2d Transforms

7/30/2019 2d Transforms

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2D Transformations


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2D Transformations

World Coordinates

Translate

Rotate

Scale

Viewport Transforms Hierarchical Model Transforms

Putting it all together


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Transformations

Rigid Body Transformations - transformations that do

not change the object.

Translate

If you translate a rectangle, it is still a rectangle

Scale

If you scale a rectangle, it is still a rectangle

Rotate If you rotate a rectangle, it is still a rectangle


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Vertices

We have always represented vertices as

(x,y)

An alternate method is:

Example:

y

xyx ),(

8.4

1.2)8.4,1.2(


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Matrix * Vector

10

01

'

'

'

'

I

dycxy

byaxx

y

x

dc

ba

y

x

100

010

001'

'

''

'

'

I

izhygxz

fzeydxy

czbyaxxz

y

x

ihg

fed

cba

z

y

x


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Matrix * Matrix

10

01?

*

,

dc

ba

dwcydzcx

bwaybzaxBA

wz

yxB

dc

baA

Does A*B = B*A? NO

What does the identity do?

AI=A


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Translation

Translation - repositioning an object along a

straight-line path (the translation distance)

from one coordinate location to another.

(x,y)

(x,y)

(tx,ty)


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Translation

Given:

We want:

Matrix form:

TPP

t

t

y

x

y

x

tyy

txx

ttT

yxP

y

x

y

x

yx

'

'

'

'

'

),(

),(

1.4'

4.3'

2.8

1.7

1.4

7.3

'

'

2.81.4'

1.77.3'

)2.8,1.7(

)1.4,7.3(

y

x

y

x

y

x

T

P


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Translation Examples

P=(2,4), T=(-1,14), P=(?,?)

P=(8.6,-1), T=(0.4,-0.2), P=(?,?)

P=(0,0), T=(1,0), P=(?,?)


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Recall

A point is a position specified with

coordinate values in some reference frame.

We usually label a point in this reference

point as the origin.

All points in the reference frame are given

with respect to the origin.


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Applying to Triangles

(tx,ty)


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What do we have here?

You know how to:


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Scale

Scale - Alters the size of an object.

Scales about a fixed point

(x,y)

(x,y)


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Scale

Given:

We want:

Matrix form:PSP

y

x

s

s

y

x

ysy

xsx

ssS

yxP

y

x

y

x

yx

'

0

0

'

'

'

'

),(

),(

6.6'

2.4'

2.24.1

3003

''

2.2*3'

4.1*3'

)3,3(

)2.2,4.1(

y

x

yx

y

x

S

P


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Non-Uniform Scale

(x,y)

(x,y)

S=(1,2)


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Rotation

Rotation - repositions an object along a

circular path.

Rotation requires an and a pivot point


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Rotation

)sin('

)cos('

sin

cos

)(

),(

ry

rx

ry

rx

R

yxp

Rpp

y

x

y

x

yxy

yxx

rry

rrx

'

cossin

sincos

'

'

cossin'

sincos'

cossinsincos'

sinsincoscos'


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Example

P=(4,4)

=45 degrees


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RotationsV(-0.6,0) V(0,-0.6) V(0.6,0.6)

Rotate -30 degrees

V(0,0.6) V(0.3,0.9) V(0,1.2)


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Combining Transformations

Q: How do we

specify each

transformation?


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Specifying 2D Transformations

Translation

T(tx, ty)

Translation distances Scale

S(sx,sy)

Scale factors

Rotation

R()

Rotation angle


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Using translate, rotation, and scale, how do

we get:


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Combining Transformations Note there are two ways to combine rotation

and translation. Why?


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Lets look at the equations

cos'sin'"

sin'cos'"

'

'

'''

')(

),(

),(

yxy

yxx

tyy

txx

PRP

TPPR

ttT

yxP

y

x

yx

y

x

yx

yx

tyxy

tyxx

yxy

yxx

TPPPRP

tytxy

tytxx

cossin"

sincos"

cossin'

sincos'

'"'

cos'sin"

sin'cos"


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Combining them

We must do each step in turn. First we

rotate the points, then we translate, etc.

Since we can represent the transformationsby matrices, why dont we just combine

them?

PSP

PRP

TPP

'

'

'


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2x2 -> 3x3 Matrices

We can combine transformations by

expanding from 2x2 to 3x3 matrices.

