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Friday, 7 March 2014

Computer Graphics Notes - Lab 19 - Other Transformations

There are some other transformations which are useful in certain applications. Two such transformations are reflection and shear.
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.



Reflection

Transformation matrix
Original image
Reflected image
Reflection about Y-axis

      -1   0    0
       0    0   0
       0    0   0


Reflection about
X-axis

       1    0   0
       0   -1   0
       0    0    1


Reflection about origin

      -1   0    0
       0    0   0
       0    0   1


Reflection about 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



Example
Find out the final coordinates of a figure bounded by the coordinates (1, 1), (3, 4), (5, 7), (10, 3) when rotated about a point (8, 8) by 30° in clockwise direction and scaled by 2 units in x direction and 3 units y direction.








Example
Show that transformation matrix for a reflection about a line Y = X is equivalent to reflection to X-axis followed by counter-clockwise rotation of 90°.




Example
Show that 2D reflection through X axis followed by 2D reflection through the line Y = - X is equivalent to a pure rotation about the origin.



Example
Prove that successive 2D rotations are additive; i.e.
R (θ1). R (θ2) = R (θ1 + θ2)



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