A point in the plane can be represented by a pair of numbers, (x, y), giving its x-coordinate, x, and its y-coordinate, y, as shown in the figure below.
More complicated figures are represented by connecting a series of points by straight lines. For example, the triangle in the left figure below is made up of three straight line segments connecting the points
Although the right figure below looks like a circle, it is actually made up of 60 straight line segments.
Each of the colored shapes in the figure below can be described in this way -- that is, as a series of lines connecting points in a list. For each one, find an appropriate list of points. Check your work in your CAS window by drawing each figure.
Because images can be represented numerically, we can simulate physical operations by arithmetic ones. For example, the physical operation of sliding an image two units to the right and four units down can be simulated numerically by adding two to the x-coordinate of each point in the image and subtracting four from each y-coordinate. Consider, for example, the red triangle in the image below.
(1, 1), (1, 3), (3, 1)
(3, -3), (3, -1), (5, -3)
More generally, we can translate each point (x, y) in an image by a units horizontally and b units vertically. Numerically, we compute (x + a, y + b). We often use the notation
to denote translation.
Similarly, we can reflect an image in the x-axis by reflecting each point in the image using the formula
The figure below shows the result of reflecting the red triangle in the x-axis. The result is colored blue..
Next consider the figure below. It has three triangles. The original triangle is red. The red image was rotated counterclockwise by Pi/2 radians pivoting around the origin to obtain the blue image. The red image was rotated clockwise 5 Pi/4 radians pivoting around the origin to obtain the green image.
All of these operations on images are called rigid transformations and can be done physically by first drawing an image on a clear piece of acetate positioned over a piece of graph paper and then picking up the acetate and laying it back down in a new position over the same graph paper.
If you would like to experiment with rigid transformations physically, click here for two differently colored pieces of graph paper. If you have a printer you can print these images -- one on paper and one on acetate and then use them for physical experimentation.
There is another rigid transformation that is worthy of a name -- the identity transformation, which does absolutely nothing and is written.
So far we have described four kinds of rigid transformations both physically and algebraically and, in your CAS window, by programs --
but you can probably think of other rigid transformations -- for example,
Sometimes when we compose two rigid transformations the order in which we do them makes a difference and sometimes it doesn't. If the order does not make a difference then we say that the two rigid transformations commute. If the order does make a difference then we say that the two rigid transformations do not commute. For each of the following pairs of rigid transformations determine algebraically whether they commute. You can also experiment in your CAS window.
One of the most important facts about rigid transformations is that any rigid transformation can be undone. That is, given a rigid transformation T there is another rigid transformation S called the inverse of T and written
(u, v) = T(x, y)
S(u, v) = (x, y)
Examples
Find the inverse of each of the following rigid transformations. You can check your answer experimentally using your CAS window.
The set of rigid transformations of the plane is an example of a very general mathematical structure called a group. Click on the icon below to look at the group module in the mathematical structure that we are building.
