Geometric Transformations

Transcript

1 hapter 6 Geometric Transformations Goal: In the first three chapters, sets of axioms provided the basis for the study of geometries. In this chapter, the concept of transformation provides a different way of studying some geometries. Definition transformation f from X to Y is a one-to-one, onto mapping. Definition group of transformations of a set X onto itself is a nonempty set S of transformations together with an operation such that: 1. If f and g are in S, then fg and gf are in S. (losure) 2. If f, g and h are in S, then (fg)h = f(gh). (ssociativity) 3. There is a unique element in S satisfying If = fi = f for all f in S. (Identity) 4. Given f in S, there exists a unique element f 1 satisfying f 1 f = ff 1 = I. (Inverses) Theorem The set of symmetries of an equilateral triangle is a group of transformations. Homework omplete the group table below. onsider Figure 6.1. Let R l be reflection about l, R m be reflection about m and R n be reflection about n. Let R 120 be rotation about the centroid by 120 counterclockwise, and let R 240 be rotation about the centroid by 240 counterclockwise. Let I be the identity transformation. Let be function composition. I R l R m R n R 120 R 240 I I R l R m R n R 120 R 240 R l R l I R 120 R 240 R m R n R m R m R 240 I R n R n R 120 I R 120 R 120 R n R 240 R 240 R 240 R m R

2 n l 3 m 3 2 R l R m R n R 120 R Figure 6.1: Symmetries of an Equilateral Triangle Definition transformation f is an isometry of onto if it preserves distances. That is, for any two points P 1, P 2 of, the distance from P 1 to P 2 is equal to the distance from f(p 1 ) to f(p 2 ). In ordinary Euclidean geometry, the isometries studied are isometries of a set of points onto itself. Isometries of this kind are called motions. Euclidean geometry involves the study of motions of the artesian plane. Question 1 What types of transformations can be applied to a set of points in the plane so that the distance between any two points is always preserved? 6.1 Plane Motions 1. Translations: translation is a correspondence between points and their image points so that each image is the same distance in the same direction from the original point. translation is determined by determining the translation vector. The set of all translations forms a group. The product of two translations is a translation. The vector for the product is the sum of the vectors for the two translations. 2. Rotations: Let R(O, α) indicate a rotation through an angle of α about the point O. onvention: a counterclockwise rotation is associated with a positive angle. rotation is determined by determining the center and angle of rotation. The set of all rotations about one fixed point is a group of transformations. 40

3 Figure 6.2: Translation α... Ȯ. Figure 6.3: Rotation 3. Reflections: reflection R l utilizes a fixed line l. point on l is its own image. ny other point P is mapped into a point P such that l is the perpendicular bisector of P P. reflection is determined by determining the line of reflection. l Figure 6.4: Reflection The product of two reflections about the same line is not a reflection. Reflection reverses orientation. 4. Glide Reflections: glide reflection is a product of a reflection and a translation parallel to the fixed line of reflection. Glide reflection is determined by determining the line of reflection and the vector of translation. One geometric application of various motions of the plane is in simple geometric designs that form plane-filling repeated patterns (Work of M.. Escher). You will 41

4 v l Figure 6.5: Glide Reflection be asked to read an article describibg how to make Escher-like diagrams, also you will be required to make your own drawing. Homework Justify the following using Sketchpad: 1. if a palne motion is the product (composition) of two reflections then: if the two lines of reflection are parallel, the motion is a translation. if the two lines of reflection are nonparallel, the motion is a rotation. 2. if the motion is the product (composition) of three reflections then: if two of the lines of reflection coincide, the motion is a reflection. if the three lines of reflection are parallel, the motion is a reflection. if two lines of reflection intersect at a point on the third, the motion is a reflection. if two lines of reflection intersect at a point not on the third, the motion is a glide reflection. Theorem If a transformation is a plane motion, then it is the product of three or fewer reflections, and conversely. Definition plane motion is a direct motion if it is the product of an even number of reflections. It is an opposite motion if it is the product of an odd number of reflections. The system of direct and opposite motions under multiplication is isomorphic to the system of even and odd numbers under addition (Z 2 ). Theorem The four types of Euclidean motions of the plane constitute a group of transformations. 42

5 Exercises: 1. onsider Figure 6.6: What are the coordinates of,, and? Draw the image of under a translation by the vector v. What are the coordinates of,, and? Generalization: What is the relationship between (x, y ) and (x, y)? Y v X onsider Figure 6.6: Figure 6.6: Translation y Vector v What are the coordinates of,, and? Draw the image of under a reflection about the line y = x. What are the coordinates of,, and? Generalization: What is the relationship between (x, y ) and (x, y)? 43

6 3. onsider Figure 6.7: What are the coordinates of,, and? Draw the image of under a rotation about (0, 0) by 90 (counterclockwise). What are the coordinates of,, and? Generalization: What is the relationship between (x, y ) and (x, y)? What is the relationship between (x, y ) and (x, y) if the rotation was about (0, 0) by -90 (clockwise)? Y X onsider Figure 6.7: Figure 6.7: Rotation bout the Origin by ±90 What are the coordinates of,, and? Draw the image of under a rotation about (0, 0) by 180 counterclockwise. What are the coordinates of,, and? Generalization: What is the relationship between (x, y ) and (x, y)? 5. onsider Figure 6.8: What are the coordinates of,, and? Draw the image of under a rotation about Q by 90. What are the coordinates of,, and? Generalization: What is the relationship between (x, y ) and (x, y)? 44

