2.1. fundamental of computer graphics

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CHAPTER 2.1 : FUNDAMENTAL OF COMPUTER GRAPHICS

Prof. RATNADEEPSINH M. JADEJAAssistant Professor

Mechanical Department

Mechanical Engineering Department – School Of Engineering

GEOMETRIC TRANSFORMATIONS

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All the changes performed on the graphic image are done by changing the database ofthe original picture.These changes are called as transformations.

Geometric transformations are used for the following purposes:1. They are a must in the creation of a model, or the purpose of viewing it.2. They are used to express location of objects relative to others.3. Geometric transformations can be used to view an object from different positions and

directions.4. They can be used to perform certain transformations such as translate/move, rotate,

scale and mirror.5. They can also be used to obtain orthographic and perspective views of a model.6. Geometrical Transformations with the help of kinematic analysis can be used to create

animation to study the motion of the model.


FORMULATION

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P2(x2,y2)

P1(x1,y1)

Y

X

X2

X1

Y2

Y1A point represented by a 1 × 2 matrix

P1 = [x1,y1]

Similarly, a line can be represented as :

L = �1�2

= �1 �1�2 �2

Line represented after transformation

L’ = L [TM]

Where, [TM] = Transformation Matrix

0


TRANSLATION

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(x1, y1)

(x2, y2)

(x1’, y1’)

(x2’, y2’) Translation Matrix T = [tx ty]

Equation of line L = �1 �1�2 �2

Transferred line represented as

L’ = L + [T] = �1 �1�2 �2

+ [tx ty]

X

Y

0


TRANSLATION

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A(1,1)

B(7,3)

C(1,4)

A(4,3)

B(4,1)

C(4,6)

A triangle translated 3 unit in x direction and 2 unit in y direction.Triangle ABC represented as

ΔABC = 1 14 11 4

Translation equation of triangle is

ΔA’B’C’ = ΔABC + [tx ty]

ΔA’B’C’ = 1 14 11 4

+ [3 2]

= 4 37 34 6

X0

Y


ROTATION

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X

Y

P(x,y)

P’(x’,y’)

r

r

x=r cosф

y=r

sin

ф

X’=r cos(Ɵ+ф)

y’=r

sin

(Ɵ+ф

)

ф

Ɵ

Point P is represented asP = [x y]

= [r cos ф r sin ф]

The rotated point be represented as :P’ = [r cos(Ɵ+ф) r sin(Ɵ+ф)]

= [r (cos Ɵcos ф – sinƟ sin ф) r (sin Ɵ cos ф + cos Ɵ sin ф )

= [(x cos Ɵ – y sin Ɵ) (x sin Ɵ + y cos Ɵ)

This can be expressed as :

P’ = [x y] cos Ɵ sin Ɵ

−��Ɵ cos Ɵ= = P × [R]

where R is rotation matrix

R = cos Ɵ sin Ɵ

−��Ɵ cos ƟIf rotation is anticlockwise then Ɵ is +VEIf rotation is clockwise then Ɵ is -VE


SCALING

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B’(4,2)

A(1,1)

C(1,2)A’(2,2)

B(2,1)

C’(2,4)

X0

Y Scaling transformation alters the size of the object.Scaling can be uniform(in X and Y direction) or non-uniform (either in x direction or in Y direction.

Scaling in X = SxScaling in Y = SyExample: A uniform scaling factor Sx = Sy = 2 ΔA’B’C’ = ΔABC · S

ΔA’B’C’ = 1 12 11 2

· �� 00 ��

= 2 24 22 4


REFLECTION (MIRROR)

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Reflection transformation equation.P’ = P · M

Where M is a reflection matrixX

Y

1 2

3

1’2’

3’

(a) Reflection about Y axis

M = −1 00 1

1 2

3

1’ 2’

3’

(b) Reflection about X axis

M = 1 00 −1 1 2

31’

2’ 3’ (c) Reflection about y = x axis

M = 0 11 0

Reflection is the process of obtaining a mirror of the original shape


REFLECTION (MIRROR)

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Reflection is the process of obtaining a mirror of the original shape

Reflection transformation equation.P’ = P · M

Where M is a reflection matrix

X

Y

(d) Reflection about y = -x

M = 0 −1

−1 0

(e) Reflection about x and y axis

M = −1 00 −1

X

Y

1 2

3

1’

2’3’

1 2

3

1’2’

3’

1’’2’’

3’’


SHEAR

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This transformation causes the image to slant.

