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Unformatted text preview: Transformational Geomepy  Part1 Three Approaches to the Development of Geometry 1) Synthetic Approach (via Axiomatic Systems)
2) Analytical Approach (via Algebraic Methods) 3) Transformational Approach (via the Study of Transformations) Example of the Analﬁical Approach: Deﬁne coordinate axes. Let P = ( 2, 0 ). The secant line y = l intersects the circle
C(P, 2) at two points A and B. Determine the coordinates of these points of intersection. Solution: The equation of C(P, 2) is (x—2)2 + y2 = 4. The ycoordinate of these points is y = l. The xcoordinate is a solution of the equation
(x—2)2+ 12 = 4 9 (x—z)2 = 3 —> x—2 =ix/3 9 x=2::\/3. The points ofintersection are: (2 — V3 , l ) and (2 +\/3 , 1 ) . Note: This (x, y) coordinatization of points gives the Euclidean plane an orientation so
that discussion of clockwise rotation and counterclockwise rotation is meaningful. The Transformational Approach : The study of functions which preserve various geometric properties. [This approach provides a logical framework that gives meaning to Euclid’s use of
superposition] The Deﬁnition of Trian 1e Con ence: What it means to sa AABC 5 A A ’ B ’ C ' Euclid: “Move AABC and place it over and
above A A l B / C ' and the ﬁrst triangle ﬁts on C .
top of the second triangle exactly.” A
In Transformational Geomeﬂ: “There is a
A B B CI transformation of the plane, t, which is an
isometry such that t(A) = A ’, t(B) = B ’, and
t(C) = C ’.” Deﬁnitions: Let f : X 9 Y be a function. onto at least one
Function f is onetoone if each element at most one element
of Y is the of X
onetoone image of exactly one
and onto If f : X 9 Y is onetoone and onto,
then the inverse fimction f ‘1 : Y 9 X exists and
f‘l(y) = x ifandonlyif f(x) = y, forall xinXandallyinY. Also, f‘1(f(x))=xforallxinX and f(f‘1(y))=yforallyinY. Note: To prove that function f : X 9 Y is onetoone and onto, it sufﬁces to exhibit
another function g : Y 9 X such that g( f(x) ) = x for all x in X and f( g(y) ) = y
for all y in Y. When such a function g is found, it will turn out that g = f ‘1 , the
inverse function of f . Deﬁnition: A transformation of the (Neutral) plane, t, is a one—toone mapping (function)
of all points in the plane onto all points in the plane. Motion is another term for a
transformation of the plane. In the following examples, the model of the Neutral plane used is the Cartesian X,Y
Plane. Example 1: t1: A = ( x , y) 9 A’ = ( 2x , 3y) is a transformation ofthe plane. (Note: X’ = t(X) is a common notation for the image of X under t .) Example 2: t2: A = ( x , y) 9 A ’ = ( — x , y) [Reﬂection about the yaxis] Theorem: The composition (serial application) of two transformations of the plane is
also a transformation of the plane. (Theorem 5.3.1) Example3: t2° t1: (x,y) 9 t2(2x,3y) = (——2x,3y)isatransformationofthe
plane. Deﬁnition: Let t be a transformation of the plane and let P be a geometric property.
Then, property P is invariant under t (or t preserves property P ) if, when a set of points
has property P , then the set of images also has property P . Note: t1 from Example 1 preserves the property of collinearity of points but it does not
preserve the distance between points. t2 from Example 2 preserves both collinearity and
distance between points. Convention: If t is a transformation of the plane and X is a point in the plane, then the
image point t(X) is denoted X’. Thus, t(A) = A’ and t( t(A)) = t(A’) = A”. Deﬁnition: An isometry t is a transformation of the plane under which distances between
points is an invariant property, that is, t preserves distances between points. Thus, if t is an isometry and A and B are any two points,
then AB = A’B' = (t(A)) (t(B)) .
In Examples 1 and 2, t1 is not an isometry, but t; is an isometry. Notes: The composition (that is, the serial application) of two isometries is an isometry
(Theorem 5.3.2). Also, if t is an isometry, then t'1 is also an isometry. An isometry preserves betweenness and collinearity (Theorems 5.3.3 and 5.3.4) The quality of being a line, or a line segment, or a ray, or an angle, or a circle, is
invariant under an isometry. (Corollary 5.3.6, Theorem 5.3.9) Theorem 5.3.7: The image of a triangle under an isometry is a congruent triangle. Proof: Let triangle A A B C and isometry the given. Since t preserves
collinearity and betweenness, the image of A A B C under t is the triangle A A ' B l C ’. Now, AB = A'B', BC = B’C’, and AC = A'C’, sincetpreserves distance.
