2D and 3D Transformation

April 19, 2023204481 Foundation of Computer Graphics 1

2D and 3D Transformation

Pradondet Nilagupta

Dept. of Computer Engineering

Kasetsart University


Transformations and Matrices

Transformations are functions Matrices are functions representations Matrices represent linear transformation {2x2 Matrices} {2D Linear Transformation}


Transformations (1/3)

What are they? changing something to something else via rules mathematics: mapping between values in a range set

and domain set (function/relation) geometric: translate, rotate, scale, shear,…

Why are they important to graphics? moving objects on screen / in space mapping from model space to screen space specifying parent/child relationships …


Transformation (2/3)

Translation Moving an object

Scale Changing the size

of an object

ty

tx

wold wnew

hold

hnew

xnew = xold + tx; ynew = yold + ty

sx=wnew/wold sy=hnew/hold

xnew = sxxold ynew = syyold


Transformation (3/3)

To rotate a line or polygon, we must rotate each of its vertices

Shear

(x,y)

Original Data y Shear x Shear


What is a 2D Linear Transform?

.y and x vectorsand ascalar for

,)y(T)x(aT)yxa(T:Definition

)y,x2()y,x2(yy),xx(2

:say 2,by x,in Scale

11001010

Example


Example

), 00( yx ), 002( yx

), 11( yx ), 112( yx

y

x x

y

), 002( yx

), 112( yx

yyxx 1010 ,22

y

), 00( yx

), 11( yx

yyxx 1010 ),(2

yyxx 1010 ),(

x

y

yyxx 1010 ),(2

yyxx 1010 ),(

y

x

yyxx 1010 ,22 Scale in x by 2

yyxx 1010 ),(2 yyxx 1010 ),(2


Transformations: Translation (1/2)

A translation is a straight line movement of an object from one position to another.

A point (x,y) is transformed to the point (x’,y’) by adding the translation distances Tx and Ty:

x’ = x + Tx

y’ = y + Ty


Transformations: Translation(2/2)

moving a point by a given tx and ty amount

e.g. point P is translated to point P’

moving a line by a given tx and ty amount

translate each of the 2 endpoints

)10,5(P

)10,15(P

0

10

y

x

t

tT

)20,5(1P

)10,5(2P )10,5(1P

)0,5(2P

10

0

y

x

t

tT


Transformations: Rotation (1/4)

Objects rotated according to angle of rotation theta ()

Suppose a point P(x,y) is transformed to the point P'(x',y') by an anti-clockwise rotation about the origin by an angle of degrees, then:

Given x = r cos , y = r sin x’ = x cos – y sin y’ = y sin + y cos



Rotation P by anticlockwise relative to origin (0,0)

)0,0(

),( yxP

),( yxP

x

yr

)0,0(

),( yxP



Rotation about an arbitary pivot point (xR,yR)Step 1: translation of the object by (-xR,-yR)

x1 = x - xR

y1 = y - yRStep 2: rotation about the origin

x2 = x1 cos() - y1sin ()

y2 = y1cos() - x1sin ()Step 3: translation of the rotated object by (xR,yR)

x’ = xr + x2

y’ = yr + y2



object can be rotated around an arbitrary point (xr,yr) known as rotation or pivot point by: x' = xr + (x - xr) cos() - (y - yr) sin ()

y' = yr + (x - xr) sin ()+(y - yr) cos()


Transformations: Scaling (1/5)

Scaling changes the size of an object Achieved by applying scaling factors

sx and sy Scaling factors are applied to the X

and Y co-ordinates of points defining an object’s



uniform scaling is produced when sx and sy have same value i.e. sx = sy

non-uniform scaling is produced when sx and sx are not equal - e.g. an ellipse from a circle. i.e. sx sy

x2 = sxx1 y2 = syy1



Simple scaling - relative to (0,0)

General form:

y*sy

x*sx

y

x

),(1 yxP),(1 yxP

)3,2(1P

)1,3(2P

)3,4(1P

)1,6(2P

Ex: sx = 2 and sy=1



If the point (xf,yf) is to be the fixed point, the transformation is:

x' = xf + (x - xf) Sx

y' = yf + (y - yf) Sy

This can be rearranged to give:

x' = x Sx + (1 - Sx) xf

y' = y Sy + (1 - Sy) yf

which is a combination of a scaling about the origin and a translation.




