2. 2D Transformations
What is transformations?
Changes in orientation , size , and shape are accomplished
with geometrical transformations that alters the coordinate
description of object.
Why the transformations is needed?
To manipulate the initially created object and to display the
modified object without having to redraw it.
3. 2 ways
Object Transformation
• Alter the coordinates descriptions an object
• Translation, rotation, scaling etc.
• Coordinate system unchanged
Coordinate transformation
• Produce a different coordinate system
2D Transformations
4. 4
Transformations and Matrices
Transformations are functions
Matrices are functions representations
Matrices represent linear transformation
{2x2 Matrices} {2D Linear Transformation}
What are they?
changing something to something else via rules
mathematics: mapping between values in a range set and
domain set (function/relation)
geometric: translate, rotate, scale, shear,…
Why are they important to graphics?
moving objects on screen / in space
mapping from model space to screen space
specifying parent/child relationships
5.
6.
7.
8.
9.
10.
11. Basic Transformations : Translation (1/2)
Translation
Definition : repositioning objects along a straight line path
from one position to another
Let : original position, : new position
We need translation distance (translation vector)
for x direction
for y direction
Then,
y
x
=P
y
x
=P
T =
t
t
x
y
x' x t
y' y t
x
y
12. Translation (2/2)
In matrix form
Translation is rigid-body Transformation that moves
objects without deformation
• every point on the object is translated by the same amount
For straight line
• applying translation distance to each line end points
For polygon, curves
P' P T
x
y
t
t
x
y
16. Basic Transformations : scaling (1/2)
Scaling
Definition : alters the size of an objects
we need scaling factors
: for x value
: for y value
In matrix form
Then
if : Uniform scaling
: differential scaling
S
S 0
0 S
X
y
x x sx' y y sy'
y
x
s0
0s
PSP
y
x
s s
s s
x y
x y
17. scaling (2/2)
if : uniform compression
• move objects to the coordinate origin
if : uniform Enlargement
• move objects farther from the origin
Fixed point scaling : scaling based on a fixed point
• An object is scaled relative to the fixed point by scaling distance
from each vertex to fixed point
• when
0 s s 1x y
1 s sx y
( ),x yf f
x' x (x x )s x s x (1 s )
y' y (y y )s y s y (1 s )
f f x x f x
f f y y f y
(x ,y) (0,0)f f
18. 18
Inverse 2D - Transformations
y
-
y
x
-
x
),(
-
(sx,sy)
(-θ
-
(θ
(-dx,-dy)
-
(dx,dy)
MM
MM
SS
RR
TT
sysx
1
1
1
)
1
)
1
:RefMirror
:Sclaing
:Rotation
:nTranslaito
11
19. 19
Homogeneous Co-ordinates (1/3)
Translation, scaling and rotation are expressed
(non-homogeneously) as:
translation: P = P + T
Scale: P = S · P
Rotate: P = R · P
Composition is difficult to express, since translation
not expressed as a matrix multiplication
Homogeneous coordinates allow all three to be expr
essed homogeneously, using multiplication by 3 ´ 3
matrices
W is 1 for affine transformations in graphics
20. 20
Homogeneous Co-ordinates (2/3)
P2d is a projection of Ph onto the w = 1
plane
So an infinite number of points correspon
d to : they constitute the whole line
(tx, ty, tw)
x
y
w Ph(x,y,w)
P2d(x,y,1)
w=1
21. Homogeneous Coordinates (3/3)
For Translation, Rotation, Scaling
1
y
x
100
t10
t01
s
y
x
y
x
1
y
x
100
0cossin
0sin-cos
s
y
x
1
y
x
100
0s0
00s
s
y
x
y
x
Translation:
Scaling:
Rotation:
22. 22
Classification of Transformations
1. Rigid-body Transformation
Preserves parallelism of lines
Preserves angle and length
e.g. any sequence of R() and T(dx,dy)
2. Affine Transformation
Preserves parallelism of lines
Doesn’t preserve angle and length
e.g. any sequence of R(), S(sx,sy) and T(dx,dy)
unit cube 45 deg rotaton Scale in X not in Y
23. 23
Properties of rigid-body transformation
100
2221
1211
y
x
trr
trr
The following Matrix is Orthogonal if the upper left 2X2 matrix has the
following properties
1.A) Each row are unit vector.
sqrt(r11* r11 + r12* r12) = 1
B) Each column are unit vector.
sqrt(c11* c11 + c12* c12) = 1
2.A) Rows will be perpendicular to each other
(r11 , r12 ) . ( r21 , r22) = 0
B) Columns will be perpendicular to each other
(c11 , c12 ) . (c21 ,c22) = 0
e.g. Rotation matrix is orthogonal
100
0cossin
0sincos
• Orthogonal Transformation Rigid-Body Transformation
• For any orthogonal matrix B B-1 = BT
24. Concatenation properties.
Matrix multiplication : associative
• ex) For three matrices A, B and C
• Translation or Rotation : additive property commutative
• scaling : multiplicative property commutative
However, Translation and Rotation : non commutative
• order of transformation matrix multiplication is important
A B C (A B) C A (B C)
ABBA
25.
