Notes — Chapter 8: Transformations
To the Student. Trace ashape onto a transparent sheet and see whether you can move the sheet so that the tracing matches a second shape. If you can, the shapes are congruent. ’The motion of the sheet gives a transformation of the plane. If the motion is a sliding of the sheet without turning, the transformation is a translation. If you could pin the sheet at one point and turn it to make the match, the transformation is a rotation about the pinned point, its center.
To match a left handprint with a right one, you must flip the sheet over. You would reverse it left-to-right about the symmetry line between the two handprints. The handprints are congruent, but they look different because each is a mirror image of the other. Flipping
the sheet has the same effect as reflecting one handprint through the line of symmetry between the two prints.
To show that a left footprint is congruent to the next right footprint, trace the left footprint, slide the sheet forward a pace, then flip it over. A flip is always needed when matching shapes that are mirror images. This slide and reflection gives a glide reflection.
Two shapes in the plane are congruent precisely if one can be carried onto the other by one of the transformations described above. These are the transformations that preserve size and shape in the plane, the isometries of the plane. This chapter shows you what these transformations are, how they are defined by reflections, and how they relate to symmetry.
▶ On Transformations. |The material in this chapter has only been studied systematically for about 150 years. The ideas are basic and can be grasped intuitively, but the details could fill a course by themselves. Jacobs’s approach in the text is intuitive. This note sketches the origins of the transformational approach, how it relates to coordinates, and how one would prove the facts seen in this chapter.
Before geometry became a science, people recognized that some pairs of different things had the same or similar shapes. If two shapes are the same, then we could move one on top of the other so that they would coincide. Euclid used this notion for figures; it is called superposition in older texts. Postulates give us conditions for triangles to be congruent, meaning they are superposible. A full appreciation superposition came only with the systematic use of transformations in geometry. This was launched 1872 by Felix Klein in a lecture referred to as “Kline’s Erlanger Programm.”
For Euclidean geometry, the key transformations are the zsometries, the transformations of the plane onto itself that preserve Euclidean distances. A physical model these isometries is given by tracing figures onto sheets to see whether you can make one figure match another.
We can imagine our transparent sheet extends as far as we wish, so the isometry that shows that two figures are congruent can be applied to other figures traced onto the same sheet. Before there were flexible transparent sheets, geometers had already classified the isometries of the plane and shown how these transformations can be produced by single reflections or combinations of two or three reflections. Knowing what the isometries are allows us to make precise the Euclidean idea of superposition and to define congruence for any pair of figures.
In this chapter the properties of isometries are developed geometrically. In the exercises, students discover how transformations affect the coordinates of the vertices of figures. More elaborate coordinate methods are used for applications such as computer graphics or map making, but the link between transformations and coordinates is already clear from these exercises. (See
Matrices and Computer Graphics in the supplemental notes for Chapter 12 here.)
Jocobs lets students discover what transformations tell us about congruence and symmetry. He omitted proofs, but many facts about isometries are easy to estab- ‘lish. For example, if an isometry carries △ABC onto AA’B’C’, then △ABC & AA‘B’’C’, by SSS, so ∠ABC = ∠A’B‘C’, by CPCTE. This proves that isometries preserve angles because they preserve distances. For convenience, Jacobs made “preserving angles” part of his definition of zsometry, but this property is a consequence of preserving distances.
The properties of reflections are also taken for granted here, but they are also easily proven. Intuition tells us that when A and B are reflected in line ℓ to give A’ and B’ we have equality of lengths: AB = A’B’. This follows from the definition of reflection and the congruence postulates. It is also intuitive that if two transformations both preserve distances, then their composition also preserves distances. Because of this, any composition of reflections is an isometry. Jacobs left this as an unstated assumption, but is easy to state and prove.
Isometries of the plane are either reflections, translations, rotations, or glide reflections — plus the unmentioned identity transformation. Each sort of isometry is characterized by certain properties. For instance, reflections leave just the points on their mirror line fixed and rotations leave only their centers fixed, except for rotations by multiples of 360°, which leave every point fixed. Translations by a positive distance move lines onto lines parallel to themselves and leave no point fixed.
Conversely, a 1:1 transformation of the plane onto itself that sends lines onto lines parallel to themselves and that leaves no point fixed must be a translation. Here is why: Pick a test point P, it moves to some other point P’. Look at the line PP’. The image of this line is parallel to the original line and contains P’, so this line is sent into itself. We say this line is znvariant. The same argument applies to any other point Q, so the line QQ’ is also invariant. If these lines were to intersect, their point of intersection would have to be a fixed point. Because our transformation has no fixed points, QQ’ ∥ PP’. Also, PQ ∥ P’Q’, because our transformation sends lines into lines parallel to themselves. So PQQ’P’ is a parallelogram, and it follows that an arbitrary point Q moves in the same direction and through the same distance as our test point P. This shows that our transformation is a translation.
