CE-201CE201_Week9-10_2010

INDIAN INSTITUTE OF TECHNOLOGY ROORKEEDEPARTMENT OF CIVIL ENGINEERING

CE‐201: Computer Aided GraphicsWeek 09‐10

Rajat Rastogi

[email protected]@iitr.ernet.in

Previous LecturePrevious Lecture

• Output PrimitivesOutput Primitives– Line

Fill– Fill

– Text

Ch t– Character

Lecture outlineLecture outline

• 2‐D Geometric Transformation2 D Geometric Transformation– Basic Transformation

Composite Transformation– Composite Transformation

– Other Transformation

TransformationsTransformations

• Transformation of an object means transforming allTransformation of an object means transforming all its points

• Transformation of a polygon can be achieved by p yg ytransforming its vertices

4

Basic TransformationsBasic Transformations

• ScaleScale

• Rotate

l• Translate

Basic Transformations…1Basic Transformations…1

ScaleModeling Coordinates

ScaleTranslatey

x

ScaleRotateTranslateTranslate

World Coordinates

Basic Transformations…2Basic Transformations…2Modeling Coordinates

y

x

Initial locationat (0, 0) withx- and y-axesaligned

Basic Transformations…3Basic Transformations…3Modeling Coordinates

y

x

Scale .3, .3Rotate -90Translate 5, 3


Modeling Coordinates

y

x



Modeling Coordinates

y

x


World Coordinates

ScalingScaling

• A scaling transformation alters the size of an object.A scaling transformation alters the size of an object.

• Scaling a coordinate means multiplying each of its components by a scalar or scaling factor (sx and sy)p y g ( x y)

• Transformed coordinates will be:x’ = x . sx y’ = y . sy (1)x y y y ( )

• This can be written in matrix form as

⎤⎡⎤⎡⎤⎡ xsx 0'(2)⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡yx

ss

yx

y

x .0

0''

Scaling…1Scaling…1

• Uniform scaling means this scalar is assigned the g gsame value for all the components

× 2


• Non‐uniform scaling: Different scalars are used for gthe two coordinates

X × 2,Y × 0.5

• Scaling also reposition the transformed objects with respect to the coordinate originrespect to the coordinate origin.


• A fixed point is used to control the change in positionA fixed point is used to control the change in position of an object after Scaling operation. This fixed point may be a vertex or a centre or any other position on an object. If (xf, yf) are the coordinates of a fixed point, then,

x’ = xf + (x – xf). sx y’ = yf + (y – yf). sy (3)

x’ = x . sx + (1 – sx). xfy’ = y s + (1 s ) y (4)y = y . sy + (1 – sy). yf (4)

– This is a combination of multiplicative and additive terms


• General form of a scaling matrix with respect to aGeneral form of a scaling matrix with respect to a fixed point P(h, k).

TSTS vbavPba TSTS .. ,,, −=

⎟⎞

⎜⎛

⎟⎞

⎜⎛

⎟⎞

⎜⎛ − 010001 hah

⎞⎛

⎟⎟⎟

⎠⎜⎜⎜

⎝⎟⎟⎟

⎠⎜⎜⎜

⎝⎟⎟⎟

⎠⎜⎜⎜

⎝

−=100

1010000

10010 kbk

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−−

=100

00

kbkbhaha

⎠⎝ 100

2‐D Rotation2 D Rotation

• Two dimensional rotation is applied to an object byTwo dimensional rotation is applied to an object by repositioning it along a circular path in the xy plane.

• Two information are required: P΄q– Rotation angle θ– Pivot point or rotation point (xr, yr)

θ

P(Xr yr) P

• Positive values of rotation angle define counterclockwise rotation about the pivot point and negative values rotate object in clockwise direction

2‐D Rotation…1

x = r cos (φ)

2 D Rotation…1

y = r sin (φ) (5)

x’ = r cos (φ + θ)y’ = r sin (φ + θ) (6)

(x’, y’)

• Trig Identity…x’ = r cos(φ) cos(θ) – r sin(φ) sin(θ)(x, y)

r x r cos(φ) cos(θ) r sin(φ) sin(θ)y’ = r sin(φ) sin(θ) + r cos(φ) cos(θ) (7)

