Point Transformations

Each point on the plane (2D) correspond with two coordinates, which are defined as elements of the matrix size of 1×2 [ x y ]. In three dimensions (3D) we must use a matrix of size 1×3 [ x y z ]. In other words, the point can be specified as a column vector

(1.1)

or

(1.2)

To form such a vector row-matrix is used, i.e. the set of points, that defines a coordinate vector in a measurement system. This set is stored in a computer as a matrix or array of numbers. Position of points can be controlled by manipulating the corresponding matrix. The lines connecting the points, form segments, curves and pictures.

Consider the results of multiplying the matrix [ x y ] (containing the coordinates of point P) to the general transformation matrix T:

(1.3)

This means that the original coordinates x and y are transformed into x* and y*, where x* = ax + ​​cy, y* = bx + dy. Let us consider some special cases.

If a = d = 1 and c = b = 0, the transformation reduces to the identity matrix

(1.4)

and coordinates of point P are unchanged. As well-known from linear algebra, multiplication by the identity matrix is equivalent to multiplying by 1 in ordinary algebra.

If d = 1, b = c = 0 then

(1.5)

where x*= ax – the result of scaling the x coordinate. The effect of this transformation is shown in Figure 1.1(a).

Fig.1.1. Point transformations

If b = c = 0 then

(1.6)

This transformation changed both x and y coordinates of the vector P (Fig. 1.1(b)). If a <> d, then the coordinates are scaled differently (differential scaling). When a = d, there is uniform scaling.

If the value of a or d is negative, the vector is reflected about the coordinate axes or to the plane. To verify this, we take b = c = 0, d = 1 and a = 1, then

(1.7)

and the result is symmetrical about the axis y (Fig. 1.1(c)). If b = c = 0, a = 1, d = -1, then performed a symmetrical reflection about the x axis. If b = c = 0, a = d <0, it is a reflection about the origin, is shown in Figure 1.1(d), where a=-1, d = 1. Note that both the reflection and scaling operations depend only on the diagonal elements of the transformation matrix.

Consider now the case with off-diagonal elements. Take first the values ​​of a = d = 1, c = 0, then

(1.8)

Note that the x -coordinate of P is unchanged, while the y coordinate is linearly dependent on the initial coordinates. This transformation is called the shear (Fig. 1.1(e)). Similarly, in the case when a = d = 1, b = 0, the transformation will lead to a shear proportional to the coordinate y (Fig. 1.1(f)). Thus, it is clear that the off-diagonal elements of the transformation matrix create the effect of shearing the coordinates of the point P.

Before completing the transformation of points, let us analyze the effect of total transformation, when the initial vector lies at the origin, i.e.

(1.9)

It is clear that the origin is invariant to general transformation form. This restriction eliminates the use of homogeneous coordinates.

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