Transefermation

Computer Graphics
12/28/16 T.L. SAHU CSE SRIT II RAIPUR 1
• Computer graphics are visual
representations of data displayed on a
monitor made on a computer. Computer
graphics can be a series of images (most
often called video) or a single image.

Transformations
Using Transformation
Parameters reposition and resize
the two dimensional objects

Contents
12/28/16 T.L. SAHU CSE SRIT II RAIPUR
1. 2D Transformations
2. 2D Translation
3. 2D Scaling
4. 2D Shearing
5. 2D Rotations
3

2D
Transformations

2D Transformations
Problem:
• Given a 2D object, the transformation is the change
in the object:
– Position (translation)
– Size (scaling)
– Orientation (rotation)
– Shapes (shear)
Solution:
• Construct a sequence of matrices that can be applied
to all the points of the object.12/28/16 T.L. SAHU CSE SRIT II RAIPUR 5

2D Transformations
• World Coordinates
• Translate
• Rotate
• Scale
• Viewport Transforms
• Putting it all together

Transformations
• Rigid Body Transformations - transformations that do not
change the object.
• Translate
– If you translate a rectangle, it is still a rectangle
• Scale
– If you scale a rectangle, it is still a rectangle
• Rotate
– If you rotate a rectangle, it is still a rectangle

Vertices
• We have always represented vertices as
(x,y)
• An alternate method is:
• Example:






=
y
x
yx ),(






=
8.4
1.2
)8.4,1.2(

2D Transformation and Matrices
(Matrix Multiplication)










++
++
++
=










×










ziyhxg
zfyexd
zcybxa
z
y
x
ihg
fed
cba
***
***
***
• Representation of points :
• How matrix multiplication takes place:
9

2D
Translations

Translation
• Translation - repositioning an object along
a straight-line path (the translation
distances) from one coordinate location to
another.
(x,y)
(x’,y’)
(tx,ty)

Translation
Translation is a process of changing the
position of an object in a straight line path
from one coordinate location to another.
x’= x+tx
y’=y+ty

Translation: Initial

Translation: Final

Translation Operation

Translation Operation
Coordinates:
Matrix Form:
y
x
tyy
txx
+=
+=
'
'






+





=





y
x
t
t
y
x
y
x
'
'
16

Translation
• Given:
• We want:
• Matrix form:
TPP
t
t
y
x
y
x
tyy
txx
ttT
yxP
y
x
y
x
yx
+=






+





=





+=
+=
=
=
'
'
'
'
'
),(
),(
1.4'
4.3'
2.8
1.7
1.4
7.3
'
'
2.81.4'
1.77.3'
)2.8,1.7(
)1.4,7.3(
=
=






+





−
−
=





+−=
+−=
=
−−=
y
x
y
x
y
x
T
P

Applying to Triangles
(tx,ty)

2D
Scaling

Scale
• Scale - Alters the size of an object.
• Scales about a fixed point
(x,y)
(x’,y’)

Scaling
• A scaling transformation changes the size of an object.
• Any positive numeric values are valid for scaling factor Sx
and Sy.
• Sx and Sy values <1 reduces the size of object.
• Sx and Sy values >1 produce an enlarged object.
• Sx and Sy values =1 size of object does not change.
• Sx = Sy : uniform scaling
• Sx ≠ Sy : differential scaling

Scaling: Initial

Scaling: Final

Scaling Operation

Scaling Operation
Coordinates:
Matrix Form:
y
x
syy
sxx
×=
×=
'
'












=





y
x
s
s
y
x
y
x
0
0
'
'
P(x, y) ->P’ (x’, y’)
P’ -> S . P
25

Non-Uniform/Differential
Scalin’
(x,y)
(x’,y’)
S=(1,2)

Scale
• Given:
• We want:
• Matrix form:
PSP
y
x
s
s
y
x
ysy
xsx
ssS
yxP
y
x
y
x
yx
⋅=












=





=
=
=
=
'
0
0
'
'
'
'
),(
),(
6.6'
2.4'
2.2
4.1
30
03
'
'
2.2*3'
4.1*3'
)3,3(
)2.2,4.1(
=
=












=





=
=
=
=
y
x
y
x
y
x
S
P

2D
Rotations

Rotation
• Rotation - repositions an object along a
circular path.
• Rotation requires an Θ and a pivot point

Rotation
A two dimensional rotation is applied to an
object by repositioning it along a circular
path in the xy plane. To generate a rotation
angle θ and the position of the rotation
point about object is to be rotated.

Rotation: Initial

Rotation: Final

Rotation: Operation

Where does it come from?

Rotation
)cos('
)cos('
sin
cos
)(
),(
Θ+=
Θ+=
=
=
=
=
φ
φ
φ
φ
θ
ry
rx
ry
rx
R
yxP
PRP
y
x
y
x
yxy
yxx
rry
rrx
⋅=











 −
=





+=
−=
+=
−=
'
cossin
sincos
'
'
cossin'
sincos'
cossinsincos'
sinsincoscos'
θθ
θθ
θθ
θθ
θφθφ
θφθφ

Rotation Operation
Coordinates:
Matrix Form:
yxy
yxx
)cos()sin('
)sin()cos('
φφ
φφ
+=
−=











 −
=





y
x
y
x
)cos()sin(
)sin()cos(
'
'
φφ
φφ
P(x, y) ->P’ (x’, y’)
P’ -> R(φ) . P
36

Example
• P=(4,4)
• Θ=45 degrees

What is the difference? Revisited
V(-0.6,0) V(0,-0.6) V(0.6,0.6)
Translate (1.2,0.3)
V(0,0.6) V(0.3,0.9) V(0,1.2)
Translate (1.2,0.3)
V(0.6,0.3) V(1.2,-0.3) V(1.8,0.9)
V(0,0.6) V(0.3,0.9) V(0,1.2)

Rotations
V(-0.6,0) V(0,-0.6) V(0.6,0.6)
Rotate -30 degrees
V(0,0.6) V(0.3,0.9) V(0,1.2)

Combining Transformations
Q: How do we
specify each
transformation?

Specifying 2D Transformations
• Translation
– T(tx, ty)
– Translation distances
• Scale
– S(sx,sy)
– Scale factors
• Rotation
– R(θ)
– Rotation angle

• Using translate, rotation, and scale, how
do we get:

• Note there are two ways to combine
rotation and translation. Why?

2D
Shearing

Shearing

Shearing Operation
Coordinates:
Matrix Form:
xhsyy
yhsxx
yxy
xyx
+×=
+×=
'
'












=





y
x
sh
hs
y
x
yyx
xyx
'
'
46

Q: How do we
specify each
transformation?

Specifying 2D Transformations
• Translation
– T(tx, ty)
– Translation distances
• Scale
– S(sx,sy)
– Scale factors
• Rotation
– R(θ)
– Rotation angle

• Using translate, rotation, and scale, how
do we get:

• Note there are two ways to combine
rotation and translation. Why?

Rotation about an arbitrary point
cosѳ sinѳ 0
-sinѳ cosѳ 0
-xpcosѳ +ypsinѳ +xp -xpsinѳ-ypcosѳ+yp 1
T1.R.T2 =
51

Transformation matrix (original and
reflected image)
1 0 0
0 -1 0
0 0 1
Reflection about x axis

reflected image)
-1 0 0
0 -1 0
0 0 1
Reflection about origin

reflected image)
-1 0 0
0 1 0
0 0 1
Reflection about Y axis

reflected image)
0 1 0
1 0 0
0 0 1
Reflection about line y = x

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