Therefore, AA’B'C' is congruent to AABC by S S S . QED Corollary 5.3.8: The property of angle measure is preserved by an isometry. Proof: Given 1 ABC and isometry t, the angle is an angle in AABC. AABC is
congruent to AA’B’C’, so A A’B'C' is congruent to 4 ABC. QED The Transformational Deﬁnition of Congruence: Two ﬁgures are said to be congruent if
one is the image of the other under some isometry. The above deﬁnition of the congruence of two ﬁgures is a rigorous formulation of the
method of rigid translation, reﬂection, and superposition that Euclid used in a non
rigorous fashion to establish the congruence of triangles and other ﬁgures in his
Elements. Deﬁnitions of the Isometries: Translation Reﬂection Rotation and Glide Reﬂection Deﬁnitions for the four isometries are presented here in the context of Euclidean
Geometry. These isometries also have deﬁnitions in Hyperbolic Geometry, but certain
adjustments need to be made to the deﬁnitions in some cases. However, the definition of the “Reﬂection” isometry is exactly the same both in
Hyperbolic Geometry and in Euclidean Geometry. Deﬁnition: If P and Q are two points in the plane, we call the segment 732? a directed line segment or a vector when we distinguish endpoints P and Q as the beginning point
and the endpoint, respectively. Deﬁnition: Let 51’ represent the Euclidean Plane. Let 70—5 be any vector.
Deﬁne the TRANSLATION TPQ : 9 ——) 5’ as follows:
For each point X in (1’, deﬁne TpQ (X) = X’ as follows: If X is on line m = W, then TPQ ()0 = that point X’ on line 775 such that
the vectors )0? and 7'? have equal length and point in the same direction. . If X is not on line m, then there is aunique line Z which is parallel to m and
passes through X and TPQ (X) = that point X’ on line Z such that the vectors XX’ and W have equal length and point in the same direction. TPQ is called the translation of the plane through the vector 75?. Also, TPQ '1 = TQp Consequences of the deﬁnition of the translation TpQ :
1) TPQ(P) = P’ = Q. 2) If vector it? is any vector which is parallel to vector 7Q such that 7471—37 and '1? have equal length and point in the same direction, then TPQ and TAB are two names for the same translation. Thus, if A’ = TpQ (A) , then TpQ and TAAI
are two names for the same translation. 3) If X is not on line m, then P Q X’ X is a parallelogram, by Theorem 4.2.14. 4) If S is the coordinatization of line m = Q" such that (P)s = 0 and (Q)s > 0, then, for any point X on 7’5, (X’)s = (X)s + PQ ; thus, to compute the
coordinate of X’ = TPQ (X) , add the length PQ to the coordinate of X . Nate: Every translation is an isometry and it maps a line segment onto a parallel (or
collinear) line segment (Theorem 5.3.13 and Corollary 5.3.14). Deﬁnition: Let E be any line in the Euclidean Plane 37‘ .
Deﬁne the REFLECTION Rt : .7’ —> 9 as follows: For each point X in 51’, deﬁne R¢(X) = X' as follows:
If X isonline t, then wedefme Re(X) = X,thatis, X’= . If X is not on line e , then we deﬁne R c (X) to be equal to that point X’ on the
other side of line Z such that line 8 is the perpendicular bisector of 7(3?’ . R e is called the reﬂection about the (mirror) line e and it is its own inverse,
that is, (Rc)'1 = Re. Theorem 5.3.16: Every reﬂection is an isometry. (The proof is left as an exercise.) Deﬁnition: Let P be a point and let 6 be a directed angle, that is, an angle of rotation.
We deﬁne the ROTATION R M : 5’ ——> 3’ as follows: For each point X in 3‘, R P,6 (X) = X’ where X’ is deﬁned as follows: IfX = P, then deﬁne Rp,9(P) = P andso P’ = P. If X 72 P, then R P,9 (X) = X’ is the point on the circle C( P, PX) such that the
directed angle A XPX’ is congruent to the directed angle 0 . Rp, 9 is called the rotation of the plane about the point P through the angle 0 . Point P is called the center of rotation for R p, 9 . Also, RP, 1800 is called a halfturn. It is readily seen that RP, e is a transformation of the plane because it has an inverse
function, namely R 13,9 '1 = R p, (36009) . Theorem 5.3.16: Every rotation is an isometry. (The proof is left as an exercise.) Deﬁnition: Let K be any line in the Euclidean Plane flj .
