Transformations

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A transformation is a mathematical function that repositions points in a one-dimensional, two-dimensional, three-dimensional, or any other n-dimensional space. In this article, only transformations in the familiar twodimensional rectangular coordinate plane will be discussed.

Transformations map one set of points onto another set of points, generally with the purpose of changing the position, size, and/or shape of the figure made up by the first set of points. The first set of points, from the domain of the transformation, is called the set of pre-images, whereas the second set of points, from the range of the transformation, is called the set of images. Therefore, a transformation maps each pre-image point to its image point.

Reflections are transformations that involve "flipping" points over a given line; hence, this type of transformation is sometimes called a "flip." When a figure is reflected in a line, the points on the figure are mapped onto the points on the other side of the line which form the figure's mirror image.

For example, in the first figure below, the triangle ABC, the pre-image figure, is reflected in the x-axis to produce the image triangle A'B'C'. Note that if triangle ABC is traversed from A to B to C and back to A, the direction of this movement is counterclockwise. If triangle A'B'C' is traversed from A' to B' to C' and back to A', the direction is clockwise. This "reversal of orientation" is similar to the way images in a mirror are reversed, and is a fundamental property of all reflections.

In the case of the reflection in the x-axis, as seen in the first figure, the first coordinate of each image point is the same as the first coordinate of its pre-image point, but the second coordinate of any image point is the opposite, or negative, of the second coordinate of its pre-image point. Mathematically, we say that for a reflection in the x-axis, pre-image points of the form (x, y) are mapped to image points of the form (x, -y), or, more compactly, r(x, y) = (x, -y), where r represents the reflection.

Such a formula is sometimes called an image formula, since it shows how a transformation acts on a pre-image point to produce its image point. A reflection in the y-axis leaves all second coordinates the same but replaces each first coordinate with its opposite; therefore, an image formula for a reflection in the y-axis may be written r(x, y) = (-x, y). The image formula r(x, y) = (y, x) represents a reflection in the line y = x. When this reflection is done, the coordinates of each pre-image point are reversed to give the image.

A second type of transformation is the rotation, also known as a "turn." A rotation, as its name suggests, takes pre-image points and rotates them by some specified angle measure about some fixed point. In the figure below, the pre-image triangle CDE has been rotated 90° about the origin of the coordinate system to produce the image triangle C'D'E'.

The image formula for a 90° rotation is R(x, y) = (-y, x). It can be shown that, in general, the image formula for a rotation of angle measure t about the origin has image formula R(x, y) = [xcos(t) - ysin(t)], [xsin(t) + ycos(t)]. If t is positive, the direction of the rotation is counterclockwise; if t is negative, then the rotation is clockwise.

Another type of transformation is the translation or "slide." Translations take pre-image points and move them a given number of units horizontally and/or a given number of units vertically. A translation image formula has the form T(x, y) = (x + a, y + b), where a is the number of units moved horizontally and b is the number of units moved vertically. If a is positive, the horizontal shift is to the right. If a is negative, then it is to the left. Similarly, if b is positive, the vertical shift is upward; but if b is negative, the vertical shift is downward.

In the figure below, the pre-image triangle CDE has been translated 4 units to the left and 2 units upward to give the image triangle C'D'E'. The image formula would be T(x, y) = (x + (-4), y + 2) = (x - 4, y + 2).

Reflections, rotations, and translations change only the location of a figure. They have no effect on the size of the figure or on the distance between points in the figure. For this reason they are called "isometries" from the Greek words meaning "same measure." An isometry is also known as a "distance-preserving transformation." The next transformation discussed—the dilation—is not an isometry; that is, it does not preserve distance.

A dilation is also known as a "stretch" or "shrink" depending on whether the image figure is larger or smaller than the pre-image figure. The image formula for a dilation is d(x, y) = (kx, ky), where k is a real number, called the magnitude or scale factor, and where the center of the dilation is the origin. If k > 1, the image of the dilation is an enlargement of the pre-image on the same side of the center as the pre-image. Part (a) of the boxed figure illustrates this for k = 3.

If 0 < k < 1, the image of the dilation is a reduction in size of the pre-image on the same side of the center as the pre-image. Part (b) of the figure illustrates this for k = ½.

If k < -1, the image of the dilation is an enlargement of the pre-image on the opposite side of the center from the pre-image. Part (c) of the figure illustrates this for k = -3.

If -1 < k < 0, the image of the dilation is a reduction in size of the pre-image on the opposite side of the center from the pre-image. Part (d) of the figure illustrates this for k = -½.

Notice that the effect of a negative value for k is equivalent to that of a dilation with a magnitude equal to the absolute value of k followed by a rotation of 180° about the center of the dilation.

All of the transformations discussed above can easily be extended into 3-dimensional space. For example, a dilation in the 3-dimensional xyz-coordinate system would have an image formula of the form d(x, y, z) = (kx, ky, kz). When transformations—reflections, rotations, translations, and dilations—are expressed as matrices (as is taught in linear algebra), they can then be combined to create the movement of figures in computer animation programs.

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