Transformation as Matrices

Scale:

x’ = sxx

y’ = syy

Rotation:

x’ = xcos - ysin y’ = xsin + ycos

Translation:

x’ = x + tx

y’ = y + ty

ys

xs

y

x

s

s

y

x

y

x

0

0

cossin

sincos

cossin

sincos

yx

yx

y

x

y

x

y

x

ty

tx

y

x

t

t


Transformations: Shear (1/2)

y

ayx

y

xaShx 10

1Shear in x:

)0,1(

)1,( a)1,0(

)0,1(

)1,1(


Transformations: Shear (2/2)

Shear in y:

)1,0(

)0,0(

),1( b

)1,0(

)0,1()0,0(

)1,1(

ybx

x

y

x

bShy 1

01


Shear in x then in y

)1,0(

)0,0(

)1,0(

)0,0(

)1,0(

)0,0()0,1( )0,1(

),(1 bab

),( 1a

),( 1 aba

),( 11 baba

),( 11 a

)1,0(

)0,0(

)1,1( b

)1,1(

),1( b


Shear in y then in x

)1,0(

)0,0(

)1,0(

)1,0(

)0,0()0,1( )0,1(

),(1 b

),( 1a

),( 1 aba

),( 11 abba

),( 11 a

)1,0(

)0,0(

)1,1( b

)1,1(

),1( b


Homogeneous coordinate

As translations do not have a 2 x 2 matrix representation, we introduce homogeneous coordinates to allow a 3 x 3 matrix representation.

The Homogeneous coordinate corresponding to the point (x,y) is the triple (xh, yh, w) where:

xh = wx yh = wy

For the two dimensional transformations we can set w = 1.


Matrix representation

1),( y

x

P yx

100

00

00

, y

x

yx s

s

S

100

0cossin

0sincos

R

100

10

01

, y

x

yx t

t

T


Basic Transformation (1/3)

Translation



Rotation

P x y x y

t t

t t PS t( , , ) ( , , )

cos sin

sin cos ( )1 1

0

0

0 0 1



Scaling


Composite Transformation

Suppose we wished to perform multiple transformations on a point:

P2 T3,1P1

P3 S2, 2P2

P4 R30P3

M R30S2,2T3,1

P4 MP1


Example of Composite Transformation(1/3)

A scaling transformation at an arbitrary angle is a combination of two rotations and a scaling:

R(-t) S(Sx,Sy) R(t)

A rotation about an arbitrary point (xf,yf) by and

angle t anti-clockwise has matrix:

T(-xf,-yf) R(t) T(xf,yf)



Reflection about the y-axis Reflection about the x-axis

100

010

001

100

010

001



Reflection about the origin Reflection about the line y=x

100

010

001

100

001

010


3D Transformation

Z

X

YY

X

Z


Basic 3D Transformations

Translation Scale Rotation Shear

As in 2D, we use homogeneous coordinates (x,y,z,w), so that transformations may be composited together via matrix multiplication.


3D Translation and Scaling

TP = (x + tx, y + ty, z + tz)

SP = (sxx, syy, szz)

1000

100

010

001

z

y

x

t

t

t

1

z

y

x

1000

000

000

000

z

y

x

s

s

s

1

z

y

x


3D Rotation (1/4)

Positive Rotations are defined as follows:

Axis of rotation is Direction of positive rotation isx y to zy z to xz x to y

Z

Y

X


3D Rotation (2/4)

Rotation about x-axis Rx(ß)P

1000

0cossin0

0sincos0

0001

1

z

y

x

y

z

)0,1,0(

)1,0,0(


3D Rotation (3/4)

Rotation about y-axis Ry(ß)P

1000

0cos0sin

0010

0sin0cos

1

z

y

x

x

z

)0,0,1(

)1,0,0(


3D Rotation (4/4)

Rotation about z-axis Rz(ß)P

1000

0100

00cossin

00sincos

1

z

y

x


3D Shear

xy Shear: SHxyP

1000

0100

010

001

y

x

sh

sh

1

z

y

x

x

z

y

x

z

y


Rotation About An Arbitary Axis (1/3)

1. Translate one end of the axis to the origin

2. Rotate about the y-axis and angle

3. Rotate about the x-axis through an angle

Z

P1

P2

Y

X

b

a

c

u1

u2

u3

ß

U



Z

P1

P2

Y

X

b

a

c

u1

u2

u3

ß

U

Z

Y

X

b

a

c

u1

u2

u3

ß

U Z

Yµ

a

u2

X

4. When U is aligned with the z-axis, apply the original rotation, RR, about the z-axis.

5. Apply the inverses of the transformations in reverse order.



T-1 Ry(ß) Rx(-µ) R Rx(µ) Ry(-ß) T P

Documents

2D and 3D Transformation