26. 26
Commutative of Transformation Matrices
• In general matrix multiplication is not commutative
• For the following special cases commutativity holds i.e. M1.M2 = M2.M1
M1 M2
Translate Translate
Scale Scale
Rotate Rotate
Uniform Scale Rotate
• Some non-commutative
Compositions:
Non-uniform scale, Rotate
Translate, Scale
Rotate, Translate
Original
Transitional
Final
27. Composite Transformation (1/2)
Translation
If two successive translation factor (tx1, ty1) and (tx2, ty2) are
applied to a coordinate point P
• then
• ex)
i.e,
Two successive Translation are additive
1 0 t
0 1 t
0 0 1
1 0 t
0 1 t
0 0 1
1 0 t t
0 1 t t
0 0 1
x2
y2
x1
y1
x1 x2
y1 y2
P' T (t ,t ) {T (t ,t ) P} {T (t ,t ) T (t ,t )} P2 X2 y2 1 x1 y1 2 X2 y2 1 x1 y1
T(t ,t ) T(t ,t ) T(t t ,t t )x2 y2 x1 y1 x1 x2 y1 y2
29. General pivot-point Rotation (1/2)
Rotation about arbitrary point p (xr, yr)
step1) Translate P to origin
step2) Rotation about origin
step3) Retranslation to position P.
30. General pivot-point Rotation (2/2)
Composite transformation matrix
x
y
x
y
x y
y x
r
r
r
r
r r
r r
cos sin
sin cos
cos sin ( cos ) sin
sin cos ( cos ) sin
),y,R(x),-yT(-x)R()y,T(x rrrrrr
32. General Fixed-Point Scaling
Scaling about arbitrary point p(xr, yr)
step 1) translate p to origin
step 2) scaling about origin
step 3) Retranslate to position P
x
y
s
s
x
y
s x s
s y s
r
r
x
y
r
r
x r x
y r y
( )
( ) )ss,y,S(x),-yT(-x)sS(s)y,T(x yxffffyxff ,,
33. Other Transformations : Reflection (1/3)
Reflection : produce a mirror images of an object
We need axis of reflection
rotating the objects 180°about reflection axis
34. Reflection about the x axis: Just flip the y coordinate
Reflection about the y axis: Just flip the x coordinate
Reflection about the origin, flip both coordinates
1
32
1
32
1
32
1’
3’2’
1’
2’3’
1’
2’3’
Reflection about x axis Reflection about y axis Reflection about origin
Reflection (2/3)
R
1 0 0
0 1 0
0 0 1
y
R
1 0 0
0 1 0
0 0 1
X
100
010
001
R0
35. Reflection (3/3)
Reflection through arbitrary point P (xr, yr)
• translate point p to origin : Tr
• perform reflection about origin : Ro
• retranslate to original position : Tr
Reflection through y=x line
100
y10
x01
100
010
001
100
y10
x01
r
r
r
r
0 1 0
1 0 0
0 0 1
38. Shear Transformation
Shear : distorts the shape of an object
Shear transformation that distorts the shape of an object such that the
transformed shape appears as if the object were composed of internal
layers that had been caused to slide over each other.
Shearing (slide over) can be in either x or y direction
In X Shear, x-values changes along x-direction by shearing factor of shx
• point is shifted by horizontally by an amount proportional to its
distance from x-axis
• y values are unchanged
In Y Shear, y-values changes along y-direction by shearing factor of shy
• point is shifted by vertically by an amount proportional to its distance
from y-axis
• x values are unchanged
42. Transformation matrix of X- Shear with y- reference line
X-direction shearing relative to liney yref
1 sh sh y
0 1 0
0 0 1
x x ref
yy'
)y(yshxx' refx
44. Y – Shear relative to x reference line
y-direction shearing relative to linex xref
1 0 0
sh 1 sh x
0 0 1
y y ref
x' x
y' y sh (x x )y ref
45.
46. The area of the parallelogram can be calculated as,
Area of the trapezium =(sum of two parallel side)*h / 2
Parea = (a+c) (b+d) – (ab) – (cd) – (b+b+d) – (c+a+c)
= ad – bc
= det
Thus, area of any parallelogram is formed by transforming
a square which is a function of the transformation matrix
determinant and is related to the area of the initial square
Parea = Sarea (ad-bc) = Sarea det [T]
Area of any transformed figure Tarea is related to the ar
ea of the initial figure Iarea by,
Tarea = Iarea (ad-bc)
2
1
2
1
2
c
2
b
dc
ba
51. ASSIGNMENT -2 TWO –D TRANSFORMATION
Assigned Date : 26-02-13
Submission Date : 05-03-13
Fo r a l l t h e b e l o w g i v e n pr o b l e m Or i g i n a l a n d
Fi n a l po s i t i o n d i a g r a m i s n e c e s s a r y
1. Scale a square at [(2, 2), (6, 6)] along one of its diagonal and find the
new vertices.
2. Perform a clockwise 30 degree rotation of a triangle A (2, 3), B (5, 5),
C(4,3) about the point (1,1)
3. Find a transformation of triangle A (1, 0), B (0, 1), C (1, 1) by
a) Rotating 30 degree about the origin and then translating one
unit in x and y direction.
b) Translating one unit in x and y direction and then rotating 30
degree about the origin.
4. Represent a triangle at [(3, 3), (5, 6) , (7,3) ] in a new coordinate
system with origin at (3,3) and its x-axis making an angle +45 degree
with the x-direction of the base coordinate system.
5. Find out the co-ordinate of a figure bounded by (0,0) , (1,5) , (6,3) (-
3,-4) when reflected along the line whose equation is y=1 (x+5) and
sheared by 2 units in
x –direction and 2 units in y-direction.
[HINT: Ref -Page No. 3-46 in Godse Book]
6. Compute the new reflected coordinates of a point (-4,-4), when
reflected with respect to a line that passes through (-4, 0) and (0,-4)
[HINT: Ref -Page No. 3-62 in Godse Book]
7. For a point (2, 4) in a window at [(1, 1), (5, 5)], find the equivalent
point in the View Port at [(1, 1), (5, 10)].
