Transformational Geometry is the study of transformations of the plane to the plane and how they affect the geometric figures which we studied in Euclidean geometry (such as lines, polygons, circles, etc).  The subject has very old roots (recall the use of “transformations” made by Euclid in The Elements such as Proposition 4?).  In these notes we will examine transformations of the plane in detail, classifying those which are distance-preserving (isometries), and studying some of the algebraic groups which they form.
     As is the custom we will identify the Euclidean plane with the set R² = RxR = {(x, y): x,y in R}, so that we can use the Cartesian-coordinate system notation and results from analytic geometry whenever it is convenient or desirable.
     First, we review some set theoretic results on functions.
     Recall that a function f:R²-->R² is said to be one-to-one iff f(P) = f(P’) implies P = P’, for all P, P’ in R² (or in coordinate notation, f(x, y) = f(x´, y´) implies (x, y) = (x´,y´), for all points (x, y), (x´, y´) in R²).  Recall that this is equivalently to “P & P’ different implies that f(P) & f(P’) are different”, for all points P, P’ in R²?  Thus distinct points are associated with distinct points by a one-to-one function f:R²-->R².
     A function f:R²-->R² is called onto iff for each P in R² there exists a point P’ in R² such that f(P’) = P (or in coordinate notation, for each (x, y) in R² there exists a point (x´, y´) in R² such that f(x´, y´) = (x, y)).  Hence each point of the plane is an "image" point of some point of the plane under an onto function f.
     A function which is both one-to-one and onto is called a bijection.

Definition 1.  A transformation f of the plane to the plane, f:R²-->R², is a bijection of the plane onto the plane .

     Several examples of transformations of the plane to the plane follow (these may be checked experimentally on GSP, see the "Transform Menu").

Examples
 1.  The identity function, i:R²-->R², i(x, y) = (x, y), for all points (x, y) of the  plane, is obviously a transformation of the plane onto the plane(?), which  "fixes" every point of the plane.  The function f(x, y) = (x², y²) on the other  hand, is not a transformation of the plane (why not?).

 2.  The mapping n:R²-->R², given by n(x, y) = (x, - y), for all points (x, y) in R²,  is a transformation of the plane (see figure 1) which reflects the plane about  the x-axis, (explain why n is a bijection).  Since n(x, - y) = (x, y), the  composition of n with itself is the identity transformation i, that is n(n(x, y)) = n(x, - y) = (x, y) = i(x, y), or n_°n = i.  Note that the points on the x- axis are "fixed" by this mapping for all x in R? (see "Reflect" in the GSP "Transform Menu").

 Fig. 1

 3.  The mapping t:R²-->R², given by t(x, y) = (x+1, y), for all (x,y) in R², is a  transformation of the plane (see figure 2) which translates the plane one  unit to the right (why is t a bijection?).  Observe that t moves all points in the plane, hence t has no "fixed points"?  (see "Translate" in the GSP  "Transform Menu").

 Fig. 2

 4.  The mapping r:R²-->R², given by r(x, y) = (-y, x), for all (x,y) in R², is a  transformation of the plane (see figure 3) which rotates the plane 90°  counter-clockwise about the origin (why is r a bijection?).  The only point  which r "fixes" is the origin 0  (see "Rotate" in the GSP Transform Menu)?

  Fig. 3

  Fig. 4

  Fig. 5

Problem 1.   In Examples 2 and 6 above we noticed that the composition of two  transformations was again a transformation.  Show that this is always true,  that is, show that the composition f_°g of any two transformations f and g of  the plane is again a transformation of the plane (recall the results from set  theory on the composition of one-to-one functions and of onto functions).

     We now take up the general problem of classifying the "distance preserving"  transformations (isometries) of the plane to the plane.

Definition 2.  A reflection in line m (or flip) of the plane is a transformation  R_m  such that if m is the given line and if P´ is the image of point P under  R_m, that is R_m(P) = P´, then the line m is the perpendicular bisector of the  line segment PP´.

 Fig. 6

        If R_m (P) = P´, then it follows, by the definition, that R_m (P´) = P and thus  (R_m °R_m )(P) = R_m (R_m (P)) = P = i(P), for any fixed line m and all points P in R².
      The transformation in Example 2 above is an example of a reflection  about the line y = 0 (explain).

