(

yR

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COEN 290 - Computer Graphics I

Shear

Scaling in one dimension depends on another

For example

No OpenGL command• load custom matrix

zfyfxx xzxy

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COEN 290 - Computer Graphics I

Shear

Making a shear matrix

1000

01

01

01

zyzx

yzyx

xzxy

ff

ff

ff

H

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COEN 290 - Computer Graphics I

Other OpenGL Matrix Commands

glMultMatrix( m )• multiples the current matrix by m

glLoadMatrix( m )• replaces the current matrix by m

glLoadIdentity()• replace the current matrix with an identity

matrix

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COEN 290 - Computer Graphics I

OpenGL Matrix Format

GLfloat m[16];

151173

141062

13951

12840

mmmmmmmmmmmmmmmm

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COEN 290 - Computer Graphics I

Modeling Transforms and our Pipeline

WorldCoordinates

EyeCoordinates

ClipCoordinates

NDC’s

PerspectiveDivideProjection

Transformation

ModelCoordinates

ModelingTransformations

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COEN 290 - Computer Graphics I

Using Multiple Modeling Transforms

Multiple modeling transforms form a composite modeling transform

Individual transform matrices are multiplied together• for example

),,()(),,( zyxvzyx sssSRtttTM

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COEN 290 - Computer Graphics I

Multiplying Matrices

Matrix multiplication is not commutative• in general,• order of operations is important

OpenGL multiples matrices on the right

BAAB

DCBADCBA

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COEN 290 - Computer Graphics I

Independent vs. Dependent Transforms

Every modeling transform affects the model coordinate system

Transformations accumulate• we record every transformation every made• to undo a transform, we’d need to do the

inverse transform– this is very inconvenient

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A Stack Based Solution

We create a stack containing matrices Any transform multiples the top of stack Push makes a copy and pushes it onto the

stack Pop discards the top of stack

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A Stack Based Solution ( cont. )

wzyx

ProjectionTransformation

1wzwywx

ModelingTransformations

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COEN 290 - Computer Graphics I

OpenGL Matrix Stack Commands

Top of stack matrix is called the current matrix

glPushMatrix()• copy the current matrix and pushes it

glPopMatrix()• pop the current matrix

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COEN 290 - Computer Graphics I

OpenGL Matrix Stacks

Why multiple matrix stacks?• certain techniques are done in different spaces

glMatrixMode( mode )• choose which stack to manipulate

–GL_MODELVIEW–GL_PROJECTION

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COEN 290 - Computer Graphics I

Current Transformation Pipeline

WorldCoordinates

EyeCoordinates

ClipCoordinates

NDC’s

PerspectiveDivideProjection

Transformation

ModelCoordinates

ModelCoordinates

ModelCoordinates

ModelingTransformations

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COEN 290 - Computer Graphics I

Eye coordinates are nice, but ...

Eye coordinates are restrictive• eye’s positioned at the origin• looking down the -z axis

What if we want to look at the scene from somewhere else?• use a viewing transformation

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COEN 290 - Computer Graphics I

Viewing Transformations

Reorient world coordinates to match eye coordinates

Basically just a modeling transform• affects the entire scene• usually a translation and a rotation

Usually set up after the projection transform, but before any modeling transforms

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COEN 290 - Computer Graphics I

The Simplest Viewing Transform

“Push” the origin into the viewing frustum

z

near

far

y y

farneartz 21

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COEN 290 - Computer Graphics I

polarview()

Useful if you want to view an entire objectpolarview( dist, elev, azim, twist ){glTranslatef( 0, 0, -dist );glRotatef( -twist, 0, 0, 1 );glRotatef( -elev, 1, 0, 0 );glRotatef( azim, 0, 0, 1 );}• update elev and azim for interactive viewing

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COEN 290 - Computer Graphics I

Transforming World to Eye Coordinates

Viewing transformgluLookAt( eyex, eyey, eyez,

lookx, looky, lookz,

upx, upy, upz ); Creates an orthonormal basis

• a set of linearly independent vectors of unit length

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COEN 290 - Computer Graphics I

Creating an Orthonormal Basis

1000000

ˆˆˆ

ˆ

ˆ

ˆ

ˆ

zyx

zyx

zyx

upn

upn

eyelook

eyelook

nnnvvvuuu

nuv

u

n

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COEN 290 - Computer Graphics I

gluLookAt( eyex, eyey, eyez,

lookx, looky, lookz,

upx, upy, upz ){

/* Create matrix m */ glMultMatrixf( m ); glTranslatef( -eyex, -eyey, -eyez );}

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COEN 290 - Computer Graphics I

Our Final Transformation Pipeline

WorldCoordinates

EyeCoordinates

ClipCoordinates

NDC’s

PerspectiveDivideProjection

Transformation

ViewingTransformation

ModelCoordinates

ModelCoordinates

ModelCoordinates

ModelingTransformations

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COEN 290 - Computer Graphics I

Retrieving Matrix information

GLfloat m[16];glGetFloatv( which, m );

which is which matrix stack to access• GL_MODELVIEW_MATRIX• GL_PROJECTION_MATRIX

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COEN 290 - Computer Graphics I

Putting it all Together

General flow of transformations projection transformation viewing transformation push matrix modeling transformation render pop matrix goto as necessary