07 Transform

7/30/2019 07 Transform

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Modeling Transformations

Tom Funkhouser

Princeton University

COS 426, Spring 2006

Modeling Transformations

Specify transformations for objects

Allows definitions of objects in own coordinate systems Allows use of object definition multiple times in a scene

Mike Marr, COS 426

Overview

2D Transformations Basic 2D transformations

Matrix representation

Matrix composition

3D Transformations

Basic 3D transformations

Same as 2D

Transformation Hierarchies Scene graphs

Ray casting

2D Modeling Transformations

ScaleRotate

Translate

ScaleTranslate

x

y

World Coordinates

ModelingCoordinates


x

y

World Coordinates

ModelingCoordinates

Lets lookat this indetail


x

y

ModelingCoordinates


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2


x

y

ModelingCoordinates

Scale .3, .3Rotate -90

Translate 5, 3


x

y

ModelingCoordinates


Translate 5, 3


x

y

ModelingCoordinates


Translate 5, 3

World Coordinates

Basic 2D Transformations

Translation: x = x + tx y = y + ty

Scale: x = x * sx y = y * sy

Shear: x = x + hx*y y = y + hy*x

Rotation:

x = x*cos

- y*sin

y = x*sin + y*cos

Transformationscan be combined

(with simple algebra)


Translation: x = x + tx

y = y + ty

Scale: x = x * sx

y = y * sy


Rotation: x = x*cos - y*sin y = x*sin + y*cos


Translation: x = x + tx

y = y + ty

Scale: x = x * sx

y = y * sy



x = x*sx

y = y*sy

x = x*sx

y = y*sy

(x,y)

(x,y)


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Translation:

x = x + tx y = y + ty


Shear: x = x + hx*y

y = y + hy*x


x = (x*sx)*cos (y*sy)*siny = (x*sx)*sin + (y*sy)*cos

x = (x*sx)*cos (y*sy)*siny = (x*sx)*sin + (y*sy)*cos

(x,y)


Translation:

x = x + tx y = y + ty


Shear: x = x + hx*y

y = y + hy*x


x = ((x*sx)*cos (y*sy)*sin) +txy = ((x*sx)*sin + (y*sy)*cos) + ty

x = ((x*sx)*cos (y*sy)*sin) + txy = ((x*sx)*sin + (y*sy)*cos) + ty

(x,y)


Translation: x = x + tx y = y + ty



Rotation:

x = x*cos

- y*sin

y = x*sin + y*cos

x = ((x*sx)*cos (y*sy)*sin) + txy = ((x*sx)*sin + (y*sy)*cos) + ty

x = ((x*sx)*cos (y*sy)*sin) +txy = ((x*sx)*sin + (y*sy)*cos) + ty

Overview



Matrix composition

3D Transformations


Same as 2D


Ray casting

Matrix Representation

Represent 2D transformation by a matrix

Multiply matrix by column vector apply transformation to point

=

yx

dcba

yx

''

dcba

dycxy

byaxx

+=

+=

'

'

Matrix Representation

Transformations combined by multiplication

=

yx

lk

ji

hg

fedcba

yx

''

Matrices are a convenient and efficient wayto represent a sequence of transformations!


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4

=

yx

dcba

yx

''

2x2 Matrices

What types of transformations can be

represented with a 2x2 matrix?

2D Identity?

yyxx

=

=

''

=

yx

yx

1001

''

2D Scale around (0,0)?

ysyyxsxx

*'*'

=

=

=

yx

dcba

yx

''

=

yx

sysx

yx

00

''

2x2 Matrices

What types of transformations can be

represented with a 2x2 matrix?

2D Rotate around (0,0)?

yxyyxx

*cos*sin'*sin*cos'

+=

=

=

yx

dcba

yx

''

=

yx

yx

cossinsincos

''

2D Shear?

yxshyyyshxxx

+=

+=

*'*'

=

yx

dcba

yx

''

=

yx

shyshx

yx

11

''

2x2 Matrices

What types of transformations can berepresented with a 2x2 matrix?

2D Mirror over Y axis?

yyxx

=

=

''

=

yx

dcba

yx

''

=

yx

yx

1001

''

2D Mirror over (0,0)?

yyxx

=

=

''

=

yx

dcba

yx

''

=

yx

yx

1001

''

=

yx

dcba

yx

''

2x2 Matrices

What types of transformations can berepresented with a 2x2 matrix?

2D Translation?

tyyytxxx

+=

+=

''

Only linear 2D transformationscan be represented with a 2x2 matrix

NO!

