The Geometry of Transformations



Classification and Identification of Isometries



Now that we know how to identify the fundamental isometries, we can see what happens if we combine two or moreof them. That is, what do we get if we first rotate about a point and then reflect across a line? What do we get if we first rotate about one point and then rotate about another? You should treat these questions just like you treated the mystery problems from the last lesson.

  1. A Translation followed by a Translation


    1. Open the sketch Trans + Trans and copy it to your currrent workplace.

    2. You will see two segments that have been used to define translations. If you look under the Transform menu you will see that three custom transformations have been defined, corresponding to the two individual translations and the composite.

    3. Just as you did with the previous mysteries, identify the result of (A -> B) then (C -> D). If it is a translation can you identify the displacement vector, if it is a rotation can you identify the center and angle, if it is a reflection, can youidentify the mirror?

  2. Rotation followed by a Rotation


    1. Repeat the previous exercises, using the sketch Rot'n + Rot'n to identify the result of back-to-back rotations. Again, try to identify the defining properties as well as the type of the resulting isometry.

  3. Reflection followed by a Reflection


    1. Repeat the previous exercises, using the sketch Refl'n + Refl'n to identify the result of back-to-back reflections. Again, try to identify the defining properties as well as the type of the resulting isometry.

    2. In the previous exercise, the mirrors for the two reflections intersected. It is possible that the mirrors are parallel. This special case could lead to a different isometry. Open the sketch Refl'n + Refl'n(II) and repeat the analysis for this case.

  4. Rotation followed by a Translation


    1. Repeat the previous exercises, using the sketch Rot'n + Trans to identify the result of a rotation followed by a translation (or vice versa). Again, try to identify the defining properties as well as the type of the resulting isometry.

Back to the Workshop Outline Next Previous