Affine Transformation (contd...)

Let be the vector which transforms to a vector .

If and be the end points of vector , then

(1.5)

Similarly

(1.6)

Where and , etc.

Subtracting one obtains

(1.7)

It is easy to see that or in the short form are components of a tensor and since is a tensor, then is a tensor.

Note that the components are constants.

From equation 1.7, we can conclude that two equal vectors remain equal after transformation and two parallel vectors remain parallel after transformation with the ratio of their lengths remaining unaltered due to the transformation.

Hence, it follows that two identical and identically oriented polygons remain identical and identically oriented after transformation. Since every geometric figure is some form of polygon, it follows that all parts of a body, independent of their deformations will deform in an identical manner. That is called homogeneous transformation.