Proposition:  A reflection Rm is a transformation (bijection) of  the plane.
Proof:  We show that R_m is a one-to-one and onto function.
     A.  The reflection R_m is one-to-one.
     Let  P,Q in R², such that R_m(P) = R_m(Q).  Then R_m(R_m(P))=R_m(R_m(Q)), hence P = Q   (see the discussion above).
    B.  The reflection R_m is onto.
      If P in R², either P is on m, or it's not on m.  If P is on m, then R_m(P) = P,  and if P is not on m, then R_m(P) = P´, and P´ & P differ.  Now, since R_m°R_m = i  (see the discussion above) the point P´ satisfies R_m(P´) = P, and R_m is onto.

     We see that the points on the line m are fixed by the reflection Rm in line m (as noted in Example 2 above).  Such points are formally called fixed points of the transformation.

Definition 3.   Let f:S-->S be a function on the set S*ø to itself. Then a point  P in R² is called a fixed point of f iff f(P) = P.

Example 7.  In coordinate notation, if P = (x, y) and m is the line x = c, then  R_m(x, y) = (2c-x, y); while if m is the line y = c, then R_m(x, y) = (x, 2c-y); and  if m is the line y = x, then R_m(x, y) = (y, x), (check these claims).
    In general, given any line Ax+By+C = 0, it is possible to give a coordinate  definition of the reflection about this line (can you see how?).
    For example, if A and B are non-zero, then it may be shown that
        R_m(x, y)  =  [-(A2 -B2)x -2Aby -2AC, -2Abx +(A2 -B2)y -2BC]/(A2 +B2),
is the equation for the reflection about m (show this?).

     It may be shown that reflections take lines to lines, angles are preserved, and triangles are taken to (congruent) triangles (see Problem 2 and figure 7).

  Fig. 7

Problem 2.  Show that the length of any line segment is unchanged by a  reflection in a line.  Show that this implies that angles are unchanged by  reflections.  Is a circle taken to a circle under a reflection?  Give examples to  support your claims.

Definition 4.  A reflection in a point P is similar to reflection in a line.  If P = (a, b) is a fixed point and A = (x, y) is any other point in R², then the  reflection of point A about point P, R_P(A), is that point B = (x´,y´) such that P  is the midpoint of the line segment AB, that is, a=(x+x´)/2 and b=(y+y´)/2  (see figure 8).
    Hence, x´= 2a-x, y´= 2b-y, and R_P(x, y) = (2a-x, 2b-y).
    The point P is clearly the only point fixed by R_P (right?).

  Fig. 8

    Note in the above discussion that there is an orientation-reversal under reflection about a line, but not under reflection about a point?
    For example, in figure 9 below triangle A´B´O is the reflection of triangle ABO about the origin O, while triangle A´B´´O is the reflection of ABO in line m.  Each of the triangles ABO and A´B´O is traversed clockwise when the vertices are followed in the order A-B-O, while A´B´´O is traversed counter-clockwise when the vertices are followed in the order A-B-O.

 Fig. 9

Problem 3. Show that reflection in a point P is equivalent to a reflection in a  line m, where m passes through point P, followed by another reflection in a  line m’, where m’ also passes through point P and is perpendicular to line  m (see figure 10 below).

  Fig. 10

Definition 5.  Given points A and B in the plane, A & B different, the line segment AB  determines a transformation T_AB which moves each point P of the plane to  a new point P´ such that the line segment PP´ = AB, and segment PP´ is  parallel to and in the direction of line segment AB.  This transformation is  called a translation (or slide) of the plane in the direction of line segment  AB (figure 11).

  Fig. 11

     It should be obvious that if AB and CD are line segments which have the same length and are parallel, then T_AB = T_CD?
     A translation (also known as a parallel displacement) is clearly a bijection of the plane which has no fixed points and is orientation-preserving (explain)?
     Example 3 above is an example of a translation in the direction of OP, where P = (1, 0).
     In general, if A = (x, y) and B = (u, v), then u = x+h and v = y+k, for unique numbers h,k in R?  Hence, given any point P = (s,t), the transformation T_AB of the plane in the direction AB determines the point T_AB(P) = P´= (s+h, t+k).  In Example 3 above, h=1 and k=0?  The numbers h & k are called the translation numbers of T_AB and determine T_AB completely (given h & k we can determine T_AB and given T_AB we can determine h & k?).  Therefore we will also denote T_AB by T_(h,k), especially if h and k are known.  This will be of particular importance when we consider groups of translations below.

