Page 91 - DCAP504_Computer Graphics

Computer Graphics

The scaling factor s is given by

2
s= a + b 2
The angle of rotation is given by,
( )
α = tan − 1 − b a

Therefore,
The affine transformation of R is of the form R .
n
n
i.e., f(x): R → R n
n
F(p)=Ap+B
Where, A is the transformation applied to the point P and B is a constant.
Thus, the affine transformation can be used to perform transformation by combining two or more
transformations simultaneously.
6.5 Summary

• 2-D transformations are carried out by representing coordinate values of the 2-D object in matrix
form.

• Various transformations that can be applied to 2-D objects are translation, scaling, reflection and
shearing.
• Translation transformation moves the object from one position to another in the 2-D space.

• The transformation is carried out by multiplying the coordinate values of the object individually
with the respective transformation matrix.
• Scaling transformation is done to resize and reshape the object. The object's dimension can be
varied using scaling transformation.

• Reflection transformation is used to obtain the mirror image of the object.
• The object can be rotated with respect to the origin, clockwise or counterclockwise using rotation
transformation.
• Shear transformation is used to slant the image in X and Y directions of the 2-D space.

• Homogeneous transformation matrix is derived using the homogeneous coordinate system. This
matrix represents 2-D objects as 3-D objects by including a non-zero number as third element.
• Homogeneous transformation matrix is used to combine any two transformations by simply
multiplying the transformation matrices.
• The inverse of the transformation matrix is used to derive the original object from transformed
object.

• Linear transformations such as translation, scaling, reflection, rotation, and shearing are called
affine transformations.
6.6 Keywords

Affine Transformation: This is the combination of same transformations such as translation, rotation or
reflection on an axis.
Degree of a Polynomial: It is the highest degree for a term with non-zero coefficient in a polynomial that
is expressed as the sum or difference of terms.
Homogeneous: The different parts or elements that are of same kind.
Shear Transformation: This is the transformation in which one coordinate is fixed and other coordinate
or coordinates are shifted.

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