7 Y Q X Figure 6.8: Rotation bout Q by Similarity Transformations Definition plane similarity is a transformation of the plane onto itself such that if, and, are corresponding points, the length of is r times the length of, where r s some nonzero real number. Theorem The set of all similarities of the plane is a transformation group Dilations Definition dilation (homothety) H(O, r) is a plane similarity that maps a point P to a point P, where OP = rop, the points P, O, P are collinear, and r 0. If r < 0, then O is between P and P, while if r > 0, the points P and P are on the same side of O. The point O is called the center, and the number r is called the ratio of the dilation. dilation is determined by determining its center and ratio. 2 H (O, 1) (P ) O P H (O,2) (P ) 3 Figure 6.9: Dilation 45

8 Exercise: onsider the transformation f(x, y) = ( 2 3 x + 1, 2 3 y + 3). Plot the images of,, under f. Is f a dilation? If so determine it. If not what can you say about f? Y X Figure 6.10: Dilation in the XY Plane Theorem The image of a circle under a dilation is another circle. Figure 6.11) (See P P O 1 1 Figure 6.11: Dilation of a ircle Proof: Let P be an arbitrary point on 1 noncollinear with O. Now OP O P. Hence, O = O. Hence, P P P = P O O 46

9 ut P, O, and O are all fixed lengths. Therefore P is fixed implying that the image of 1 is a circle 1. What if P is collinear with O? Theorem The Nine-Point ircle: The nine points listed below lie on a circle called the Nine-Point circle. The midpoints, G, E, F, of the sides of the triangle. The feet,,,, of the three altitudes of this triangle. The midpoints, J, K, L, of the segments formed by joining the orthocenter to the vertices of the triangle. The center of the Nine-Point circle is the midpoint of the segment joining the orthocenter and the circumcenter of, and the length of the radius of this circle is one-half the length of the radius of the circle that circumscribes. L F E F O E J K.... G D... R Figure 6.12: The Nine-Point ircle Theorem Proof: onsider a dilation H (O,2) centered at O the orthocenter and of ratio 2. learly, H (0,2) (J) = J, H (0,2) (K) = K, and H (0,2) (L) = L where J, K, L are points on the circumcircle. Now let R be a diameter of the circumcircle. Hence R = R = 90. Now R and R. Hence OR is a parallelogram. ut the diagonals of a parallelogram bisect each other. Hence G is the midpoint of OR, implying that H (0,2) (G) = R a point on the circumcircle. Similarly, H (O,2) takes E and F to points E and F on the circumcircle. Now let D be the intersection of O with the circumcircle. Hence DR = 90. Hence ODR O G. ut OR = 2 OG, hence OD = 2 O. Therefore, H (O,2) ( ) = D a point on the circumcircle. Similarly, H (O,2) ( ) = and 47

10 H (O,2) ( ) = are points on the circumcircle. Now a dilation H (O, 1 ) takes the 2 points,, D, E, F, R,,, which lie on the circumcircle to,,, E, F, G, J, K, L respectively. The image of the circumcircle under this dilation is the Nine Point ircle. Homework Prove that collinearity is invariant under an isometry. 2. Prove that an isometry that leaves each of three noncollinear points fixed leaves every point of the plane fixed. 3. Prove that each point in the plane is uniquely determined by its given distances from three noncollinear points. 4. Prove that an isometry is uniquely determined by three noncollinear points and their images under that isometry. 5. In the figure below, determine a motion which will take to. Figure 6.13: HW - Problem 5 6. In the figure below, determine a motion which will take to. Figure 6.14: HW - Problem 6 7. In the figure below, determine a motion which will take to 8. How can you determine where to place the vertices of an equilateral triangle so as to inscribe it in the square shown, with one vertex of the triangle on a vertex of the square and the other two vertices lying on the sides of the square opposite that vertex? (See Figure 6.16) 48

11 Figure 6.15: HW - Problem 7 Figure 6.16: Equilateral Inside a Square 9. onsider three concentric circles. If it exists, construct an equilateral triangle having three vertices, and on the given circles, respectively. Determine when the problem does not have a solution. (See Figure 6.17) (Hint: Let be any point on 3 and rotate 1 about by 60. Let be one of the points of intersection of 1 and 2 (if such an intersection exists). Now rotate 1 about by 60, let be the image of under this rotation.) Figure 6.17: Equilateral on Three ircles 49

12 ibliography 1. Donald M. Davis, The Nature and Power of Mathematics, Princeton University Press, New Jersey, Thomas L. Heath, Euclid s Elements, Dover Publications, Inc., New York, Howard Eves, ollege Geometry, Jones and artlett, oston, David. Kay, ollege Geometry Discovery pproach, Harper ollins, E.. Wallace & S. F. West, Roads To Geometry, Prentice Hall, New Jeresy,