• Substituter

r

• Substitute…x’ = x cos(θ) ‐ y sin(θ)y’ = x sin(θ) + y cos(θ) (8)

φθφ

2‐D Rotation…22 D Rotation…2

• This is easy to capture in matrix form:y p

(9)( ) ( )( ) ( ) ⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ −=⎥

⎦

⎤⎢⎣

⎡yx

yx

θθθθ

cossinsincos

''

• Even though sin(θ) and cos(θ) are nonlinear functions of θ,

( ) ( ) ⎦⎣⎦⎣⎦⎣ yy

– x’ is a linear combination of x and y– y’ is a linear combination of x and y

• If pivot point is (x y ) instead of (0 0) then• If pivot point is (xr, yr) instead of (0, 0), thenx’ = xr + (x – xr) cosθ ‐ (y – yr) sinθy’ = yr + (x – xr) sinθ + (y – yr) cosθ (10)

Example 1Example 1

• Find the matrix that ⎞⎛represent rotation of an

object by 30o about origin and the new coordinates of ⎟

⎟⎟⎟⎞

⎜⎜⎜⎜⎛

−=⎥

⎦

⎤⎢⎣

⎡ −

3121

23

30cos30sin30sin30cos

and the new coordinates of a point (2, ‐4) after rotation

⎟⎞

⎜⎛ +⎟

⎞⎜⎛⎟⎟⎞

⎜⎜⎛

−

⎟⎠

⎜⎝

23221

23

22

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎟

⎠⎜⎜⎜

⎝32123

42

23

21

22

Example 2Example 2

• Describe the transformation that rotates an objectDescribe the transformation that rotates an object point, Q(x,y), θ0 about a fixed center of rotation P(h, k)– Transformation Rθ,p is done in 3 steps:

1. Translate so that the centre of rotation P is at the origin,

2. Perform a rotation of θ degrees about the origin, and

3. Translate P back to (h, k).

Find R b sing transformation• Find Rθ,p by using transformation:

Rθ,p = T‐v × Rθ × Tv

Example 2…1Example 2…1

• Describe the transformation that rotates an objectDescribe the transformation that rotates an object point, Q(x,y), θ0 about a fixed center of rotation P(h, k)– General form of the matrix would be:

⎟⎞

⎜⎛

⎟⎞

⎜⎛ −

⎟⎞

⎜⎛ − 010sincos01 hh θθ

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛×⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛×⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−=

10010

1000cossin

10010, kkR p θθθ

⎟⎟⎞

⎜⎜⎛

+−−++−−

= )cossin(cossin)sincos(sincos

kkhhkh

θθθθθθθθ

⎟⎟

⎠⎜⎜

⎝ 100)(

TranslationTranslation

• Translation transformation reposition an object alongTranslation transformation reposition an object along a straight line path

• Translation distance is added to the original gcoordinates to move to a new position

• A point moves from (x, y) to (x’, y’), thenp y y– x’ = x + tx, y’ = y + ty (11)

• Translation distance pair (tx, ty) isP’

T tx ycalled a translation vector or a shift vector

P

ty

tx

Translation…1Translation…1

• In matrix form, it can be defined as:In matrix form, it can be defined as:

(12)⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

y

xtt

yx

yx

'

'

• This is a rigid body transformation that moves objects without deformation.

⎦⎣⎦⎣⎥⎦⎢⎣ yyy

2

1

2D Transformation2D Transformation

⎤⎡⎤⎡ +⎤⎡⎤⎡ '(x’,y’)

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡++

=⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡''

yx

byax

ba

yx

),( ba(x,y)

( ) ( )( ) ( ) ⎥

⎤⎢⎡⎥⎤

⎢⎡ −

=⎥⎤

⎢⎡ xx θθ sincos'

⎥⎤

⎢⎡⎥⎤

⎢⎡

=⎥⎤

⎢⎡ xax 0'