Let vector FQ be any vector that is parallel to line Z or lies on a line B itself. Deﬁne the GLIDE REFLECTION GL pQJ : 51’ —> 57' as follows:
GLPQJ = TPQ ° Rt . Thus, the Glide Reﬂection GL pQ, e is deﬁned to be the composition of the reﬂection Re
about the mirror line i followed by the translation TpQ by the vector P_Q. The same glide reﬂection G pre results if the translation TpQ is applied ﬁrst, followed by the reﬂection Re , that is,
GPQJ = Rt ° TPQ , also. Theorem 5.3.17: Every glide reﬂection is an isometry. (The proof is left as an exercise.) Deﬁnition: Let t be a transformation of the plane. Let A be a point. Point A is called a ﬁxed point of t (or an invariant point of t) if t(A) = A’ = A .
We also say that the transformation t ﬁxes pointA if t(A) = A’ = A . For example, the center of rotation P for rotation RP, e is the only ﬁxed point of R1), 9.
If every point of a particular geometric ﬁgure is a ﬁxed point t, then we say that the ﬁgure is pointwise ﬁxed by t ( or pointwise invariant under I ).
For example, the mirror line b of a reﬂection Re is point—wise ﬁxed by the reﬂection Rt . A More General Deﬁnition: Let t be a transformation of the Euclidean Plane. A geometric ﬁgure (as a set of points) is invariant under t if the set of images under t
of points in the ﬁgure is the same as the set of points in the ﬁgure itself. We also say that the transformation t ﬁxes the geometric ﬁgure, but not necessarily
pointwise. For example, if the transformation t is the translation TPQ , then the line containing the
vector PQ is invariant under t. Note that this line is not pointwise invariant under t. Thus, it is possible that a ﬁgure (as a set) is ﬁxed by transformation t, but the ﬁgure is not
pointwise ﬁxed by t . Examples:
1) For the rotation R p, 9 , the center of rotation P is the only ﬁxed point of RP, 9 . 2) For the reﬂection Re , the mirror line Z of the reﬂection is pointwise ﬁxed by Re ,
and a line perpendicular to the mirror line i is invariant under the reﬂection Re . 3) For the glide reﬂection G pQ, e , no point in the plane is ﬁxed by G an .
4) For the translation TpQ , no point in the plane is ﬁxed by TpQ . Deﬁnition: Let AABC be a given triangle. Note that “AACB”, “ABAC”, and “ACBA”
are different symbols or labelings for the same triangle. The orientation of a labeling of
a triangle as “AXYZ” is a clockwise orientation (or a counterclockwise orientation) if,
when we travel along the triangle so as to encounter vertices X, Y and Z in that order, the
direction traveled around an interior point of the triangle is a clockwise direction (or a counterclockwise direction).
(See Figure 5.3.7 on page 268 of the book.) If the orientation of a triangle labeling is a clockwise (or counterclockwise) orientation,
we say that the labeling is a clockwise labeling (or a counterclockwise labeling) of the
triangle. Also, the notion of orientation of labeling generalizes to other polygons, such as
quadrilaterals, pentagons, etc. For example, if the orientation of the labeling “AABC” of a triangle is a clockwise
orientation, then the labeling of that triangle as “AACB” is a counterclockwise labeling
of the triangle, and the labeling of that triangle as “ABCA” is a again a clockwise
labeling of the triangle. The situation is reversed if “AABC” is a counterclockwise labeling. Deﬁnition: Let t be a transformation of the plane. Then, t is called a direct transformation if application of the transformation t preserves
the orientations of ﬁgures (as evidenced by its preserving of the orientation of the
labelings). Thus, if t is a direct transformation, then the labelings “A A B C” and
“A A ’ B ’ C ’ ” have the same orientation (i.e., both are clockwise or both are
counterclockwise) for any triangle AABC and writing A’ = t(A), B ’ = t(B), and
C’ = t(C) . And, t is called an opposite transformation if application of the transformation t
reverses the orientations of ﬁgures (as evidenced by its reversing of the orientation of the
labelings). Thus, if t is an opposite transformation, then the labelings “A A B C” and
“A A ’ B ’ C ’ ” have the opposite orientations (i.e., one is clockwise and the other is counterclockwise) for any triangle AABC Examples: A translation is a direct transformation.
A reﬂection is an opposite transformation.
A rotation is a direct transformation, A glide reﬂection is an opposite transformation. Note: Translations, Rotations, Reﬂections and Glide Reﬂections can be characterized by:
1) whether they are direct or opposite transformations,
2) the number of invariant points they have, and 3) the conditions under which the image of line segment is a parallel line segment. See Table 5.3.1 for the characteristics of each type of isometry in these dimensions. The Three Fixed—Points Theorem and the Three Points Agreement Theorem Theorem B 5.1 The Three FixedPoints Theorem : Let t be an isometry in
Neutral Geometry. Suppose that there exist three distinct noncollinear points A, B,
and C such that t leaves each point A, B, and C ﬁxed. Then, t ﬁxes every point in the
plane. Thatistosay, if t(A) = A’ = A, t(B) = B’ = B, and t(C) = C' = C,forthree
noncollinear points A, B, and C, then X’ = X for every point X inthe plane, that is, the only isometry t that ﬁxes three noncollinear points is
t = “the identity function” and every point of the plane is ﬁxed by t. Proof: Let t be an isometry and let points A, B and C be three distinct points which
are noncollinear. Suppose that t leaves points A, B, anc C ﬁxed.
