Composite Transformations. With the help of matrix operations for the coordinate vectors defining the vertices of the figures, we can control the shape and position of the surface

With the help of matrix operations for the coordinate vectors defining the vertices of the figures, we can control the shape and position of the surface. However, to obtain the desired orientation may require more than one transformation. Since matrix multiplication is not commutative and the conversion order is important.

To illustrate the effect of non-commutative matrix multiplication operations consider the rotation and reflection transformations for the coordinate vector [ x y ]. If it is rotated by 90° (by using matrix [ T₁ ]) and then reflect about the line y =-x (by using matrix [ T₂ ]), these two consecutive transformations give following results:

(1.40)

and then:

(1.41)

On the other hand, if the reflection follows the rotation, we obtain the following results:

(1.42)

and

(1.43)

Because results are different, therefore conversion order of matrix transformations is important.

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