100

0cossin

0sincos

cossin

sincos

10000

00

0

0

,

100

10

01

,

R

s

s

s

s

ssS

t

t

t

t

y

xttT

y

x

y

x

yx

y

x

y

x

yx


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Homogenous Coordinates

We need to do something to the vertices

By increasing the dimensionality of the

problem we can transform the additioncomponent of Translation into

multiplication.

2

22

147

263

2

14

6

7

3.,

1

2

4

2

4. ExEx

h

hh

yy

hxx

h

y

x

y

xP h

h

h

h


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Homogenous Coordinates

Homogenous Coordinates - term used in

mathematics to refer to the effect of this

representation on Cartesian equations. Convertinga pt(x,y) and f(x,y)=0 -> (xh,yh,h) then in

homogenous equations mean (v*xh,v*yh,v*h) can

be factored out.

What you should get: By expressing thetransformations with homogenous equations and

coordinates, all transformations can be expressed

as matrix multiplications.


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Final Transformations -

Compare Equations

PRPPRP

y

x

y

x

y

x

y

xR

PssSPPssSP

y

x

s

s

y

x

y

x

s

s

y

xssS

PttTPPttTP

y

x

t

t

y

x

y

x

t

t

y

xttT

yxyx

y

x

y

x

yx

yxyx

y

x

y

x

yx

1100

0cossin

0sincos

1

'

'

cossin

sincos

'

'

,,

1100

00

00

1

'

'

0

0

'

',

,,

1100

10

01

1

'

'

'

',


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1

'

'

100

10

01

100

0cossin

0sincos

1

"

"1

'

'

100

0cossin

0sincos

1

"

"

1100

10

01

1

'

'

)60(),2,4641.0(),4,3(

"'",'

y

x

t

t

y

x

y

x

y

x

y

x

t

t

y

x

RTP

PBAPPBPPAP

y

x

y

x

1100

cossin

sincos

1

"

"

1100

0cossin

0sincos

100

10

01

1

"

"

1100

cossincossin

sincossincos

1

"

"

y

x

t

t

y

x

y

x

t

t

y

x

y

x

tt

tt

y

x

y

x

y

x

yx

yx


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How would we get:


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How would we get:


33/47

Coordinate Systems

Object Coordinates

World Coordinates

Eye Coordinates


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Object Coordinates


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World Coordinates


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Screen Coordinates


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Coordinate Hierarchy

Object #1

Object Coordinates

Transformation

Object #1 ->World

Object #2

Object Coordinates

Transformation

Object #2 ->World

Object #3

Object Coordinates

Transformation

Object #3 ->World

World Coordinates

Transformation

World->Screen

Screen Coordinates


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Lets reexamine assignment 1


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Transformation Hierarchies

(See chapter 10 for details)

For example, a robot arm


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Lets examine:


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What is a better way?


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What is a better way?

W ld


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We can have transformations be

in relation to each other

World

Coordinates

Upper Arm

Coordinates

Lower Arm

Coordinates

Hand

Coordinates

Transformation:

Upper Arm -> World

Transformation:

Lower -> Upper

Transformation:

Hand-> Lower


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44Angel: Interactive Computer

Graphics 5E Addison-Wesley 2009

Rotation about a Fixed Point

Start with identity matrix: CI

Move fixed point to origin: CCT

Rotate: CCR

Move fixed point back: CCT -1

Result: C = TR T 1which is backwardsCp

This result is a consequence of doing postmultiplications.Lets try again.


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Reversing the Order

We want C = T 1 R Tso we must do the operations in the following order

C

ICCT -1CCRCCT

Each operation corresponds to one function call in theprogram.

Note that the last operation specified is the first executed inthe program


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OpenGL Example

Rotation about z axis by 30 degrees with a fixed

point of (1.0, 2.0, 3.0)

Remember that last transform specified in the

program is the first applied

glMatrixMode(GL_MODELVIEW);

glLoadIdentity();glTranslatef(1.0, 2.0, 3.0);

glRotatef(30.0, 0.0, 0.0, 1.0);

glTranslatef(-1.0, -2.0, -3.0);

glBegin(GL_TRIANGLES);...


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Angel: Interactive Computer

Matrix Stacks In many situations we want to save

transformation matrices for use later

Traversing hierarchical data structures (Chapter 10)

Avoiding state changes when executing display lists

OpenGL maintains stacks for each type of

matrix

Access present type (as set by glMatrixMode)by glPushMatrix()

glPopMatrix()

Date post:	14-Apr-2018
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