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Section III:

TRANSFORMATIONS

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2 x 1 matrix:

X

YGeneral Problem: [B] = [T] [A]

[T] represents a generic operator tobe a lied to the oints in A. T is the

geometric transformation matrix.

If A & T are known, the transformed

.

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General Transformation of 2D points:

xcax

' baxx TT

'

ydby ' dcyy 'cyaxx

' cyaxx

'

dybxy ' dybxy 'Solid bod transformations the above

equation is valid for all set of points and lines of

the ob ect bein transformed.

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S ecial cases of 2D Transformations:

1) T = identity matrix:a=d=1, b=c=0 => x'=x, y'=y

b=0, c=0 => x' = a.x, y' = d.y;, .

If a = d > 1 we have enlar ementIf, 0 < a = d < 1, we have compression;

If a = d, we have uniform scaling,else non-uniform scaling.

x

= = y

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YExam le

6

5of Scaling4

3

9

=21

1

1

6 Sy = 2

0

1 2 3 4 5 6 7 8 9

X

What if Sx and/ or Sy < 0 (are negative)?

.

reflections.

ote : ouse s ts pos t on re at ve to or g n

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Special cases of Reflections (|T| = -1)

Y=0 Axis (or X-axis) 10X=0 Axis (or Y-axis)

01

10

Y = X Ax s

01

Y = -X Axis

01

10

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Off dia onal terms are involvedin SHEARING;

xxca 'a = d = 1;

yydb

'let, c = 0, b = 2

cyaxx

'x' = x

y' = 2x + y ; dybxy 'y' depends linearly on x ; This effect is called

s ear.

=shear in this case is proportional to y-coordinate.

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ROTATION Y

5 sincos' xX 4 =cossin' yxY

In matrix form, this is : 2

)sin(-)cos( 1

)cos()sin( 1 2 3 4 5 XPositive Rotations: counter clockwise about

For rotations, |T| = 1 and [T]T

= [T]-1

.Rotation matrices are orthogonal.

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Special cases of Rotations

90 01

180

01 10

270 or -90 01

360 or 0

01

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Example - Transformation of a Unit Square

b+dY

d(1, 1)

xb

x

00 00

11

01S

babaSS '

10

dc

Area of the unit square after transformation

= - =Extend this idea for any arbitrary area.

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Y

6

4

2tx = 5

2

3

7y=

0 X

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Translations

B = A + Td , where Td = [tx ty]T

Where else are translations introduced?

1) Rotations - when objects are not centeredat the ori in.

2) Scaling - when objects/lines are notcen ere a e or g n - ne n ersec s eorigin, no translation.

Origin is invariant to Scaling, reflectionand Shear not translation.

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Note: we cannot directly represent translationsas matrix multiplication, as we can for:

Can we represent translations in ourgeneral transformation matrix?

Yes, by using homogeneous coordinates

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HOMOGENEOUS COORDINATES

'

db

x

*'Use a 3 x 3 matrix:

ww 100' x' = ax + cy + t

x' y

(x, y, w).

(x/w, y/w) are called the Cartesian

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Interpretation of

Homogeneous Coordinates

Ph (x, y, w)

1 2d x w, y w,

x

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1, 1, 1

& (x2, y2, w2) may represent the same point,

say, (1,2,3) & (3,6,9).

There is no unique homogeneous

All triples of the form (t.x, t.y, t.W) form aline in x,y,W space.

w=1 in this space.

W=0, are the points at infinity

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General Purpose 2D transformations in

snm Parameters involved in scaling, rotation,reflection and shear are: a, b, c, d

. , . ,

parameters:

parameters:

What about

, S ?

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COMPOSITE TRANSFORMATIONS

If we want to apply a series of, , ,We can do it in two ways:

1) We can calculate p'=T1*p, p''= T2*p','''=T * ''

2) Calculate T= T1*T2*T3, then p'''= T*p.

Method 2, saves large number of additionsand multi lications com utational time

needs approximately 1/3 of as many operations.Therefore we concatenate or com ose thematrices into one final transformation matrix,

and then apply that to the points.

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Translate the points

1021

tt 1, 1, 2, 2 100ca ng:

Similar to translations

Rotations:

1, 2(i) stick the (1+2) in for, or ca cu ate

1

or1

, t en2

or2multiply them.

Exercise: Both gives the same result work

u

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Rotation about an arbitrar

point P in space

As we mentioned before, rotations area lied about the ori in. So to rotate about

any arbitrary point P in space, translate sothat P coincides with the origin, then rotate,then translate back. Steps are:

Translate by (-Px, -P )

Rotate

x, y

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Rotation about an arbitrary

point P in space

P1 House at P1 Rotation by

P1

Translation of Translation

1 ac to 1

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point P in space

- - * *x

, y x, y

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Scalin about an arbitrar oint in S ace

Again, Translate P to the origin

ca e

rans a e ac

T = T P P * T S S *T -P -P

*

*x

100

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Reflection throu h an arbitrar line

Steps:

Reflection matrix

Reverse the rotation

Translate line back

1 T rflGenRfl

C ti it f T f ti

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Commutivity of Transformations

If we scale, then translate to the origin,

translate to origin, scale, translate back?

When is the order of matrixmultiplication unimportant?

When does T1 * T2 = T2 * T1?

1 2 2 1

T T

translation translation

rotation rotation

scale(uniform) rotation

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O r d e r : - -

,Trans ate, sca e

Rotate differentialDifferential scale,

scale rotate

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COORDINATE SYSTEMS

Screen Coordinates: The coordinate s stemused to address the screen (device coordinates)

World Coordinates: A user-defined application

units of measure, axis, origin, etc.

Window: The rectangular region of the worldthat is visible.

Viewport: The rectangular region of the screen

s ace that is used to dis la the window.

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Viewport2V

ewpor

X U

World Screen

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The overall transformation:

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The overall transformation:

Translate the window to the origin

Scale it to the size of the viewport

Translate it to the viewport location

);,(*),(*),( minminminmin yxTSSSVUTM yxWV ;minmaxminmax

VVS

xxUUSx

)*(0 minmin USxS xx

)*(0 minmin VSySM yyWV

Window Viewport Transformation

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Window Viewport Transformation

(Xmax, Ymax)Y

X Y

Window in WorldX

Window translatedX

Coordinates to origin

VMaximumScreenCoordinates

(Umin, Vmin)

Window Scaled to

U U

View ort Translatedsize to Viewport to final position

Exercise -Transformations of Parallel Lines

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Exercise -Transformations of Parallel Lines

Consider two parallel lines:1, 1 2, 2 (ii) C[X3, Y3] to B[X4, Y4].

Slope ofthe lines:

3412 YYYYm 3412

If the lines are

aT

m

m'transformed lines is:

cma