( ) ( ) ⎥⎦

⎢⎣⎥⎦

⎢⎣

⎥⎦

⎢⎣ yy θθ cossin'⎥

⎦⎢⎣⎥⎦

⎢⎣

⎥⎦

⎢⎣ yby 0'

scaling matrix rotation matrixscaling matrix rotation matrix

24

2D Transformation…12D Transformation…1

⎤⎡⎤⎡ +⎤⎡⎤⎡ '(x’,y’) vector addition

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡++

=⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡''

yx

byax

ba

yx

),( ba(x,y)

( ) ( )( ) ( ) ⎥

⎤⎢⎡⎥⎤

⎢⎡ −

=⎥⎤


⎥⎤

⎢⎡⎥⎤

⎢⎡

=⎥⎤

⎢⎡ xax 0'

matrix multiplicationmatrix multiplication

( ) ( ) ⎥⎦

⎢⎣⎥⎦

⎢⎣

⎥⎦


⎦⎢⎣⎥⎦

⎢⎣

⎥⎦

⎢⎣ yby 0'


25

2D Transformation…22D Transformation…2

⎤⎡⎤⎡ +⎤⎡⎤⎡ '(x’,y’) vector addition

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡++

=⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡''

yx

byax

ba

yx

),( ba(x,y)

( ) ( )( ) ( ) ⎥

⎤⎢⎡⎥⎤

⎢⎡ −

=⎥⎤


⎥⎤

⎢⎡⎥⎤

⎢⎡

=⎥⎤

⎢⎡ xsx x 0'

matrix multiplicationmatrix multiplication

( ) ( ) ⎥⎦

⎢⎣⎥⎦

⎢⎣

⎥⎦


⎦⎢⎣⎥⎦

⎢⎣

⎥⎦

⎢⎣ ysy y0'


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡''

yx

yx

dcba

26

⎦⎣⎦⎣⎦⎣ yydc

translation multiplication matrix??

Matrix RepresentationMatrix Representation

• Represents 2D transformation with matrixRepresents 2D transformation with matrix

– Multiply matrix by column vector to apply transformation to a pointtransformation to a point

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡ xdbax

''

dbyaxx +=

''

• In case a sequence of transformations have to be li d th h t b d i t b t

⎥⎦

⎢⎣⎥⎦

⎢⎣

⎥⎦

⎢⎣ ydcy' dycxy +='

applied, these have to be done in step by step procedure, applying one transformation at a time

27

Matrix Representation…1Matrix Representation…1

• A more efficient approach would be to combine theA more efficient approach would be to combine the transformations so that the final coordinates can be obtained directly from the initial coordinates.

• Transformations combined by multiplication

⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡ ihdb'⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡yx

kjih

gfed

dcba

yx

''

– This needs replacement of a 2x2 matrix by a 3x3 matrix

28

Homogeneous CoordinatesHomogeneous Coordinates

point in 2D cartesian• point in 2D cartesian

yy⎤⎡

29xx

yy⎥⎦

⎤⎢⎣

⎡yx

Homogeneous Coordinates...1Homogeneous Coordinates...1

• ‘w’ is a non‐zero value

• Infinite homogenous representations

• Simplest form (x, y, 1)

– Forms a line L in 3D

All homogeneous points– All homogeneous points on L represent same 2D Cartesian point

– Example: (2,2,1) = (4,4,2) = (1,1,0.5)HomogeneousHomogeneous CartesianCartesian

30

),,( wyx hh ),(wy

wx hh

Homogeneous Coordinates…2Homogeneous Coordinates…2

• The basic transformation matrices would be:The basic transformation matrices would be:

⎥⎥⎤

⎢⎢⎡ −

= 0)cos()sin(0)sin()cos(

θθθθ

otationR⎥⎥⎤

⎢⎢⎡

= 0000

y

xs

scaleS

⎥⎥⎦⎢

⎢⎣

=1000)cos()sin( θθotationR

⎥⎤

⎢⎡ 01 xt

⎥⎥⎦⎢

⎢⎣ 100

00 yscaleS

• Example: Translation⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

=100

10 y

xtranslationT

• Example: Translation

⎥⎥⎤

⎢⎢⎡

++

=⎥⎥⎤

⎢⎢⎡

∗+∗∗+∗

=⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

1111

1001

byax

byax

yx

ba

31

⎥⎥⎦⎢

⎢⎣⎥

⎥⎦⎢

⎢⎣⎥

⎥⎦⎢

⎢⎣⎥⎥⎦⎢

⎢⎣ 1111000 bybyyb


• Add a 3rd coordinate to every 2D pointAdd a 3rd coordinate to every 2D point– (xh, yh, w) represents a point at location (xh/w, yh/w)