Example 8.  If ABC is a triangle, where A=(0, 0), B=(1, 1), C=(1, 2), h=2, and k=3,  then the translation T_(2,3) transforms ABC to A´B´C´, where A´=(2, 3),  B´=(3, 4), and C´=(3, 5).  So ABC is congruent to A´B´C´ (see figure 12).

  Fig. 12

Problem 4. Prove or disprove:  Every translation is the composition of two  reflections in lines (hint: draw two lines and consider the associated line reflections).

Definition 6.  A glide-reflection (or slide-flip) G_(m,AB) is the composition of a  reflection R_m and a translation T_AB, G_(m,AB) = R_m°T_AB, where AB is  parallel to line m (figure 13).
    Notice that the order of the composition in a glide-reflection doesn't  matter, that is, R_m°T_AB = T_AB°R_m?

 Fig. 13

     If a translation is indeed the composition of two reflections (that is, Problem 4 is true), then a glide reflection is the composition of three reflections (?).

Definition 7.  A rotation (or turn) about point P through angle ø, R_(P,ø), is a  transformation of the plane (figure 14), such that if A is a given point in the  plane, A & P different, and A´ is that point after the rotation, R_(P,ø)(A) = A´, then PA =  PA´ and the angle determined by line segments PA and PA´ equals ø (the  rotation is counter-clockwise if ø>0 and clockwise if ø<0).

  Fig. 14

     Example 4 above is an example of a rotation about the origin 0 with angle  ø =Pi/2.  Observe that rotations are clearly bijections and are orientation-preserving (why?).
    Also the point P is the only fixed point of the rotation R_(P,ø) whenever  ø is not 2kPi, where k in Z?

Problem 5.  Show that a rotation R_(P,ø) is a compositions of two reflections  about lines m and m', respectively, where m and m' intersect at point P  (hint: draw a picture, and show that the lines m & m' exist).

    Note that a rotation is not uniquely determined by ø (we could rotate 2Pi - ø radians, or add any multiple of 2Pi to ø, and get the same result?).
    Also, since every point P in the plane has polar coordinates P = (r, Ø), where 0 <Ø < 2Pi and r not 0, it is possible to express all rotations about the origin through an angle ø by the polar formula R_(0,ø)(P) = (r, ø+Ø) (explain).
     In coordinate form, R_(0,ø)(x, y) = ((cosø)x -(sinø)y, (sinø)x +(cosø)y)?

Problem 6.  Prove that rotations preserve distances, hence they take lines to  lines, preserve angles, and take circles to circles.

Problem 7.  Give a scheme for computing R_(P,ø)  in terms of R_(0,ø)  and the  translations T_0P and T_P0  (hint: draw a picture of these transformations).

     Using the results from Prob. 10 and the coordinate form for R_(0,ø)   it is  possible to show that the coordinate form of a general rotation of the plane  about point P=(a, b) through angle ø has form:
             R_(P,ø)(x, y) = ((cosø)(x-a) -(sinø)(y-b) +a, (sinø)(x-a) +(cosø)(y-b) +b)?

Problem 8.  Prove that R_(P,Pi)(P) = R₀(P) for all points P in the plane.  Hence a  reflection about a point is just a rotation about the point through an angle of  Pi radians (see Problem 4 above).

Problem 9.  Consider the composition of a translation T_AB and a rotation  R_(P,ø).  Does this determine a new type of transformation?  Does the order  of composition matter?  (hint: see figure 15).

 Fig. 15

Definition 8.   A transformation f of the plane is called an isometry of the plane  if it preserves distances, that is the segment f(P)f(Q) = PQ, for all P, Q in  R².

Problem 10.   Show that all reflections, translations, glide-reflections and  rotations are isometries (hint: see Prob's. 2, 4, 5, 6, and Defn. 6).

Problem 11.  Some of the isometries in Problem 10 preserve  orientations (for  example triangle ABC goes to triangle A'B'C'), and some of them reverse  orientations  (triangle ABC goes to triangle A'C'B').  Which are which? (Hint: draw pictures.)

