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Affine transformations

Transformations of the plane.

Uniform “contraction” of the plane toward a line. One of the simplest transformations of the plane is what we can describe as a uniform “contraction” of the plane toward a line. We describe now exactly what we mean by this. Suppose we are given a line a in the plane and a positive coefficient k. Assume, for example, that k = 3/4. By a uniform contraction, with coefficient k = 3/4, of the plane toward line a we mean that every point P of the plane is sent into point P' in such a way that P' lies on the same side of the plane as P, lies on the perpendicular from P to a, and is k = 3/4 of the distance from P to a. See Fig. 1. In the example we have just used k is less than one, and we have a proper contraction of the plane toward the line. If k is greater than one we would have an expansion of the plane from the line, instead of a contraction. We will, however, use the word “contraction”, putting it in quotation marks, to represent both cases.

Properties of uniform “contraction” transformations.

Under a uniform “contraction” of a plane toward a line:

1)        Any straight line of the plane is transformed into a straight line.

2)        Parallel lines remain parallel.

3)        If a point divides a segment in a given ratio, the image of the point divides the image of the segment in the same ratio.

“Contractions” of the plane to a line are a special case of more general, so-called affine transformations of the plane.

Affine transformation of the plane. The linear transformation

x' = a1 x + b1y + c1

y' = a2 x + b2y + c2

is an affine transformation of the plane provided the matrix

is non-singular.

.

Affine transformation of three-dimensional space. The linear transformation

is an affine transformation of space provided the matrix

is non-singular.

Generalized Cartesian coordinate systems. We call a pair of vectors e1, e2, starting from a common origin O and not lying on the same line, a “coordinate frame” of the plane. See Fig. 2. The coordinates of a point P relative to this frame Oe1e2 are given by the numbers x, y such that in order to reach the point P from the origin O one lays off from the point O x-times the vector e1 and then y-times the vector e2. Such a coordinate frame represents a generalization of the usual Cartesian rectangular coordinate system. The usual Cartesian system is a special case of this generalized system in which the coordinate vectors e1, e2 are mutually perpendicular and

of unit length. A similar generalized coordinate system can be introduced in three-dimensional space using three vectors e1, e2, e3 emanating from a common point O.

In an affine transformation of the plane a given coordinate frame Oe1e2 is transformed into a certain other coordinate frame Oe1' e2' (generally speaking, with another “metric” i.e. with different lengths for the vectors e1' and e2' and a different angle between them) and an arbitrary point M is sent into the point M' having the same coordinates relative to the new frame as M had relative to the old. See Fig. 3. In such a transformation a given net of equal parallelograms is transformed into another arbitrary net of equal parallelograms.

In an affine transformation of space a given coordinate frame Oe1e2e3 is transformed into a certain other coordinate frame Oe1' e2' e3' (generally speaking, with another “metric” i.e. with different lengths for the vectors e1', e2' and e3' and different angles between them) and an arbitrary point M is sent into the point M' having the same coordinates relative to the new frame as M had relative to the old.

Mathematics, Its Content, Methods and Meaning. Vol. 1, p. 231- 232

Properties of affine transformations.

- straight lines are mapped into straight lines, parallel lines are mapped into parallel lines, and if a point divides a segment in a given ratio, the image of the point divides the image of the segment in the same ratio.

Theorem. Any affine transformation of the plane can be obtained by performing a certain rigid motion of the plane onto itself, and then, in general, two uniform “contractions” with different coefficients k1 and k2 toward two mutually perpendicular lines. Similarly, any affine transformation of three-dimensional space can be obtained by performing a certain rigid motion of the space onto itself, and then, three uniform “contractions” toward three mutually perpendicular planes with certain coefficients k1, k2, k3.

References.

Mathematics, Its Content, Methods and Meaning. Vol. I