Transformations as Functions

Most high school geometry textbooks will proclaim that there are four basic transformations. Three of these (translations, reflections, and rotations) are rigid transformations; the resulting copy (image) is congruent to the original version (pre-image). Here are examples:

The first example is a simple translation, which can be written algebraically as \((x, y) \mapsto (x + 3, y + 2)\). This means that the pre-image moves over two units to the right and up three units.

The second example is a 90° rotation counterclockwise around the origin, that is, \((x, y) \mapsto (-y, x)\). The third image can be characterized as a rotation, but was created using a translation and a reflection, \((x, y) \mapsto (y – 3, x – 3)\); this is generally called a glide reflection.

All congruent transformations can be described in terms of no more than two of these three basic transformations. The fourth transformation, dilation, results in a similar image, i.e., one that has the same angles and proportional sides as the original pre-image. We can combine this with other transformations, such as this dilated reflection, \((x, y) \mapsto (2y, 2x)\).

This is the basic presentation of transformations: Translations, reflections, rotations, glide reflections, and dilations. Presenting them this way allows for a scaffolded exploration of the way in which congruent and similar shapes can be interrelated. Glide reflections allow for discussing the composition of functions.

At the same time, though, it’s not completely accurate on at least two levels.

First, rotations and translations are special cases and can be combined into a single rule. We could group these into “parity isometries”; all such isometries can be described by \((x, y) \mapsto (x \cos(\theta) – y \sin(\theta) + h, y \sin(\theta) + x \cos(\theta) + k)\). It’s obvious why we wouldn’t want to present this as the first formula for transformations.

Reflections and glide reflections, meanwhile, can be grouped as “non-parity isometries”, with the mapping function \((x, y) \mapsto (y \sin(\theta) – x \cos(\theta) + h, y \sin(\theta) + x \cos(\theta) + k)\). Dilations, meanwhile, result from rigid transformations multiplied by a scale factor.

Secondly, though, this is not the complete set of transformations. These are merely the transformations that lead to a similar image. Transformations on the Cartesian plane in general can be seen as a graphical representation of a function that takes two values and returns two values. Plugging in different functions can yield some interesting results. Here are some examples, where the blue is the pre-image and the red is the image:

\((x, y) \mapsto (x + 3y, y)\):

\((x, y) \mapsto (x – y, y – x)\):

\((x, y) \mapsto (y^2, x^3)\):

\((x, y) \mapsto (x^2, x – y^2)\):

\((x, y) \mapsto (y^2, \sqrt{x})\):

\((x, y) \mapsto (\Gamma(x), \Gamma(-y))\):

Seen this way, a transformation is a graphical (geometric) representation of a(n algebraic) function. The congruence and similarity transformations are analogous to linear functions, an observation that allows us to represent linear functions as transformations on a number line, rather than as graphs on the Cartesian plane. More on this in a future post.

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