1. Lecture 7 Geometric Transformations (Continued)
Computer Graphics
Lecture 7
Geometric Transformations (Continued)

2. Some examples of translation, rotation and scaling
Suppose a 2-D point is translated by an amount tx in x-direction and ty in y-direction. Then the matrix used to represent this is:
T = tx x
ty . y

3. For scaling the matrix is given as:
S = sx x
0 sy 0 . y

4. Example 1
Using the matrix, translate the point (2, 6) by 4 in x-direction and -3 in y-direction.
T = tx x
ty . Y
T =
= 3
3X3 . 3X1 = 3X1

5. That is,
(2, 6) = (6, 3, 1) = (6, 3)
(2, 6) 5 4
(6, 3) 2 1

6. Example 2
Using a 2-D transformation matrix rotate the point (2, 6) by 90 degrees anti-clockwise about (0,0).
Rθ = cosθ -sinθ 0
sinθ cosθ 0 θ = 90
R90 =
= 2
= (2, 6) = (-6, 2)

8. Example 3
A triangle is defined by points a(2, 6), b(2,10) and c(6, 8). Firstly translate the triangle by 4 in x-direction and -3 in y-direction and then rotate it anti-clockwise by 90 degrees about (0,0).
T = tx
ty =

9. Rθ = cosθ -sinθ 0
sinθ cosθ 0 θ = 90
R90 =
1 0 0
0 0 1
a = b = c = 6

10. = = = = = =

11. = = = a(2, 6) = a’(-3, 6) = b(2, 10) = b’(-7, 6) = c(6,8) = c’(-5, 10)

13. Example 4
Scale the line joining the points (2, 6) and (6, 8) by 2 in the x-direction and 0.5 in the y-direction.
S = sx
0 sy 0 = = 3 =
= 4

15. Reflection
It is a transformation that produces a mirror image of an object.
It is generated relative to an axis of reflection by rotating the object 180 degrees about the reflection axis.
Reflection about x-axis
y = 0 and the matrix is given as
It keeps x-values same but flips y values of coordinate positions.

16. It moves out of the xy-plane and rotates 180 degrees about the x-axis and back into the xy-plane on the other side of x-axis

17. 2. Reflection about y-axis x = 0 and the matrix is gives as It keeps y-values same but flips x-values.

19. Reflect the point (2, 6) in x-axis.
= -6
(2, 6) = (2, -6)

21. The points of a triangle are given as (2, 3), (4,3) and (4,6)
The points of a triangle are given as (2, 3), (4,3) and (4,6). Reflect the triangle around y-axis. = =

22.
= 6
(2,3) = (-2,3)
(4,6) = (-4,6)
(4,3) = (-4,3)