A dilation about the center (0,0) by scale factor A sends point (x, y) to point (λx, λy). I chose the center (0,0) so that it would be easy to express this transformation in coordinate form. Dilations also send lines into lines that are parallel to themselves. Here I allow the possibility of a negative A, so that reflections through the center of dilation are also considered dilations. I allow centers other than (0,0); the geometry is the same, but the
algebraic formulas are more complicated. A dilation with A = −1 is the point symmetry about its center. This point symmetry is also a rotation by 180° about the same center. Dilations have unique fixed points, except when A = 1 and every point stays fixed. If = 1, we have the identity transformation.
The composition of a translation and a dilation sends lines onto lines parallel to themselves. If A # 1, this combination must be equivalent to a single dilation with the same A, but with a new center. This shows that the composition of a translation and dilation with A 4 1 again gives us a dilation. This is a general result in the field of transformational geometry.
We could make distance the basis for our study of the geometry. This is done analytically when we look at the coordinate plane with the usual Pythagorean distance. A circle is the locus of points equidistant from its center, so distance alone lets us define circles. Every line is the locus of points equidistant from the end points of any segment that that line bisects, so lines can also be defined using only distance. Isometries preserve distance, and so they necessarily preserve lines and circles.
Furthermore, any isometry will send a pair of parallel lines, 2 and m, into lines @’ and m’ that must again be parallel. Using these properties of isometries we can see why an isometry must be of one of the types already described. This requires some care, but no new ideas are involved. I outline a possible approach following my lesson-by-lesson comments below.
A good general source on geometry and transformations is H. S. M. Coxeter’s Introduction to Geometry, Second Edition, John Wiley & Sons, New York, 1961. You might also look at books having “transformational geometry” in their titles or descriptions.
Lesson 1. Transformations. Transformations and symmetries are essential tools in art, crafts, and technology. We begin with a comic strip that must be read both right-side-up and upside-down to be appreciated. This lesson defines transformation, and describes basic types of geometric transformations: translations, rotations, reflections, and dilations.
The exercises are concrete: drawing and observing, analyzing the operation of an escalator in terms of translations and rotations. We see transformations used in computer graphics, and we see how transformations and coordinates are related.
Lesson 2. Reflections. Points A and B are symmetric with respect to line ℓ if @ is the perpendicular bisector of the segment AB. This we saw at the start of Chapter 6. Here we define the reflection of P through the line 2 to be the point P’ such that @ is the perpendicular bisector of PP’, if P is not on @. If P is on @, we define P’ = P. Reflections generate figures that are symmetric with respect to the mirror line.
In the chapter opener, two successive reflections in parallel mirrors gives an image moved in a direction perpendicular to the mirror planes by a distance twice the distance separating the mirrors. This double reflection is our definition of a translation. The direction of the translation is perpendicular to the mirrors, and the magnitude of the translation is the distance the image moves. Every point moves the same distance and in the same direction. See Triangle Reflections (44-47). After two parallel reflections a line returns to its original direction, so the translate of a lines is parallel to the original line. This is characteristic of translations, as noted above.
Rotations are defined as the compositions of reflections in intersecting lines. The point at the intersection of the lines of reflection remains fixed and is the center of the rotation. The magnitude of the rotation is the angle through which the image is rotated. This angle is twice the angle between the lines of reflection that define the rotation. (See Exercises 48-52.) A characteristic of rotations is that they have a single fixed point. The angle between a line ℓ and its image ¢’ under a rotation is always equal to the magnitude of the rotation.
Lesson 3. Isometries and Congruence. Two very different looking shapes turn out to be congruent in the chapter opener. In this lesson we give the general definition of congruence based on isometries. We see how shapes that are congruent can be made to coincide by transforming them by one, two, or three reflections. The three-reflection transformation introduced here is a glide reflection.
The exercises include: art, piano-moving, rowing, children’s swings, and coordinate methods for transformation problems. The Set III exercise makes interesting use of the transparency of tracing paper.
Lesson 4. Transformations and Symmetry. A figure has a symmetry if it can be mapped onto itself by an isometry. We have seen line and point symmetries in Chapters 6 and 7. Now we return to these symmetries from point of view of transformations. We add n-fold rotation symmetry and translation symmetry to our list of symmetry types.
