Unit Square Transformation

So far we have considered the behavior of points and lines for simple matrix transformations. However, we can properly consider the application of the matrix to any point on the plane. As shown previously, the only point that remains invariant under the action of matrix transformations is a point of origin. All other points are subject to the transformation of the plane, which can be thought of as stretching the original plane, the coordinate system and transfer to a new form.

Consider a grid from unit squares in the coordinate plane xy (Fig.1.5). Four vertices of the coordinate vector of the unit square, passing from one angle to the origin, are as follows:

(1.44)

Applying the (2×2) the total transformation matrix, we obtain:

(1.45)

Fig.1.5. Unit square transformation

The results of this transformation you can see in Fig.1.5. From the previous expression it is followed that the origin is not subject to transformation, i.e. [ A ] = [ A* ] = [0 0]. Further, we note that the coordinates of B* are the first line of the transformation matrix and the coordinates of D* – the second one. Thus, the transformation matrix is determined if the coordinates B* and D* are defined (conversion of the unit vectors [1 0] [0 1]). As sides of the unit square parallel to the original and previously has been shown that parallel lines are converted into parallel, the resulting figure is a parallelogram.

The influence of the elements a, b, c and d can be installed separately. Elements b and c, as shown in Fig. 1.5, causing a shift of the original square by the y and x, respectively. As noted earlier, the elements a and d play the role of scaling factors. Thus, the 2×2 matrix gives a combination of translation and scaling.

It is easy to define as the square of the parallelogram A*B*C*D* in Fig. 1.5, which can be computed as follows:

As a result, we obtain:

(1.46)

One can show that that square S* of any parallelogram formed by converting the unit square is a function of the determinant of the transformation matrix and is associated with the original square S by a simple expression:

(1.47)

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