The y shear preserves all the x coordinate values but shifts the y values.

The x shear preserves all the y coordinate values but shifts the x values.

P’ = P · SH

SH = 1 �0 1

for y shear

SH = 1 0� 1

for x shear

X

Y

A B

CD

A’ B’

C’D’


HOMOGENEOUS COORDINATES

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P (x, y) which can be expressed as P (x’ y’ h)

Where x = x’/h and y = y’/h

Where h is non zero number. A convenient value for h = 1.

So in homogeneous coordinate system point P is expressed as P (x, y, 1)

The general transformation matrix is also modified to a 3 × 3 matrix expressed in the form :

� � 0� � 0� � 1


e). Shear :

HOMOGENEOUS COORDINATES

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The important matrix transformations in the homogeneous coordinate system will now be modified as under.

T = 1 0 00 1 0�� 1

a). Translation : b). Rotation :

R = �� Ɵ sin Ɵ 0

− sin Ɵ cos Ɵ 00 0 1

c). Scaling :

S = �� 0 00 �� 00 0 1

d). Reflection / Mirror :

M = � � 0� � 00 0 1

For y shear, SHy = 1 � 00 1 00 0 1

For x shear, SHx = 1 0 0� 1 00 0 1


INVERSE TRANSLATIONS

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T = � � 0� � 0� � 1

Find out the inverse matrix of T

inv � � 0� � 0� � 1

= �

��

� −� 0−� � 0

�� − �� − �� − ��

inv 1 0 00 1 0�� 1

= 1 0 00 1 0

−�� −�� 1

Inverse Transformation for Translation:

Inverse Transformation for Rotation:

inv cos Ɵ sin Ɵ 0

− sin Ɵ cos Ɵ 00 0 1

= cos Ɵ − sin Ɵ 0sin Ɵ cos Ɵ 0

0 0 1

= cos(− Ɵ) sin(−Ɵ) 0

−sin(−Ɵ) cos(− Ɵ) 00 0 1

Inverse Transformation for Scaling:

inv �� 0 00 �� 00 0 1

= 1/�� 0 0

0 1/�� 00 0 1


CONCATENATION OR COMPOSITE TRANSFORMATION

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With homogeneous coordinates, composite transformations (more than one) can be done by calculating the matrix product for all individual transformations.

First let us consider some properties of matrices.1. As general rule, matrix multiplication is not commutative. A·B ≠ B · A

2. Matrix multiplication is associative. For any three matrices, A, B and C,

A · B · C = (A · B) · C = A · (B · C)


ROTATION ABOUT ANY POINT

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1

1’

2

2’

33’

(a) Rotation about Origin

+

(a) Rotation about any given point


ROTATION ABOUT ANY POINT

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+

P (x, y)

x

Y

a) Original shape

+

+

b) Translate to Origin

+

+

c) Rotation about Origin

+

d) Translate back to Original Point

P’ = P · T · R · (inv T)


SCALING ABOUT ANY POINT

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+

P(X, Y)

X

Y

a) Original shape

+

b) Translation to Origin

+ +

+

c) Scaling about Origin

+

d) Translate back to Original Point

P’ = P · T · S · (inv T)


REFLECTION ABOUT ANY AXIS y = mx + c

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a) Original Shape b) Translation to Origin

c) Rotation

d) Reflection

e) Inverse Rotation f) Inverse Translation

Original Image

Mirror Image

P’ = P · T · R · M · [inv R] · [inv T]


Example

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Ex 1. A rectangle is formed by four points ABCD whose coordinates are: A (50, 50), B(100, 50),C(100, 80) and D(50,80). Calculate the new coordinates of the rectangle in reduced size usingthe scaling factors Sx = 0.5 and Sy = 0.6

Ans. =

25 30 150 30 15025

4848

11


Example

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Ex 2. Write a 3 × 3 transformation matrix for the following effects:a). Scale the image to be twice as large and then translate it 1 unit to the left.b). Scale X direction to be half as large and then rotate anticlockwise by 90° about the originc). Rotate anticlockwise about origin by 90° and then scale the X direction by half as large.d). Translate down 0.5 unit, right 0.5 unit, and then rotate anticlockwise by 45°.

