A transformation is a one-to-one function that maps a set of points onto another set of points. The "points" mentioned here may be one-dimensional points on the real number line, two-dimensional points in the plane, three-dimensional points in Euclidean space, or any higher dimensional points in abstract space. We shall confine ourselves to one and two dimensions in this article. The points in the domain of the translation are often called "pre-images," while points in the range are called "images." The mathematical formula that describes how to map pre-images to images is sometimes called a "transformation image formula." From a practical point of view, we may say that a transformation is a function that repositions points. For example, the one-dimensional transformation T(x) = x + 3 repositions any point on the real number line three units to the right. This particular transformation is called a translation. The transformation T(x,y) = (x - 3, y + 2) is also a translation but in two-dimensions. It repositions points in the plane three units horizontally to the left and two-units vertically upward. The transformation T(x,y) = (x,y) "repositions" points right back where they were. This is called the identity transformation in two-dimensions. T(x) = x is the identity transformation in one-dimension. Some transformations reposition points without changing distances; these are called "isometries" or distance-preserving transformations. Other transformations have as their purpose to change distances; these are the scale changes. We will consider both types in the paragraphs below.
There are three types of isometries or distance-preserving transformations: translations, rotations, and reflections. Translations were mentioned in the preceding paragraph. They slide points in same direction. The general mathematical formula for a translation in the plane is T(x,y) = (x + h, y + k). This slides points h units horizontally and k units vertically. Rotations reposition points by turning them about some central point called the center of rotation. Rotations about (0,0), called origin-centered rotations, give image formulas of the form R(x,y) = (xcos(t) - ysin(t),xsin(t) + ycos(t)), where t is the angle of rotation. For a rotation of 90 degrees about the origin, this formula becomes R(x,y) = (-y,x). Reflections are transformations which "flip" points over some given line. The image formula in the general case is complicated, but in special cases it is less cumbersome. For instance, if r represents a reflection over the line y = x, the formula is r(x,y) = (y,x). This says that when points are reflected over the y = x line, the pre-image coordinates are reversed to obtain the image points. Sometimes a fourth transformation, the glide reflection, is included among the isometries. The glide reflection slides points in one direction and then flips them over a line. Thus it is not a simple "one-step" transformation; rather it is a "composite" of a translation followed by a reflection. It is also interesting to note that both translations and rotations can be created by the composition of two reflections. A composite of two reflections across parallel lines produces a translation, whereas a composite of two reflections across intersecting lines produces a rotation. In this sense, reflections are the only isometries really needed. Rotations and translations are introduced for convenience of notation.
Scale changes do not preserve distance and so are not isometries. The two basic types of scale changes are the similarity scale change, also called a "dilation" and the non-similarity scale change. The dilation centered at the origin has an image formula of the form S(x,y) = (ax,ay) where a is not 0 or 1. The number a is called the scale factor and is applied to both coordinates of the pre-image to produce the image. The fact that the same scale factor is applied to both coordinates means that all figures mapped with a dilation will be "similar" to their images, hence the name similarity transformation. In geometry, two figures are said to be similar if the lengths of pieces in one figure are proportional to lengths of the corresponding pieces in the other. So a triangle with sides of length 3, 4, and 5 meters is similar to a triangle with sides of length 6, 8, and 10 meters. In this case, the scale factor is 2. Non-similarity scale changes centered at the origin have image formulas of the form S(x,y) = (ax,by) where a and b are not equal in absolute value. The number a is called the horizontal scale factor and the number b is called the vertical scale factor. A scale change of this type will not produce similar figures. To illustrate this, suppose the unit circle, a circle with radius 1 centered at the origin, is transformed with the dilation S(x,y) = (3x,3y). The image will be another circle centered at the origin but with radius 3. Since both the pre-image and image are circles, they are similar. Now let the unit circle be transformed by the two-dimensional scale change S(x,y) = (4x,3y). The result will be an ellipse with semi-major axis of length 4 and semi-minor axis of length 3. An ellipse is not similar to a circle, illustrating that S(x,y) = (4x,3y) is not a similarity transformation.
With the exception of translations, all the transformations discussed above fall into the category of "linear" transformations, so called because they all have the general form T(x,y) = (ax + by, cx + dy) where a, b, c, and d are real numbers. The forms ax+by and cx + dy have the form of the left side of the general equation for a straight line, Ax+By=C. For example, the transformation S(x,y)=(4x,3y) can be written as S(x,y) = (4x + 0y, 0x + 3y). Such transformations can be associated with 2 by 2 matrices with the numbers a and b in the first row and the numbers c and d in the second row. Representing transformations as matrices is advantageous for use in computer programs. Computer programmers often create animations using software that manipulates figures by means of transformations written in matrix form.
This is the complete article, containing 998 words
(approx. 3 pages at 300 words per page).
