What is a transformation?

A transformation of a collection of objects is a rule that describes how to transform or "move" each object in the collection. For us, the objects are the points in the familiar coordinate plane and the collection is the plane itself (composed of all its points). A transformation, then, is a rule that describes how to move each of the points in the plane to new locations. Every point has to go somewhere -- even if it is just to where it was in the first place. No two different points can go to the same place.

Examples: On a piece of graph paper, draw a set of axes, a scalene triangle (all sides of different length), an isosceles triangle (two sides equal), and a rectangle. Place a piece of tracing paper on top of the graph paper and trace the axes, the triangles, and the rectangle.

1. Translation: All points move the same distance in the same direction. On your graph paper draw an arrow. Place the tracing paper on top, lining up all the figures. Put a dot on the tracing paper over the tail end of the arrow. Now move the tracing paper, keeping the axes on the tracing paper parallel to the axes on the graph paper, until the dot is over the point of the arrow. Every point of the tracing paper has moved the same amount. Observe how the figures you traced have all moved the same distance and direction from their original positions (on the graph paper).
   

2. Rotation: Line up the figures on the tracing paper with those on the graph paper. Stick a pushpin through the two sheets of paper. Holding the graph paper still, rotate the graph paper around the push pin. Observe that some points move further than others. What determines this? Observe that, since we have not stretched or crumples the paper, the figures stay intact -- the triangles are still triangles, the rectangle still a rectangle. Observe also that there is one point that doesn't move at all. What point is that?
   

3. Reflection: Draw a long line across the graph paper. Line up the figures on the tracing paper with those on the graph paper and trace the line you just drew. Lift up the tracing paper, flip it over, and put it back down, placing the new line exactly over its original on the graph paper. Notice that what was on one side of this line on the graph paper is now on the other side on the tracing paper. Observe also that some points move farther than others. What determines this? Are there any points that don't move at all?

4. Dilation: Mark a point on your graph paper. Draw line segments from each of the vertices of the figures you have to this point. Mark the mid-point of each of these segments. Join up these new points in the same way their "ancestors" were joined in the original configuration. Measure the sides of the original figures and the sides of the new ones. What do you see. Are there points that have moved further than others? What determines this? Are there points that do not move at all?

5. Identity: This seemingly trivial transformation "moves" every point nowhere! In other words, all points stay put. This doesn't seem like much of a transformation, but it does fit the description in that it provides a rule telling you what to do with each and every point on the plane. Moreover, there are good theoretical reasons why we want to consider this as a transformation, albeit a very special one.

What is an Isometry?

Of particular interest to us will those transformations that preserve distance. This means that if two points are distance d apart before they are moved, they are still distance d apart after the transformation. Which of the examples above are isometries?

We will prove later that, in an isometry, not only are distances preserved, but also angles. Thus, although all points may move, the plane is not stretched, or crumpled in any way. Thus isometries are often referred to as rigid motions of the plane. The first three exercises with tracing paper illustrate this well. You can perform a translation, rotation, or reflection by just sliding, turning, or flipping the paper, without distorting it in any way. You obviously cannot do this with a dilatation.

Why should we be interested in transformations of the plane?

• We will use our study of transformations to help us understand and describe symmetric patterns.

• Historically, transformations have been the foundation for a deeper study of the underlying principles of geometry.

• The way transformations can be classified and combined is an example of a very important mathematical structure known as a group.

• Transformations of the plane lead to transformations of space, which are crucial to understanding how modern computer graphics systems work. The computer animations that have been so popular recently (Jurassic Park, Star Wars, Bug's Life, Toy Story, ...) could not be developed without extensive use of geometric transformations.

• The topic provides a good opportunity to experiment with The Geometer's Sketchpad. This software package has revolutionized the way many educators, at all levels, teach their students to explore geometry.