– (x, y, 0) represents a point at infinity

– (0, 0, 0) is not allowed

2

y

1

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

Convenient coordinate system to represent many useful transformations

1 2 x


• Basic 2D transformations as 3x3 matricesBasic 2D transformations as 3x3 matrices

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

=⎥⎥⎤

⎢⎢⎡

1001

''

yx

tt

yx

y

x

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

=⎥⎥⎤

⎢⎢⎡

0000

''

yx

ss

yx

y

x

⎥⎥⎦⎢

⎢⎣⎥⎥⎦⎢

⎢⎣⎥

⎥⎦⎢

⎢⎣ 11001

yy y

Translate [P’ = T(tx, ty).P]

⎥⎥⎦⎢

⎢⎣⎥⎥⎦⎢

⎢⎣⎥

⎥⎦⎢

⎢⎣ 11001

yy y

Scale [P’ = S(sx, sy).P]

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

ΘΘΘ−Θ

=⎥⎥⎤

⎢⎢⎡

0cossin0sincos

''

yx

yx

x y x y

⎥⎥⎦⎢

⎢⎣⎥⎥⎦⎢

⎢⎣

ΘΘ⎥⎥⎦⎢

⎢⎣ 1100

0cossin1

yy

Rotate [P’ = R(θ).P][ ( ) ]

Composite TransformationsComposite Transformations

• Products of transformation matrices is termed asProducts of transformation matrices is termed as composition or concatenation of matrices

• This is formed by multiplying matrices in order, i.e. y p y g ,from right to left

• This means that each successive transformation matrix premultiplies the product of the preceding transformation matrices.

• Concatenation propertyA.B.C = (A.B).C = A. (B.C)

34

Composite Transformations…1Composite Transformations…1

• TranslationTranslation– If two successive translation vectors (tx1, ty1) and (tx2, ty2) are applied, then

P’ = T (tx2, ty2). {T(tx1, ty1).P}

P’ = {T (tx2, ty2). T(tx1, ty1)}.P = {T(tx1+tx2, ty1+ty2)}.P

⎥⎥⎥⎤

⎢⎢⎢⎡

++

=⎥⎥⎥⎤

⎢⎢⎢⎡

⎥⎥⎥⎤

⎢⎢⎢⎡

=• 1001

1001

.1001

12 21

21

1

1

2

2

yy

xx

y

x

y

xtttt

tt

tt

TT⎥⎦⎢⎣⎥⎦⎢⎣⎥⎦⎢⎣ 100100100

35


• ScalingScaling– Similarly, two successive scaling operations are dealt as

⎤⎡⎤⎡ 0000 12 ss

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

1000000

.1000000

),().,( 1

1

2

2

1122 y

x

y

x

yxyx ss

ss

ssSssS

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

1000.000.

21

21

yy

xxss

ss

⎥⎦⎢⎣ 100

36


• General Fixed Point ScalingGeneral Fixed Point Scaling– Translate object so that the fixed point coincides with the coordinate origin

– Scale the object with respect to the coordinate origin

– Use inverse translation to return on the original position

( ) ( ) ( ) ( )T(xf, yf).S(sx, sy).T(‐xf, ‐yf) = S(xf, yf, sx, sy)

⎥⎤

⎢⎡ −

⎥⎤

⎢⎡ −

⎥⎤

⎢⎡

⎥⎤

⎢⎡ )1(0010001 xfxfxf sxsxsx

⎥⎥⎥

⎦⎢⎢⎢

⎣

−=⎥⎥⎥

⎦⎢⎢⎢

⎣

−⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣ 100)1(0

10010.

10000.