Linear Transformations

Linear transformations are combinations of Scale,

Rotation,

Shear, and

Mirror

Properties of linear transformations: Satisfies:

Origin maps to origin

Lines map to lines

Parallel lines remain parallel

Ratios are preserved

Closed under composition

)()()( 22112211 pppp TsTsssT +=+

=

y

x

dc

ba

y

x

'

'

2D Translation

2D translation represented by a 3x3 matrix Point represented with homogeneous coordinates

tyyy

txxx

+=

+=

'

'

=

1100

10

01

1

'

'

y

x

ty

tx

y

x


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Homogeneous Coordinates

Add a 3rd coordinate to every 2D point

(x, y, w) represents a point at location ( x/w, y/w) (x, y, 0) represents a point at infinity

(0, 0, 0) is not allowed

1 2

1

2(2,1,1) or (4,2,2) or (6,3,3)

Convenient coordinate system torepresent many useful transformations

x

y


Basic 2D transformations as 3x3 matrices

=

1100

0cossin

0sincos

1

'

'

y

x

y

x

=

1100

10

01

1

'

'

y

x

ty

tx

y

x

=

1100

01

01

1

'

'

y

x

shy

shx

y

x

Translate

Rotate Shear

=

1100

00

00

1

'

'

y

x

sy

sx

y

x

Scale

Affine Transformations

Affine transformations are combinations of Linear transformations, and

Translations

Properties of affine transformations: Origin does not necessarily map to origin

Lines map to lines

Parallel lines remain parallel

Ratios are preserved

Closed under composition

=

w

yx

fedcba

w

yx

100''

Projective Transformations

Projective transformations Affine transformations, and

Projective warps

Properties of projective transformations: Origin does not necessarily map to origin

Lines map to lines

Parallel lines do not necessarily remain parallel

Ratios are not preserved (but c ross-ratios are) Closed under composition

=

w

yx

ihg

fedcba

w

yx

'''

Overview



Matrix composition


Same as 2D


Ray casting

Matrix Composition

Transformations can be combined bymatrix multiplication

=

w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001

'

''

p = T(tx,ty) R() S(sx,sy) p


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Matrix Composition

Matrices are a convenient and efficient way to

represent a sequence of transformations General purpose representation

Hardware matrix multiply

Efficiency with premultiplication

Matrix multiplication is associative

p = (T * (R * (S*p) ) )p = (T*R*S) * p

Matrix Composition

Be aware: order of transformations matters

Matrix multiplication is not commutative

p = T * R * S * p

Global Local

Matrix Composition

Rotate by around arbitrary point (a,b) M=T(a,b) * R() * T(-a,-b)

Scale by sx,sy around arbitrary point (a,b) M=T(a,b) * S(sx,sy) * T(-a,-b)

(a,b)

(a,b)

The trick:

First, translate (a,b) to the origin.Next, do the rotation about origin.Finally, translate back.

(Use the same trick.)

Overview



Matrix composition

3D Transformations


Same as 2D


Ray casting

3D Transformations

Same idea as 2D transformations Homogeneous coordinates: (x,y,z,w)

4x4 transformation matrices

=

wz

yx

ponm

lkji

hgfedcba

wz

yx

''''


=

wz

yx

wz

yx

1000010000100001

'

''

=

wz

yx

tz

tytx

wz

yx

1000100

010001

'

''

=

wz

yx

sz

sysx

wz

yx

1000000

000000

'

''

=

wz

yx

wz

yx

1000010000100001

'

''

Identity Scale

Translation Mirror over X axis


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=

wzy

x

wzy

x

1000010000cossin

00sincos

''

'Rotate around Z axis:

=

wz

yx

wz

yx

10000cos0sin00100sin0cos

'

''

Rotate around Y axis:

=

wz

yx

wz

yx

10000cossin00sincos00001

'''

Rotate around X axis:

Overview

2D Transformations

Basic 2D transformations Matrix representation

Matrix composition


Same as 2D


Ray casting

Angel Figures 8.8 & 8.9

Transformation Hierarchies

Scene may have hierarchy of coordinate systems Each level stores matrix

representing transformationfrom parents coordinate system

Base[M1]

Upper Arm[M2]

Lower Arm[M3]

Robot Arm

Transformation Example 1

Rose et al. `96

Well-suited for humanoid characters

Root

LHip

LKnee

LAnkle

RHip

RKnee

RAnkle

Chest

LCollar

LShld

LElbow

LWrist

LCollar

LShld

LElbow

LWrist

Neck

Head


Mike Marr, COS 426,

Princeton University, 1995


An object may appear in a scene multiple times

Draw same 3D data withdifferent transformations


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Building

Floor 4 Floor5Floor 3Floor 2Floor 1

Chair KBookshelf 1

Desk ChairBookshelf

Desk 1 Desk 2 Chair 1

Floor Furniture

Office NOffice 1

Office Furniture

DefinitionsDefinitions

InstancesInstances


H&B Figure 109


IcosahedronTetrahedronCube Octahedron Dodecahedron

Level B

Instance 4Instance 1

Level A

Instances

Definitions


Scene



Ray Casting With Hierarchies

How interesect a ray withhierarchy of transformedprimitives?

Base[M1]

Upper Arm[M2]

Lower Arm[M3]

Robot Arm

Ray

World Coordinates


Ray Casting With Hierarchies

Transform rays, not primitives For each node ...

Transform ray by inverse of matrix

Intersect with primitives

Transform hit by matrix

Base[M1]

Upper Arm

[M2]

Lower Arm[M3]

Robot Arm

World Coordinates

Summary

Coordinate systems World coordinates

Modeling coordinates

Representations of 3D modeling transformations 4x4 Matrices

Scale, rotate, translate, shear, projections, etc.

Not arbitrary warps

Composition of 3D transformations Matrix multiplication (order matters)

Transformation hierarchies

Documents

07 Transform