Problem 12.  Show that the set of fixed points and the orientation properties of  the isometries in Prob. 10 (see also Prob. 11) above determine completely the  type of the transformation by completing the following chart.

Problem 13.  Show that all isometries take lines to lines, triangles to congruent  triangles (and thus angles to congruent angles) and circles to congruent  circles (hint: see Prob's. 2, 4, 5, 6, & Defn. 6).

Problem 14.  If an isometry T is different from i and T has exactly one fixed point P, then T is  a rotation (hint: draw a picture and consider the possibilities).

Problem 15.  Prove the following.
     a.  If an isometry T fixes two points P and Q, then T fixes all points on the  line m through P and Q.
     b.  If an isometry T fixes three non-collinear points P, Q and R, then T = i.

Problem 16.  If an isometry T (not i) fixes two points A and B, then T is a reflection  in  the line m through A and B (use Prob. 15 and consider the points on an  arbitrary line m’ perpendicular to m).

    We are now ready to answer the question: “Are there any isometries, other than the four considered in Problems 10 and 12 above?" .
    The  following series of results will show that the answer is a definite NO!

Problem 17.  Given congruent triangles ABC and A'B'C', prove that there is  exactly one isometry of the plane which takes triangle ABC to triangle A'B'C' (hint: draw  a picture and translate point A to A', then rotating the translated result to get the two triangles to coincide, to show uniqueness, assume there are two  such functions f and g and consider g^-1_°f).

    All of the isometries considered above are compositions of at most three reflections in lines (Problems 4 & 5)?  If this holds true for an arbitrary isometry, then we're done(?).  This is the consequence of the following problem, which is one of the main results of these notes.

Problem 18.  Show that any isometry of the plane is equal to at most three  reflections in lines (consider the possible images of three non-colinear  points, see also Prob's. 13 and 17).  Hence the only isometries of the plane  are those listed in Problem 10.

Definition 9.  Another type of transformation of the plane (which is not an  isometry) is a dilation.  Given a fixed point P in the plane and a real number  n>0, the dilation D_(n,P) maps a given point Q to the point Q' on the ray PQ  such that PQ' = nPQ.

    Example 5 above is the dilation D_(2,0) (check this).  A dilation is clearly a bijection, hence a transformation of the plane?  The number n is called the scale factor  of the transformation (see figure 16).

Fig. 16

Problem 19.  Any dilation of type D_(n,0) satisfies D_(n,0)(x, y) = (nx, ny) (as in  Example 5).  Hence D_(n,P) = T_0P°D_(n,0)°T_P0 for any dilation D_(n,P).  Also,  D_(n,0)°D_(1/n,0) = T_OP for some point P.

Problem 20.  Lines are transformed to (parallel) lines and angles are preserved  by  dilations.  Hence the image of a geometric figure under a dilation is  similar to the given figure.

Problem 21.  Determine the inverse of each of the transformations in Prob. 10  and the inverse of a dilation transformation (recall that a bijection f  always determines another bijection f^-1, the inverse of f, and that the  composition of any two bijections is another bijection?).

    The study of transformations of the plane nearly always includes a discussion of groups.  Why groups?  Certain collections of transformations of the plane naturally form an (algebraic) group.  This, in turn, allows the facts which are known about such groups to determine new results about the transformations of the plane.  We will briefly look at a few results in this direction, but first a definition and some examples of groups.

Definition 10.  A group <G,•> is a set G * ø with an operation "•" on G such  that:
    a.)  a•(b•c) = (a•b)•c for all a,b,c in G ("•" is an associative operation),
    b.)  there exists an element e in G such that e•a = a = a•e, for all a in G (there is an identity “e” for the group),
    c.)  for each a in G there is an element b in G such that a•b = e = b•a (all  elements have inverses, denoted by a^-1).
 

Example 9. The set of integers {0, 1, 2, ..., n-1} under addition modulo n is an example of a group, denoted by <Zⁿ, + >.  See the Cayley table below for the example of Z¹².  This is probably the most familiar example of a group and is sometimes called “clock arithmetic”, where e = 0 and the inverse of k is (12-k) modulo 12.