10010 yfy

f

f

f

yf

fsysysy


• General Fixed Point ScalingGeneral Fixed Point Scaling

(xr, yr)

a b

c

a b

d

Example 2Example 2

• Magnify the triangle with vertices A (0, 0), B (1, 1) and C (5, 2) g y g ( , ), ( , ) ( , )to twice its size while keeping C (5, 2) fixed

Homogeneous coordinates Transformation Matrix (TRT-1)

⎟⎟⎞

⎜⎜⎛

210510

of vertices

⎥⎥⎤

⎢⎢⎡

−−

=⎥⎥⎤

⎢⎢⎡

−−

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

220502

210501

020002

210501

a s o a o a ( )

⎟⎟

⎠⎜⎜

⎝ 111 ⎥⎥⎦⎢

⎢⎣⎥

⎥⎦⎢

⎢⎣⎥⎥⎦⎢

⎢⎣⎥⎥⎦⎢

⎢⎣ 100100100100

Transformation of triangle Transformed vertices

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

−−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−

111202535

111210510

100)220

502 A (-5, -2)

B (-3, 0)⎟⎠

⎜⎝

⎟⎠

⎜⎝⎥⎦⎢⎣ 111111100 C (5, 2)


• RotationRotation

RR 0)21cos()21sin(0)21sin()21cos(

)().( 12 θθθθθθθθ

θθ ⎥⎥⎤

⎢⎢⎡

+++−+

=

PRP ).(

100

21'

12

θθ +=

⎥⎥⎦⎢

⎢⎣

40


suppose we wantsuppose we want Rotate(zRotate(z 90)90) Translate(2 3 0)Translate(2 3 0)

jFFhh

suppose we wantsuppose we want Rotate(z,Rotate(z,‐‐90)90) Translate(2,3,0)Translate(2,3,0)

jFFhh

i FFhh

FFWW

FFWW

FFhhi

j FFWW

p'=R(z,−90)p p''= T(2,3,0)p'

41

p''= T(2,3,0)R(z,−90)p = TRp


• General Pivot Point RotationGeneral Pivot Point Rotation– Rotation about any selected point can be generated by performing the following operations

• Translate the object from pivot point to coordinate origin

• Rotate the object about the coordinate origin

• Translate the object so that the pivot point is returned to its j p poriginal position


43


• This transformation can be expressed asThis transformation can be expressed asT(xr, yr).R(θ).T(‐xr, ‐yr) = R(xr, yr, θ)

where, T(‐xr, ‐yr) = T‐1(xr, yr)r r r r

⎥⎥⎤

⎢⎢⎡

−−

⎥⎥⎤

⎢⎢⎡ −

⎥⎥⎤

⎢⎢⎡

1001

.0cossin0sincos

.1001

θθθθ

r

r

r

ryx

yx

⎥⎤

⎢⎡ +−−

⎥⎥⎦⎢

⎢⎣⎥

⎥⎦⎢

⎢⎣⎥

⎥⎦⎢

⎢⎣

i)(isin)cos1(sincos

100100100

θθθθθθθθ rr

rr

yx

yy

⎥⎥⎥

⎦⎢⎢⎢

⎣

−−=100

sin)cos1(cossin θθθθ rr xy


translate p translate p to originto origin

rotate about rotate about bb θ

rotate aboutrotate aboutoriginorigin

translate p translate p backback

p = (x,y)

to originto origin

θ

p by :p by :θ originorigin backback

p ( ,y)

FFWW

T(x,y,z)R(z,θ)T(−x,−y,−z)

45

Example 3Example 3

• Perform a 45o rotation of a triangle A (0, 0), B (1, 1)Perform a 45 rotation of a triangle A (0, 0), B (1, 1) and C (5, 2) about originHomogeneous coordinates Rotation Matrix

⎟⎟⎞

⎜⎜⎛

210510

of vertices

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

−

=⎥⎥⎤

⎢⎢⎡ −

022

022

22

045cos45sin045sin45cos

o a o a

⎟⎟

⎠⎜⎜

⎝ 111

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

=⎥⎥⎦⎢

⎢⎣ 100

022

100045cos45sin

Transformation Matrix

⎟⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎜⎛

=⎟⎟⎟⎞

⎜⎜⎜⎛

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

−

22720

22300

210510

022

22

022

22

Transformed vertices

A (0, 0), B (0, √2), C (3√2/2, 7√2/2)