In the exercises we see how rotational symmetry shows that vertical angles are equal and how the fact that the angles opposite the equal sides of an isosceles triangle are equal also follows from symmetry. Water wheels, piano keyboards, orchards, and quantum wave functions are all seen to possess special sorts of symmetry. The beauty of symmetrical structures goes with their useful properties. Jacobs gives some sources relating to symmetry, and your students can find many others in libraries or on the Web.
Classification of the Types of Isometries. The idea is to systematically consider various properties that
an isometry might have. We begin by looking at transforms that send some line ℓ into itself but do not leave any points fixed. We will see that these must be translations or glide reflections. We then work through all the other possibilities.
ll” Case One: Isometries that leave some line invariant. Let θ be such an invariant line.
▶ Subcase A: isometries that leave no point fixed. Putting a ruler along @, we see that there must be some constant c such that points the points of @ are shifted from a position with coordinate x to one with coordinate z +c along ℓ.
Next, establish Cartesian coordinates in the usual way using the ruler we have along @ and a ruler with the same scale on a perpendicular axis crossing ℓ at the point with coordinate 0. This point will be taken as the origin, O, of our coordinate system. Further analysis shows that this isometry must be either a translation along @ of magnitude c or a glide reflection, where the glide line is @ and the glide magnitude is c.
ll“ Subcase B: Our isometry leaves a line ℓ invariant and fizes some point P. Then our isometry must fix the point on ℓ closest point P. We make this point our origin and call it O. Again we look at what such an isometry can do by placing a ruler along @ with ruler’s zero at our origin O. Either the coordinates of points on & are unchanged by the isometry or they must be multiplied by −1. We introduce Cartesian coordinates with a second axis through O perpendicular to @. This second axis is also invariant and the isometry leaves its zero point fixed. So the coordinates of transformed points on the second axis are also multiplied by 1 or by −1. Because isometries preserve distances and angles, this pattern holds for the coordinates of all points in the plane, and the transformation is either a reflection in one axis or in the other axis, a rotation by 180° about O, or the identity. All of these transformations are on our list.
These Subcases A and B exhaust all the possibilities for an isometry that leaves some line invariant.
Case One Conclusions: Isometries that leave some line invariant must be either translations along that line, glide reflections along it, reflections in that line or in a line perpendicular to it, or rotations through 180° about some point on the invariant line
Case Two: Isometries that leave no lines invariant. I Lemma. Such isometries must leave some point fixed. Why must this be so? Pick any point P. If P is fixed we are done, so we may suppose P # P’. If P = Pe, then the midpoint of segment PP’ would be fixed, and we would be done. Continuing, if necessary, we would look at the distinct points P, P’, P’”. We note that if they were collinear, then they would determine an invariant line, contrary to hypothesis. Hence, in this case, we may conclude ∠PP’P” < 180°. Consequently, the perpendicular bisectors of segments PP’ and P’/P”
must meet at a point O interior to ∠PP’P”. Drawing the figure, we will show that O is a fixed point below.
We know that △OPP’ is sent onto AO/P’P” by our isometry. But there would seem to be two possibilities _for O’, one the point O and the other the reflection of O in the line P’P”. However, the latter is impossible, because, if it were the case, then ∠PP’P” and ∠P’P”P” would be equal alternate interior angles, and our isometry would send segment PP’ onto the equal and parallel segment P”P’”. This would give us a parallelogram: PP’P’”P”. Let M be the midpoint of PP’. its successive images under our isometry are the midpoints of successive equal segments in the path: PP’P”P’”” From what we know about parallelograms and triangles, we can conclude that these midpoints are collinear, and it would follow that the line they determine is invariant, contrary to our supposition in Case Two. Hence, O’ = O, and in Case Two there must always be some fixed point.
▶ Case Two Concluded. We are looking at an isometry without invariant lines, and we have established that there must be a fixed point, which we take to be our origin O. We center a 360° protractor on O. Recall that our isometry must preserve angles and that we are looking at an isometry that leaves no lines invariant. Using these facts, you can show that this isometry must be a rotation about O by some angle that is not a multiple of 180°.
Case Two Conclusions: An isometry that leaves no line invariant must leave some point fixed and it must be a rotation by an angle other than n x 180° about that point.
Cases One and Two are exhaustive and they give transformations of the sorts we listed, so our list of isometries is complete.
There are other possible approaches to classifying isometries. We could have begun by looking at isometries with one or more fixed points. Whatever the approach, care is needed to make sure that all possibilities have been looked at. This sketch shows that methods needed are part of school geometry. The length of this sketch also suggests why a serious study of transformational geometry would be better left for a separate course.