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transform_geom_I - Transformational Geomepy Part1 Three...

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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 y-coordinate of these points is y = l. The x-coordinate 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 one-to-one if each element at most one element of Y is the of X one-to-one image of exactly one and onto If f : X 9 Y is one-to-one 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 one-to-one 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—to-one 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 y-axis] 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 between-ness 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 between-ness, 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 half-turn. 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, (3600-9) . 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 point-wise fixed by t ( or point-wise 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 point-wise. 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 point-wise invariant under t. Thus, it is possible that a figure (as a set) is fixed by transformation t, but the figure is not point-wise 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 point-wise 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 Fixed-Points Theorem : Let t be an isometry in Neutral Geometry. Suppose that there exist three distinct non-collinear 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 non-collinear points A, B, and C, then X’ = X for every point X inthe plane, that is, the only isometry t that fixes three non-collinear 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 non-collinear. 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 non-collinear 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 non-collinear 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 t2-1(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 non-collinear points and g = t2'1 0 t1 is an isometry that fixes all three c points. Thus, by the Three F ixed-Points 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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