2IV60 Computer Graphics 2D transformations Jack van Wijk TU/e.

2IV60 Computer Graphics2D transformations

Jack van Wijk

TU/e

Overview• Why transformations?

• Basic transformations:– translation, rotation, scaling

• Combining transformations– homogenous coordinates, transform. Matrices

• First 2D, next 3D

Transformations

image

train

world

wheelmodelling…

instantiation…

viewing…

animation…

Why transformation?

• Model of objectsworld coordinates: km, mm, etc.Hierarchical models::human = torso + arm + arm + head + leg + legarm = upperarm + lowerarm + hand …

• Viewingzoom in, move drawing, etc.

• Animation

Translation

Translate over vector (tx, ty)

x’=x+ tx, y’=y+ ty

or

x

y

P

P+T

H&B 7-1:220-222

T

y

x

t

t

y

x

y

xTPP

TPP'

and , '

''

with,

Translation polygon

Translate polygon:

Apply the same operation on all points.

Works always, for all transformations of objects defined as a set of points.

x

y

T

H&B 7-1:220-222

Rotation

x

y

P

H&B 7-1:222-223

x

y

P’

y

x

y

x

yxy

yxx

PRP

RPP'

and cossin

sincos,

'

''

with,

Or

cossin'

sincos'

: anglean over Rotate

Rotation around a point Q

x

y

P

H&B 7-1:222-223

x

y

P’

Q PQ

cossin'

sincos'

:origin around Rotate

yxy

yxx

PPP

PPP

cos)(sin)('

sin)(cos)('

: anglean over around Rotate

yyxxyy

yyxxxx

QPQPQP

QPQPQP

Q

Schale with factor sx and sy:

x’= sx x, y’= sy y

or

and 0

0,

'

''

with,

y

x

s

s

y

x

y

xPSP

SPP'

Scaling

x

y

P

H&B 7-1:224-225

x

P’

Q’Q

Scaling with respect to a point F

Scale with factors sx and sy:

Px’= sx Px, Py’= syPy

With respect to F:

Px’ Fx = sx (Px Fx),

Py’ Fy = sy (Py Fy)

or

Px’= Fx + sx (Px Fx),

Py’= Fy + sy (Py Fy)

x

y

P

H&B 7-1:224-225

x

P’

Q’QF

PF

Transformations

• Translate with V:T = P + V

• Schale with factor sx = sy =s:

S = sP

• Rotate over angle R’x = cos Px sin Py

R’y = sin Px + cos Py

x

y

P

T

S

R

H&B 7-2:225-228

Transformations…

• Messy!

• Transformations with respect to points: even more messy!

• How to combine transformations?

H&B 7-2:225-228

Homogeneous coordinates 1

• Uniform representation of translation, rotation, scaling

• Uniforme representation of points and vectors

• Compact representation of sequence of transformations

H&B 7-2:225-228

Homogeneous coordinaten 2

• Add extra coordinate:

P = (px , py , ph) or

x = (x, y, h)

• Cartesian coordinates: divide by h

x = (x/h, y/h)

• Points: h = 1 (for the time being…),

vectors: h = 0 H&B 7-2:225-228

),('

or

1100

10

01

1

'

'

:nTranslatio

PTP yx

y

x

tt

y

x

t

t

y

x

Translation matrix

H&B 7-2:225-228

)('

or

1100

0cossin

0sincos

1

'

'

:Rotation

PRP

y

x

y

x

Rotation matrix

H&B 7-2:225-228

Scaling matrix

H&B 7-2:225-228

),('

or

1100

00

00

1

'

'

:Scaling

PSP yx

y

x

ss

y

x

s

s

y

x

Inverse transformations

H&B 7-3:228

)1

,1

(),(

:Scaling

)()(

:Rotation

),(),(

:nTranslatio

1

1-

1-

yxyx

yxyx

ssss

tttt

SS

RR

TT

12

12

12''

'2

''

1'

MMMMP

PMM

P)(MMP

PMP

PMP

with

:Combined

ation... transformsecond

...sformationfirst tran

Combining transformations 1

H&B 7-4:228-229

PT

P P

PTTP

PTP

PTP

''

'''

'

),(

100

10

01

100

10

01

100

10

01

),(),(

:Combined

on translatisecond ),(

slationfirst tran ),(

2121

21

21

1

1

2

2

1122

22

11

yxxx

yy

xx

y

x

y

x

yxyx

yx

yx

tttt

tt

tt

t

t

t

t

tttt

tt

tt


H&B 7-4:229

),( ),(),(

:scaling Composite

)()()(

:rotations Composite

),( ),(),(

:ons translatiComposite

21211122

2112

21211122

yyxxyxyx

yxxxyxyx

ssssssss

R

tttttttt

SSS

RR

TTT


H&B 7-4:229

back. Translate 3)

origin; around angleover Rotate 2)

origin; with coincides such that Translate 1)