Ans. = a) G = S · T

= 2 0 00 2 0

−1 0 1

Ans. = b) G = S · R

= 0 0.5 0

−1 0 00 0 1

Ans. = c) G = R · S

= 0 1 0

−0.5 0 00 0 1

Ans. = d) G = T · R

= 0.707 0.707 0

−0.707 0.707 00.707 0 1


Example

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Ex. 3 A triangle ABC has its Vertices at A(0, 0), B(4, 0), and C(2, 3). It is to be translated by4 units in X direction, and 2 units in Y direction, then it is to be rotated in anticlockwisedirection about the new position of point C through 90°. Find the new position of thetriangle

Ans. ΔA’B’C’ = ΔABC · T

= 4 2 18 2 16 5 1

ΔA’’B’’C’’ = ΔA’B’C’ · T · R · (inv T)

= 9 3 19 7 16 5 1


Example

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Ex. 4 Triangle PQR has vertices as P(2, 4), Q(4, 6) and R(2, 6). It is Desired to reflect through anarbitrary line L whose equation is y = 0.5 X + 2. Calculate the new vertices of triangle and showthe results graphically.

ΔPQR = 2 4 14 6 12 6 1

Reflection about y = 0.5 X +2

ty = -2,m = slope =tan Ɵ

Ɵ = 26.565°

ΔP’Q’R’ = ΔPQR · T · R · M · (inv R) · (inv T)

ANS = 2.8 2.4 15.6 2.8 14.4 1.2 1


THREE DIMENSIONAL TRANSFORMATIONS

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X

Z

Y

a). Translation :

[x’ y’ z’ 1] = [x y z 1] ·

1 0 0 00 1 0 00 0 1 0�� 1

b). Rotation :

Rz =

��Ɵ ��Ɵ 0 0−��Ɵ ��Ɵ 0 0

0 0 1 00 0 0 1

i). Rotation about Z axis: ii). Rotation about X axis :

Rx =

1 0 0 00 ��Ɵ ��Ɵ 00 −��Ɵ ��Ɵ 00 0 0 1

ii). Rotation about Y axis :

Ry =

��Ɵ 0 −��Ɵ 00 1 0 0

��Ɵ 0 ��Ɵ 00 0 0 1


THREE DIMENSIONAL TRANSFORMATIONS

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c). Scaling :

S =

�� 0 0 00 �� 0 00 0 �� 00 0 0 1

d). Reflection :

Mxy =

1 0 0 00 1 0 00 0 −1 00 0 0 1

i). Reflection about XY plane: ii). Reflection about YZ plane :

Myz =

−1 0 0 00 1 0 00 0 1 00 0 0 1

ii). Reflection about XZ plane :

Mxz =

1 0 0 00 −1 0 00 0 1 00 0 0 1

X

Y

Z


Example

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Ex 5. State the homogeneous transformation matrices for the following operations.a) Rotation through 120° about origin.b) Translation through 10 and -20 units along the x and y direction respectively.c) Rotation through 30° about origin.

Find concatenated matrix if the operations are done in above sequence. Will the orderin which the operations are done affect the end result?


Example

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Ex 6. A triangle ABC having coordinates A (3, 4, -2) and B (-4, 6, 3), C (-6, 4, 3) is to berotated about X axis by 20° anticlockwise. Determine the new coordinates of the triangle.


Example

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Ex 7. Given a point P (1, 3, -5), find :a) Transformed point P’ if P is Translated by 2 unit in X 3 unit in Y and -4 unit in Z and

then rotated by 30° about the Z axis.b). Same as in (a), but the point P is rotated first and then translated.c). Is the final point P’ the same in both (a) and (b) ? Explain your answer.

2.1. fundamental of computer graphics

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