2D Transformations in computer Graphics

1. AIIT
COMPUTER GRAPHICS
Two Dimensional Geometric
Transformation
Prof. (Dr.) Sonu Mittal
smittal@amity.edu
AMITY UNIVERSITY, NOIDA
INDIA

2. AIIT
TYPES OF TRANSFORMATION
1) Geometric Transformations: In this type of
transformations, the object itself is moved relative
to a stationary coordinate system or background.
2) Coordinate Transformations: In this type of
transformation, the object is held stationary while
the coordinate system is moved relative to the
object.

3. AIIT
TYPES OF TRANSFORMATION
1. Translation
2. Scaling
3. Rotation
4. Reflection
5. Shearing

4. AIIT
TRANSLATION
 Simply moves an object from one position to
another
 xnew = xold + tx ynew = yold + ty
The pair (tx , ty) is called translation vector or shift vector
Note: House shifts position relative to origin
y
x
0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6

5. AIIT
TRANSLATION EXAMPLE
y
x
0 1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
(1, 1) (3, 1)
(2, 3)
(6, 4)
tx
(7, 2)
(5, 2)
=4
=1
ty

6. AIIT
 In matrix representation we can write translation as

7. AIIT
ROTATION
 Rotation can be represented as:

8. AIIT
SCALING EXAMPLE
y
0 1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
(1, 1) (3, 1)
(2, 3)
Scaling vectors=(2,2)

9. AIIT
SCALING
Scaling transformation can be represented as

10. AIIT
REFLECTION AND SHEARING

11. AIIT
REFLECTION
 A reflection is a transformation that produces a mirror image
of an object relative to an axis of reflection. We can choose
an axis of reflection in the xy plane or perpendicular to the xy
plane.
Y
X
O
Original
object
Reflected
object

12. AIIT
Reflection Transformation matrix
Reflection about
Y-axis
-1 0 0
0 1 0
0 0 1
Reflection about
X-axis
1 0 0
0 -1 0
0 0 1
Reflection about
origin
-1 0 0
0 -1 0
0 0 1
Reflection about diagonal axis (line
y=x)
0 1 0
1 0 0
0 0 1
Reflection about
Line y = - x
0 -1 0
- 1 0 0
0 0 1

13. AIIT
SHEARING
 A transformation that slants the shape of an object is called the
shear transformation.
 Two common shearing transformations are
 X-shear
 Y-shear
X-Shear
 The X shear preserves the y coordinates, but changes the x
values which causes vertical lines to tilt right or left.

14. AIIT
Y SHEAR
 The Y shear preserves the x coordinates, but changes
the y values which causes horizontal lines to tilt up or
down.

15. AIIT

16. AIIT

17. AIIT
INVERSE GEOMETRIC TRANSFORMATION
 Each geometric transformation has an inverse
which is described by the opposite operation
performed by the transformation.

18. AIIT
COMPOSITE TRANSFORMATION
 A Composite Transformation is a transformation that consists
of more than one step to the transformation.
 For example, translate an object and then, do a second
translation on the product, to again move the object.
 There can be a combination of two different transformations.
For example, translate the object and then perform a size
transformation.

19. AIIT
NUMERICAL 1
 Find the matrix that represents rotation of an object
by 30 degree about the origin
 What are the new coordinates of the point P(2,-4)
after the rotation.

20. AIIT
NUMERICAL 2
 Q) Translate a point P (2,5) by 3 units in x direction
and 4 units in Y direction.

21. AIIT
NUMERICAL 3
 Translate a polygon with coordinates A(2,5) B(7,10)
and C(10, 2) by 3 units in X direction and 4 units in
Y direction.

22. AIIT
NUMERICAL 4
 Find the transformation that scales (w.r.t origin) by
(i) a units in x direction
(ii) b units in y direction
(iii) a units in the x direction and b units in the y
direction

23. AIIT
NUMERICAL 5
 Q) Scale the polygon with coordinates A(2,5)
B(7,10) and C(10,2) by two units in X direction and
two units in Y direction.

24. AIIT
GENERAL PIVOT POINT ROTATION
1) Translate the object so that the pivot-point position
is moved to the coordinate origin.
2). Rotate the object about the coordinate origin.
3).Translate the object so that the pivot point is
returned to its original position.

25. AIIT
GENERAL FIXED POINT SCALING
 Translate object so that the fixed point coincides
with the coordinate origin.
 Scale the object with respect to the coordinate
origin.
 Use the inverse translation of step 1 to return the
object to its original position.

26. AIIT
NUMERICAL 6
 Perform a 45 degree rotation A(0,0), B(1,1) , C(5,2)
a) About the origin b) About P(-1,-1).

27. AIIT
NUMERICAL 7
 Q) Prove that two successive 2D rotations are
additive.

28. AIIT
NUMERICAL 8
 Magnify a triangle with vertices A (0,0), B(1,1) and
C(5,2) to twice its size while keeping C(5,2) fixed.

29. AIIT
NUMERICAL 9
 Find a transformation of triangle A(1,0), B(0,1),
C(1,1) by rotating 45 degree about the origin and
then translating one unit in x and y direction.

30. AIIT
NUMERICAL 10
Q) Apply the shearing transformation to square with
A(0,0), B(1,0), C(1,1) and D(0,1) as given below:
(a) Shear Parameter value of 0.5 relative to the line
Yref=-1
(b) Shear Parameter value of 0.5 relative to the line
Xref=-1

31. AIIT
NUMERICAL 12
 Describe the transformation that rotates an object
point Q(x,y) θ degree about a fixed center of
rotation p(h,k), and find the matrix for rotation about
a point p(h,k)

32. AIIT
NUMERICAL 13
 Reflect the diamond shaped polygon whose
vertices are A(-1,0), B(0,-2), C(1,0) and D(0,2)
about
(a) The horizontal line y=2
(b) The vertical line x=2
(c) The line y=x+2

33. AIIT

34. AIIT
THANK YOU