Example 10.  Another example of a group, which is familiar from Calculus, is  the set R² with coordinate-wise (vector) addition, that is (x, y) + (x', y') =  (x+x', y+y'), for all (x, y),(x', y') in R².  Similarly, Rⁿ is a group under coordinate-wise addition, for all n > 2.

Example 11.  The set of all invertible functions of the plane to the plane,  denoted by Inv(R²), is a group under composition, since the composition of  two invertible functions is an invertible function (why?), the composition  of invertible functions is associative (why?), the identity function i is the  identity "e" for the group (explain), and each invertible function f has an  inverse (function) f^-1, such that  f_°f^-1 = f^-1_°f = i (thus inverses exist?).

    Care must be taken to make sure that the composition of two transforma-tions of a certain type yields a transformation of the same type when checking to see if a set of transformations of a certain type forms a group under composition (see Prob. 29 below).
    Using the observations in Example 11, it is straight forward to prove that the set of all transformations of the plane under composition forms a group.

Problem 22.  The set of all transformations (bijections) of the plane forms a  group, called Bij(R²), under the operation of composition.

Problem 23.  Is the set of all possible rotations of the plane a group under  composition? Explain (hint: look at rotations R_(0,Pi)).

    Recall that a translation T_AB is completely determined by its translation numbers {h, k} (see the discussion below figure 11 above).  Thus there is a one-to-one correspondence (bijection) between the set of all translations, Trans(R²), and R², given by the function f(T_AB) = (h,k), where T_AB =T_(h,k).   In addition, these sets are groups (Trans(R²) is a group by Prob. 24 below, and R² is a group by Example 10 above).
    The obvious (?) question then is "How are these groups related?".   The answer is contained in Prob. 24 below, however, first we need to examine the concept of “equivalent groups” (“isomorphic groups” in group theory terminology).

Definition 11.  Call two groups <G,• > and <K,o> isomorphic if there is a  bijection h:G-->K between them, as sets, such that the operations are  preserved, that is, h(a•b) = h(a)oh(b), for all a,b in G, (this means that the  same result occurs when going around the diagram in figure 17 below -  either way - from upper left to lower right).

 Fig. 17

Example 12.  The set {a, b} with the operation "o" defined by: aoa = a, aob =  boa = b, and bob = a, forms a group <{a,b},o> (check this) which looks  suspiciously like the group Z² (consider their Cayley tables).  In fact, if we  identify element a with 0 in Z² and element b with 1 in Z², then it follows that  these groups are indeed isomorphic?

Problem 24.  The set of all translations of the plane, Trans(R²), is a group under  composition.  It is isomorphic to the additive group R², under the bijection  f:Trans(R²) --> R², where f(T_AB) = (h, k), for h & k the translation numbers  of the translation T_AB.  Since we know a lot about R² we have thus have a  lot of information about Trans(R²)?

    We see from above that the set Bij(R²) is a subset of the set Inv(R²), and that each is a group under the same operation (composition).  Whenever such a situation occurs, we have an example of an object called a "subgroup".

Definition 12.  A subgroup <H,• > of a group <G,• > is a subset H of G, H not ø, which is itself a group under the same operation "•".

    In order to determine if <H,• > is a subgroup of <G,• > we could just show that H is a group under the operation "•", but there is a theorem in Group Theory (which is not very difficult to prove) which states that it is sufficient to show that <H,• > satisfies: H not ø, and a•b^-1 in H, whenever a,b in H.
 We will assume this result below when needed.

Problem 25.  Show that the set of isometries of the plane, Iso(R²), forms a  subgroup of Bij(R²), and that Trans(R²) is a subgroup of Iso(R²).

Definition 13.  A group <G,• > is said to be abelian iff a•b = b•a for all a,b in G.

    Most of the groups which we are considering here are of this type.

Problem 26.  Is the group Bij(R²) abelian (explain)?  Also check Iso(R²) and  Trans(R²) for the abelian property.

Problem 27.  If Rot(R²) is the set of all rotations about the origin, show that  Rot(R²) is a group under composition (is this group abelian?).  Also show  that Rot(R²) is a subgroup of Iso(R²).  Is Rot(R²) isomorphic to, say, the  group of reals modulo 2Pi, under addition modulo 2Pi, R_2Pi, by the bijection  R(0,ø) -->ø?  Argue that Rot(R²) is not isomorphic to Trans(R²) (hint: see  Prob. 24 above).