⎟⎟⎟

⎠⎜⎜⎜

⎝

⎟⎠

⎜⎝

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

1112

11110022

Example 4Example 4

• Perform a 45o rotation of a triangle A (0, 0), B (1, 1)Perform a 45 rotation of a triangle A (0, 0), B (1, 1) and C (5, 2) about point (‐1, ‐1)Homogeneous coordinates Transformation Matrix (TRT-1)

⎟⎟⎞

⎜⎜⎛

210510

of vertices

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

−

−−

=⎥⎥⎤

⎢⎢⎡

−−

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

−

⎥⎥⎤

⎢⎢⎡

−−

)12(22

122

22

110101

022

022

22

110101

a s o a o a ( )

⎟⎟

⎠⎜⎜

⎝ 111⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

⎥⎥⎦⎢

⎢⎣

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

⎥⎥⎦⎢

⎢⎣ 100

)12(22

100110

100

022

100110

Transformation of triangleTransformed

⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎛

−−−

−−−

=⎟⎟⎞

⎜⎜⎛

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

−

−−

)12

29()122()12(

)12

23(11

210510

)12(22

22

122

22

Transformed vertices

A (-1, √2-1)

⎟⎟⎟⎟

⎠⎜⎜⎜⎜

⎝

⎟⎟

⎠⎜⎜

⎝

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

111

)2

()()(111100

)(22 B (-1, 2√2-1)

C (3√2/2-1, 9√2/2-1)


• Scaling and RotationScaling and Rotation

⎥⎥⎤

⎢⎢⎡

+−−+

=− 0cossinsincos)(0sincos)(sincos

)().,().( 22

2112

122

22

1

211 θθθθ

θθθθθθ ssss

ssssRssSR

s2

• Example: s = 1 s = 2 and θ = 45o⎥⎥⎥

⎦⎢⎢⎢

⎣ 100)()(),()( 211221

θ• Example: s1 = 1, s2 = 2 and θ = 45o s1

(1/2, 3/2)(2, 2)

(0, 1) (1, 1)

(3/2, 1/2)

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


• Translation, Rotation and ScalingTranslation, Rotation and Scaling

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

=⎥⎥⎤

⎢⎢⎡

'

'

yx

trsrsrstrsrsrs

yx xxyxx

=),,,().,,().,( yxccccyx ssyxSyxRttT θ

⎥⎥⎦⎢

⎢⎣⎥

⎥⎦⎢

⎢⎣⎥

⎥⎥

⎦⎢⎢⎢

⎣ 1.

1001ytrsrsrsy yyyyx

⎥⎥⎤

⎢⎢⎡

+−−++−−

sin)cos1(cossinsin)cos1(sincos

),,,().,,().,(

yxcycyx

xycxcyx

yxccccyx

tsxsysstsysxss

ssyxSyxtt

θθθθθθθθ

θ

⎥⎥⎦⎢

⎢⎣ 100


• Rigid‐body transformation matrix (involving onlyRigid body transformation matrix (involving only Translation and Rotation)

⎤⎡⎤⎡⎤⎡ ' xtrrrx

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

1.

1001

' yx

trrrtrrr

yx

yyyyx

xxyxx

⎥⎤

⎢⎡ ++−−

=

sin)cos1(sincos

),,().,(

xrr

rryx

tyx

yxRttT

θθθθ

θ⎥⎦⎢⎣

⎥⎥⎥

⎦⎢⎢⎢

⎣

+−−100

sin)cos1(cossin yrr txy θθθθ

Other TransformationsOther Transformations

• ReflectionReflection

• Shear

ReflectionReflection

• Reflection transformation produces a mirror image of p gan object.

• Rotation is 180o about axis of reflection• This axis may be in x‐y plane or perpendicular to that. For reflection axis perpendicular to x‐y plane, the rotation path is in x y planethe rotation path is in x‐y plane.