:point around angleover Rotate

R

R

Rotation around a point 1

R

1) 2) 3)H&B 7-4:229-230

'''''

'''

'

)PT(P

)PR(P

)PT(P

R

yx

yx

,RR

R,R

3)

2)

1)

:point around angleover Rotate

Rotation around a point 2

R

1) 2) 3)H&B 7-4:229-230

)P)T()R(T(

)P)R(T(

)PT(P

)P)T(R()PR(P

)PT(P

'

'''''

'''

'

yxyx

yx

yx

yx

yx

R,R,RR

,RR

,RR

R,R

R,R

3)

2)

1)

Rotation around point 3

R

1) 2) 3)H&B 7-4:229-230

PP

)P)T()R(T(P

'''

'''

100

sin)cos1(cossin

sin)cos1(sincos

or

3)-1

xy

yx

yxyx

RR

RR

R,R,RR

Rotation around point 4

R

1) 2) 3)H&B 7-4:229-230

again.back Translate 3)

origin; w.r.t.Schale 2)

origin; with coincides such that Translate 1)

:point w.r.t. and factors with Scale

F

Fxx ss

Scaling w.r.t. point 1

F

1) 2) 3)H&B 7-4:230-231

'''''

'''

'

)PT(P

)PS(P

)PT(P

F

yx

yx

yx

,FF

ss

F,F

3)

, 2)

1)

:point w.r.t.Schale

Scaling w.r.t.point 2

F

1) 2) 3)H&B 7-4:230-231

PP

)PT()ST(P

'''

'''

100

)1(0

)1(0

or

),( 3)-1

yyy

xxx

yxyxyx

sFs

sFs

F,Fss,FF

Scaling w.r.t.point 3

F

1) 2) 3)H&B 7-4:230-231

again.back Rotate 3)

origin; w.r.t.Scale 2)

frame;- standard with coincides framesuch that Rotate 1)

:frame rotated w.r.t. and factors with Scale 21

xy

ss

Scale in other directions 1

1) 2) 3)H&B 7-4:230-231

'''''

'''

'

)PR(P

)PS(P

)PR(P

3)

, 2)

1)

:directionother in Scale

21 ss


1) 2) 3)H&B 7-4:230-231

PP

)P)R()S(R(P

'''

'''

100

0cossinsincos)(

0sincos)(sincos

or

, 3)-1

22

2112

122

22

1

21

ssss

ssss

ss


1) 2) 3)H&B 7-4:230-231

Order of transformations 1

x

y

x’y’

x’’y’’

x

yx’

y’

x’’y’’

xx )T(2,3)R(30'' xx )R(30)T(2,3'' Matrix multiplication does not commute.

The order of transformations makes a difference!

Rotation, translation… Translation, rotation…

H&B 7-4:232


• Pre-multiplication:P’ = M n M n-1…M 2 M 1 P

Transformation M n in global coordinates

• Post-multiplication:P’ = M 1 M 2…M n-1 M n P

Transformation M n in local coordinates: thecoordinate system after application of

M 1 M 2…M n-1 H&B 7-4:232


OpenGL: glRotate, glScale, etc.:• Post-multiplication of current

transformation matrix• Always transformation in local coordinates• Global coordinate version: read in reverse

order

H&B 7-4:232


xx )R(30)T(2,3'' glTranslate(…);

glRotate(…);

x

yx’’

y’’

x’

y’

Local trafo

interpretation

Local transformations:

x

yx’’

y’’

Global transformations:

x’y’

Global trafo

interpretationH&B 7-4:232

Matrices in general

rotation and scalingtranslation

H&B 7-4:233-235

Direct construction of matrix

x

y

v

AB

T

u

If you know the target frame:Construct matrix directly.

Define shape in nice localu,v coordinates, use matrixtransformation to put itin x,y space.

H&B 7-4:233-235

11001

or ,

11

or , '

v

u

TBA

TBA

y

x

v

u

y

x

vu

yyy

xxx

TBA

TBAP

Direct construction of matrix

x

y

v

AB

T

u

If you know the target frame:Construct matrix directly.