Thatis,supposethat A, = A, B’ = B, and C’ = C.
Let X be any point in the plane. [We need to show that t ﬁxes X, i.e., that X’ = X . ] If t(X) = X, then we are done. So, suppose (by way of contradiction) that t does not ﬁx X .
that is, suppose X’ at X where t(X) = X’. Let line B be the perpendicular bisector of segment 3?)? and recall that A, B, and C are not collinear. [We will reach the contradiction by showing that A, B and C are all three
on line Z , which will contradict the fact that they are not collinear] Byassumption, A’ = A, B’ = B, andC’ = C.
Since t isanisometry, A’X’ = AX. AX’ = AX, since A’ = A, so Aisequidistantfrom Xand X’. Therefore, A is on the perpendicular bisector of )0? , by Theorem 3.2.8 . A is on line 2. 10 Since t isanisometry, B’X’ = BX. BX’ = BX, since B' = B, so Bisequidistantfrom Xand X’. Therefore, B is on the perpendicular bisector of XX’ , by Theorem 3.2.8 . B is on line 8. Since t isanisometry, C’X’ = CX. CX’ = CX, since C’ = C,
so C is equidistant from X and X’. Therefore, C is on the perpendicular bisector of W, by Theorem 3.2.8 . C is on line 8. Therefore, A, B, and C are collinear (they all lie on line (3 , the perpendicular bisector of
H'), and this contradicts the assumption that A, B and C are not collinear. Therefore X' = X, and so t ﬁxes point X. Therefore, for all points X in the plane, X ’ = X , that is, t = “the identity
function” and every point of the plane is ﬁxed by t . QED 11 Deﬁnition: Two transformations of the plane, t1 and t2, are said to agree at point P in the
plane if and only if t1(P) = t2(P). Note also that, for transformations t1 and t2 and point P,
tl(P) = t2(P) if, and only if, ((t2)'1 0 t1) (P) = P , that is, titP) = ow) if, andonlyif, (t2)‘1(ti(P>)= P. For example, if t1(A) = t2(A) = D , x m P
 F then, (t2)'1 (D) = (t2)'1(t2(A)) = A p D by deﬁnition of (t2)'l . 3 t2 Therefore, (t2)'1(t1(A)) = (t2)'1(D) = A 1.1m So, (of (new = A. More generally, for any point P:
(1) if t1(P) = t2(P),
then (t2)'1(t1(P)) = (t2)'1(t2(P)) = P, by deﬁnition of (t2)'1.
(2) if (t2)'1(ti(P)) = P ,
then t2(P) = t2( (t2)'1(t1(P)) ) = mp), by deﬁnition of oz)". Thus, transformations t1 and t2 agree at a point P if, and only if, point P is a ﬁxed point of the isometry g = (t2)'1 0 t1 . 12 Theorem 5.3.12 (The Three Points Agreement Theorem): Let t1 and t2 represent isometries. If t1 and t2 agree at three non—collinear points A, B, and C , then t1 and t2 agree at
every point X in the plane , that is, for every point X, t1(X) = tzOC) , and so, t1 and t; are names for the same isometry.
We say that an isometry is determined by its images on any three noncollinear points. [Note: This means that if AABC ; ADEF , then there is at most one isometry t such
that t(A) = D, t(B) = E, and t(C) = F.] Proof: Suppose that t1 and t; are isometries and suppose that points A, B, and C are three
noncollinear points such that t1 and t2 agree at A, B, and C ; thus, MA) = t2(A), mm = L203), and MC) = t2(C).
By the comments preceding this proof, t2‘1(t1(A)) = A , tz'] (t1(B)) = B , and t21(t1(C)) = C  t1 Let g be the isometry g = t2'1 0 t1. X m p
’ W F
D g(A) = A, g(B) = B,and g(C) = C. 3 t2 So, A, B and C are three noncollinear points and
g = t2'1 0 t1 is an isometry that ﬁxes all three c points. Thus, by the Three F ixedPoints Theorem (Theorem (NIB) 5.1), the isometry g ﬁxes
every point in the plane, that is, g(X) = X , for every point X in the plane.
Thus, t2'1 0 MOO) = X, for every point X in the plane.
Thus, t1(X) = t2 (X) , for every point X in the plane. t1 = t2 as transformations. Q E D ...
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