• Reflection about x axis keeps x coordinates same but flips y coordinate. Similarly, for reflection about y p y y, yaxis, the y‐coordinates remain the same and x‐coordinates are flipped.

Reflection…1Reflection…1

• Reflection across x axisReflection across x axis

x x

– Transformation matrix

⎤⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

100010001

53

⎥⎦⎢⎣ 100


• Reflection across y axisReflection across y axisy y


⎤⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

100010001

54

⎥⎦⎢⎣ 100


• Reflection across axis perpendicular to xy planeReflection across axis perpendicular to xy planey y

x x


⎤⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−

100010001

55

⎥⎦⎢⎣ 100

Reflection…4Reflection…4y

Original y Original

positiongposition

position

Reflected position

x

Reflected

x

pos t o

y

positionOriginal positionReflection about x‐axis Reflection about y‐axis

x

Reflected position

Reflection about axis perpendicular to xy plane

Reflection…5Reflection…5Reflected position

Original position

y

Original

position y

Reflected

x

Original position

x

position

x

Reflection about axisReflection about axis perpendicular to xy plane and passing through pivot point Prfl

Reflection about line y=x

⎥⎥⎤

⎢⎢⎡

001010

⎥⎥⎦⎢

⎢⎣ 100

001


• To obtain transformation matrix for reflection about the diagonal y = x, the possible sequence of transformations is – Rotate clockwise the diagonal by 45o making it equivalent to x‐axis– Perform reflection with respect to x‐axisp– Rotate counterclockwise by 45o to bring the line y = x to its original

position

• In case of any line y = mx + b, the sequence would bey y , q– Translate the line to pass through coordinate origin– Rotate the line onto one of the coordinate axis– Reflect the object about that axisReflect the object about that axis– Inverse rotate the line– Inverse translate the line back to its original position

Example 5Example 5

• Reflect diamond shaped polygon whose vertices are A (‐1, 0), p p yg ( , ),B (0, ‐2), C (1, 0), D (0, 2), about y=2

Homogeneous coordinates Transformation Matrix (TRMR-1T-1)

⎟⎟⎞

⎜⎜⎛

20

02-0101-

of vertices

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100410001

100210

001

100010001

100010001

100010001

100210001

a s o a o a ( )

⎟⎟

⎠⎜⎜

⎝ 1111⎥⎦⎢⎣⎥⎦⎢⎣⎥⎦⎢⎣⎥⎦⎢⎣⎥⎦⎢⎣⎥⎦⎢⎣ 100100100100100100

Transformation of polygon Transformed vertices

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

120

111464101-

120

11102-0101-

100410001 A (-1, 4)

B (0, 6)⎟⎠

⎜⎝

⎟⎠

⎜⎝⎥⎦⎢⎣ 11111111100 C (1, 4)

D (0, 2)

Example 6Example 6

• Reflect diamond shaped polygon whose vertices are A (‐1, 0), p p yg ( , ),B (0, ‐2), C (1, 0), D (0, 2), about x=2

Homogeneous coordinates Transformation Matrix (TMT-1)

⎟⎟⎞

⎜⎜⎛

20

02-0101-

of vertices

⎥⎥⎤

⎢⎢⎡−

=⎥⎥⎤

⎢⎢⎡ −

⎥⎥⎤

⎢⎢⎡−

⎥⎥⎤

⎢⎢⎡

010401

010201

010001

010201

a s o a o a ( )

⎟⎟

⎠⎜⎜

⎝ 1111 ⎥⎥⎦⎢

⎢⎣

=⎥⎥⎦⎢

⎢⎣⎥⎥⎦⎢

⎢⎣⎥⎥⎦⎢

⎢⎣ 100

010100010

100010

100010


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

124

11102-0345

120

11102-0101-

100010401 A (5, 0)

B (4, -2)⎟⎠

⎜⎝

⎟⎠

⎜⎝⎥⎦⎢⎣ 11111111100 C (3, 0)

D (4, 2)

Example 7Example 7

• Reflect diamond shaped polygon whose vertices are A (‐1, 0), p p yg ( , ),B (0, ‐2), C (1, 0), D (0, 2), about y = x + 2