H&B 7-4:233-235

11001

of , '

v

u

trrr

trrr

y

x

yyyyx

xxyxx

MPP

Rigid body transformation

only rotationtranslation

x

y

u

AB

T

v

H&B 7-4:233-2350,1||,1||, and

submatrix lorthonorma :

BABABAyy

xy

yx

xx

yyyx

xyxx

r

r

r

r

rr

rr

Other 2D transformations

• Reflection

• Shear

Can also be combined

H&B 7-4:240

PP'

100

010

001

Reflection over axis

Reflext over x-axis:

x’= x, y’= yor

x

y

H&B 7-4:240-242

PRP'

PP'

)180( as Same

100

010

001

Reflect over origin

Reflect over origin:

x’= x, y’= yor

x

y

H&B 7-4:240-242

tanwith

100

010

01

f

f

PP'

Shear

Shear the y-as:

x’=x+fy, y’=y

or

x

y

H&B 7-4:242-243

Transformations coordinates

Given (x,y)-coordinates,

Find (x’,y’)-coordinates.

Reverse route as

object transformaties.x

y

x’

y’

(x0, y0)

H&B 7-8:246-248

Given (x,y)-coordinates,

Find (x’,y’)-coordinates.

Example: user points at

(x,y), what’s the position

in local coordinates?


x

y

x’

y’

(x0, y0)

H&B 7-8:246-248

Given X: (x,y)-coordinates,

Find X’: (x’,y’)-coordinates.

Standard:

X=MX’ (object trafo:

from local to global)

Here:

X’=M-1X (from global to local)


x

y

x’

y’

(x0, y0)

H&B 7-8:246-248



Here:


Approach 1:

- Determine “standard matrix” M (from local to global coordinates) and invert


x

y

x’

y’

(x0, y0)

H&B 7-8:246-248



Here:


Approach 2:

- construct transformation that maps local frame to global (reverse of usual).


x

y

x’

y’

(x0, y0)

H&B 7-8:246-248



Here:


Approach 2:

1.Translate (x0, y0) to origin;

2.Rotate x’-axis to x-axis.


x

y

x’

y’

(x0, y0)

H&B 7-8:246-248



Here:


Approach 2:

M-1 = R()T(x0, y0)


x

y

x’

y’

(x0, y0)

H&B 7-8:246-248

OpenGL 2D transformations 1

Internally:

• Coordinates are four-element row vectors

• Transformations are 44 matrices

2D trafo’s: Ignore z-coordinates, set z = 0.

H&B 7-9:248-253


OpenGL maintains two matrices:• GL_PROJECTION • GL_MODELVIEW

Transformations are applied to the current matrix, to be selected with:

• glMatrixMode(GL_PROJECTION) or• glMatrixMode(GL_MODELVIEW)

H&B 7-9:248-253


Initializing the matrix to I:• glLoadIdentity();

Replace the matrix with M:• GLfloat M[16]; fill(M);• glLoadMatrix*(M);

Matrices are specified in column-major order:

Multiply current matrix with M:• glMultMatrix*(M);

H&B 7-9:248-253

m[15]m[11]m[7]m[3]

m[14]m[10]m[6]m[2]

m[13]m[9]m[5]m[1]

m[12]m[8]m[4]m[0]

M


Basic transformation functions: generate matrix and post-multiply this with current matrix.

Translate over [tx, ty, tz]:glTranslate*(tx, ty, tz);

Rotate over theta degrees (!) around axis [vx, vy, vz]:glRotate*(theta, vx, vy, vz);

Scale axes with factors sx, sy, sz:glScale*(sx, sy, sz);

H&B 7-9:248-253

OpenGL 2D Transformations 5

OpenGL maintains stacks of transformation matrices.

Two operations:• glPushMatrix():

Make copy of current matrix and put that on top of the stack;• glPopMatrix():

Remove top element of the stack.

Handy for dealing with hierarchical models

Handy for “undoing” transformationsH&B 9-8:324-327

OpenGL 2D Transformations 6

Standard:

glRotate(10, 1, 2, 0);glScale(2, 1, 0.5);glTranslate(1, 2, 3);

glutWireCube(1);

glTranslate(1, 2, 3);glScale(0.5, 1, 0.5);glRotate(10, 1, 2, 0);

Using the stack:

glPushMatrix();glRotate(10, 1, 2, 0);glScale(2, 1, 0.5);glTranslate(1, 2, 3);

glutWireCube(1);

glPopMatrix();

Undo transformation

Shorter, more robust

H&B 9-8:324-327

2D transformations summarized

- Transformations: modeling, viewing, animation;- Several kinds of transformations;- Homogeneous coordinates;- Combine transformations using matrix multiplication.

Up to 3D!

2IV60 Computer Graphics 2D transformations Jack van Wijk TU/e.

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