Transformations

1. Computer Graphics
Dr.A.G.Thakur &
Prof.D.P.Bhaskar
Sanjivani College of Engineering, Kopargaon
SPPU Pune
BY

2. • Transformations
Translation, Rotation, Scaling and Reflection.
• Homogenous Representation(HR)
• Concatenated Transformation(CT)
• Inverse Transformations.
• Mapping of Geometric Models
• Projections: Orthographic and Isometric.

3. Definition: Manipulation of size, shape and
location of the object .
Repositions the item in the Database.
Representation of an object
(Points ,Lines, Surfaces and Solids)
Transformation of object
Computer Graphics
Transformations

4. TrianglePoint Line
2D-Representation:Point MatrixCG

5. Transformation Routines
1. Translation/ Move
2. Rotation
3. Scaling
4. Reflection/Mirror
5. Shearing

6. Translation
Matrix Form
Addition process

7. Rotation
Matrix Form
P’
Multiplication process

8. Scaling
Multiplication process

9. Reflection
Multiplication process

10. Multiplication process
Reflection

11. Multiplication process
Shear

12. Transformation Matrix

14. [P’]=[T]+[P]

15. [P’]=[S]*[P]

18. Inverse translation T1 Rotating R Translation T2
[P’]=[T2*R*T1]*[P] [P’]=[CT]*[P]
Rotation@ point(m , n)
Homogeneous Representation[HR]
Concatenation of Transformations [CT]
Multiplication of all different transformation routines.

19. Inverse translation T1 Scaling S Translation T2
[P’]=[T2*S*T1]*[P]
[P’]=[CT]*[P]
Scaling @ point(m , n)
Homogeneous Representation[HR]

20. Homogeneous Representation[HR]
Translation transformations : Addition process
Other transformations :Multiplication process.
As translation is of additive type, that can’t be taken in
multiplication.
Solution: Homogeneous Representation

21. In HR form

22. Point vertex in HR form
e.g Rotation

23. Inverse translation T1 Scaling S Translation T2
[P’]=[T2*S*T1]*[P]
[P’]=[CT]*[P]
Scaling @ point(m , n)

24. Inverse translation T1 Rotating R Translation T2
[P’]=[T2*R*T1]*[P][P’]=[CT]*[P]
Rotation@ point(m , n)

25. Mirror@ Arbitrary line y=mx + c
[P’]=[T2*R2*MX*R*T1]*[P]
[P’]=[CT]*[P]

26. 1) Inverse translation [T1]
Point matrix [P]

27. 2) Scaling [S]
Sx = Sy = 0.5
3) Forward Translation[T2]

28. Concatenation Matrix
Point of scaled triangle

29. Rotation of square
Point matrix

30. Translation CG on origin
Rotate @ origin

31. Translation to required point
Concatenation matrix

32. New point matrix

34. Mapping of Geometric Models
MCS: Model/World/
Global/Cartesian CS (x’, y’)
Geometric data is stored wrt MCS
UCS: User/Local/Working CS (x , y)
Convenient for user
• Mapping between MCS & UCS Between (x,y) and (x’ , y’)
• Mapping Done by transformation

36. Problems
The coordinates of three vertices of triangle ABC in
working coordinate system [WCS] are A (3, 2), B (7, 2)
and C (5, 6) wrt the model coordinate system [MCS] of
the software.
The coordinates of origin of WCS are (6, 4). Axes of WCS
are at 300 in CCW wrt the axes of MCS.
Determine the coordinates of the vertices of triangle ABC
in MCS.

39. • The rectangle ABCD has the length of edge AB as 6
units and is inclined to the horizontal axis by 300 .The
other edge of rectangle is 4 units.
• In world coordinate system the coordinates of are A
are (3, 4) then determine coordinates of B, C and D in
user coordinate system and world coordinate system.
• To determine the user coordinate system, the edge
AB can be considered as one of the axis with point A
as the origin.

42. 3 Dimensional Transformations
Translation













1000
100
010
001
z
y
x
t
t
t
T
Point, P=

43. Rotation Rotation@ Z-axis [On X-Y Plane]
























 













1
*
1000
0100
00cossin
00sincos
1
'
'
'
Z
Y
X
Z
Y
X



55. Projection of Geometric models
• Orthographic projection
• Isometric projection.

57. Main transformation Matrix
Front view matrix (z=0)

58. Top view matrix (y=0)

59. RH side view matrix (x=0)

60. Orthographic Matrices

63. Isometric View on 2D plane
Plane of Projection ( X-Y)