Homogeneous coordinates Transformation Matrix (TMT-1)

⎟⎟⎞

⎜⎜⎛

20

02-0101-

of vertices

⎥⎥⎤

⎢⎢⎡ −

201210

a s o a o a ( )

⎟⎟

⎠⎜⎜

⎝ 1111 ⎥⎥⎦⎢

⎢⎣ 100

201


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −

120

1113212-4-2-

120

11102-0101-

100201210 A (-2, 1)

B (-4, 2)⎟⎠

⎜⎝

⎟⎠

⎜⎝⎥⎦⎢⎣ 11111111100 C (-2, 3)

D (0, 2)

ShearShear

• A transformation that distorts the shape of an objectA transformation that distorts the shape of an object such that the transformed shape appears as if the object were composed of internal layers that have been caused to slide over each other

• Two distinct shear transformations are:– Shifting coordinate x values

– Shifting coordinate y values

Shear…1Shear…1

• Shear along x axisShear along x axis– push points to right in proportion to height if shx is positive

y y

x x

T f d di t iti d t i

x x

⎤⎡ 01 h– Transformed coordinate positions and matrix

x’ = x + shx.y y’ = y⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

10001001 xsh

63

⎦⎣

Shear…2Shear…2

• Shear along x axis Sh = 1 5Shear along x axis Shx = 1.5

– Shear in x‐direction relative to other reference lines• Transformed coordinate positions

x’ = x + shx(y ‐ yref) y’ = y

⎤⎡1 hh Shx = 0.5

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −

100010.1 refxx yshsh x

yref = ‐1

64

⎦⎣

yref = ‐1 yref = ‐1

Shear…3Shear…3

• Shear along y axis relative to line x = xrefShear along y axis relative to line x xref– Transformed coordinate positions

x’ = x y’ = y + shy(x ‐ xref)y ref

⎥⎥⎥⎤

⎢⎢⎢⎡

− .1001

refyy xshsh⎥⎥⎦⎢

⎢⎣ 100

Shy = 0.5 y xref = ‐1

65xref = ‐1

xref = ‐1

Example 8Example 8

• Shear transformation on a square A (0, 0), B (1, 0), C (1, 1) and q ( , ), ( , ), ( , )D (0, 1) when shx = 2 and shy = 3

C

CD CD

B

D

A B

D

A B

D

A

D

C

B

A

D

Affine TransformationsAffine Transformations

• affine transforms are combinations of ⎤⎡⎤⎡⎤⎡ b'affine transforms are combinations of– linear transformations

– translations ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡yx

fedcba

yx

100''

translations

• properties of affine transformations– origin does not necessarily map to origin

⎥⎦⎢⎣⎥⎦⎢⎣⎥⎦⎢⎣ ww 100

origin does not necessarily map to origin

– lines map to lines

– parallel lines remain parallelp p

– ratios are preserved

– closed under composition

67

p

Transformation HierarchiesTransformation Hierarchies

• Scene may have a hierarchy of coordinate systemsScene may have a hierarchy of coordinate systems– Stores matrix at each level with incremental transform from parent’s coordinate system

• Scene graphroadroad

stripe1stripe1 stripe2stripe2 ...... car1car1 car2car2 ......

68

w1w1 w3w3w2w2 w4w4

Transformation Hierarchy Example 1Transformation Hierarchy Example 1

worldworld

torsotorso

headheadRUarmRUarm

RLarmRLarm

RUlegRUleg

RLlegRLleg

LUarmLUarm

LLarmLLarm

LUlegLUleg

LLlegLLleg

RhandRhandRfootRfoot LhandLhandLfootLfoot

69

trans(0.30,0,0) rot(z, )trans(0.30,0,0) rot(z, ) θ

Transformation Hierarchies…1Transformation Hierarchies…1

• Hierarchies don’t fall apart when changedHierarchies don t fall apart when changed

• Transforms apply to graph nodes beneath

70

ReadingsReadings

• Hearn and Baker 204‐228Hearn and Baker, 204 228

• Xiang and Plastock, 68‐88

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