Computer Graphics, Chapter 4: 2D Geometrical Transformations

8/18/2019 Computer Graphics, Chapter 4: 2D Geometrical Transformations

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COMPUTER GRAPHICS2D GEOMETRICAL TRANSFORMATIONS

Dr. Zeeshan Bhatti

BSIT-PIV

Chapter 4

Institute of Information and Communication TechnologyUniversity of Sindh, Jamshoro BY: DR. ZEE SHAN BHATT I 1Foley & Van Dam, Chapter 5


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2D GEOMETRICAL

TRANSFORMATIONS

• Translation

• Scaling

• Rotation

• Shear

• Matrix notation

• Compositions

• Homogeneous coordinates


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2D GEOMETRICAL

TRANSFORMATIONSAssumption: Objects consist of points and lines.

A point is represented by its Cartesian coordinates:

P = ( x, y)Geometrical Transformation:

Let (A, B) be a straight line segment between the points A and B.

Let T be a general 2D transformation.

T transforms (A, B) into another straight line segment

(A’, B’), where:

A’=TA and

B’=TB


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TRANSLATION


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SCALE


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SCALE

How can we scale an object without moving its origin (lower leftcorner)?


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REFLECTION

• Special case of scale


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ROTATION

• Rotate(θ): (x, y) → (x cos(θ)+y sin(θ), -x sin(θ)+y cos(θ))


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ROTATION

• How can we rotate an object without moving its origin (lower leftcorner)?


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SHEAR

• Shear (a, b): (x, y) → (x+ay, y+bx)


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CLASSES OF TRANSFORMATIONS

• Rigid transformation (distance preserving):

Translation + Rotation

• Similarity transformation (angle preserving):Translation + Rotation + Uniform Scale

• Affine transformation (parallelism preserving):

Translation + Rotation + Scale + Shear

All above transformations

are groups where

Rigid ⊂ Similarity ⊂ Affine


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MATRIX NOTATION


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2D TRANSFORMATIONS

• 2D object is represented by points and lines that jointhem

• Transformations can be applied only to the the pointsdefining the lines

• A point ( x, y) is represented by a 2x1 column vector, sowe can represent 2D transformations by using 2x2

matrices:


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TRANSLATIONTo translate a point with coordinate (x1, y1) to a new location usingdisplacements (dx, dy), following equations can be used:

x2 = x1 +dx and

y2 = y1+dy

Here (x2, y2) is the new coordinate of the point. The above equationcan be represented in matrix notation as:

22

=

+

11

This matric notation can be put as a vector addiction:

V2 = T +V1

Where

v1 =11

V2 =22

=


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PROBLEM: CONSIDER A POLYGON WITH FOLLOWINGVERTEX. TRANSLATE IT IN (5, 7)V1= (5,6), V2=(10, 6), V3=(7, 12) V4=(14, 15)


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SCALE

The size of an object can be changed by the scaling transformation.Objects can be scaled either the X or Y direction, or both of themsimultaneously, scaling by scale factors sx and sy along the X and Ydirections can be expressed mathematically as

x2 = sxx1 and

y2 = syy1

Here (x2, y2) is the new coordinate of the point. The above equation canbe represented in matrix notation as:

22

=

x11

This matric notation can be put as a vector addiction:

V2 = T x V1

Where

v1

=1

1V

2

=2

2 =

0

0


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Reflection < 0 > Decrease Size < 1 > Increase Size

(No Object/Size 0) (Same Size)


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Scale

• Scale (a, b): (x, y)

x

y

(ax,

by)

ax

by

a

0

0

b

•If a or b are negative, we get reflection

• Inverse: S-1(a,b) = S(1/a, 1/b)


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ROTATION

Rotation can be performed in ananticlockwise or a clockwise direction.

The axis of rotation may be X-axis or Y-axisor any arbitrary line on the X-Y plane.

Rotations are performed about some fixedpoint called pivot point or centre of rotationor anchor point.

Rotations are measured by angulardisplacements, we measure coordinate valuein term of the radius of rotation and angleof rotation, i.e. polar coordinates.


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Rotation• Rotate((x, y)

(x cos()+y

sin(), -x sin()+y cos())

cos sin

cos

x x cos y sin

x sin y cos sin y

• Inverse: R-1(q) = RT(q) = R(-q)


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Shear

• Shear (a, b): (x, y) (x+ay, y+bx)

1

b

a

1

x

y

x

y

ay

bx


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Reflection• Reflection through the

1

0

the

y axis:

01

through

• Reflection x axis:

10

01

• Reflection through

y = x :

1

0-x :

0 1 • Reflection through y =

1

0

0

1


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Composition of Transformations

• A sequence of transformationscollapsed into a single matrix:

can be

x

y

x

y

A B C

D

•Note: Order of transformations is important!

translate rotate

rotate translate


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Translation

• Translation (a, b):

x

y

x

y

a

b

Problem: Cannot represent translationusing 2x2 matrices

Solution:Homogeneous Coordinates


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Homogeneous Coordinates

Rn

, Y R

n+1

:Is a mapping from( X

to y ) ,W ) ( x , ( tx , ty , t )

Note:same

All triples (tx, ty, t ) correspond to the

non-homogeneous point ( x, y )

Example (2, 3, 1) (6, 9, 3).

Inverse mapping:

X Y ) ( X , Y , W ,

W W


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TRANSLATION


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SHEAR


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THANKYOU

Q & A

BY: DR ZEE SHAN BHATT I 29

Referred Book

Computer Graphics: Principles and Practice in C,by J. D. Foley, A. Van Dam, S. K. Feiner, J. F. Hughes.Hardcover, 1200 pages, Addison-Wesley Pub Co; 2nd edition,

For Course Slides and Handouts

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Documents

Computer Graphics, Chapter 4: 2D Geometrical Transformations