Think About. Unit 5 Lesson 3. Investigation. This Situation. Name: a Where do you think the origin of a coordinate system was placed in creating this

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1 Think About This Situation Unit 5 Lesson 3 Investigation 1 Name: Eamine how the sequence of images changes from frame to frame. a Where do ou think the origin of a coordinate sstem was placed in creating this animation? b What point(s) on the shuttle image would ou use in determining how each image was transformed? c Describe the tpes of transformations that appear to have been used in creating the animation. d Computer animations are frequentl used in movies and video games. Are there other applications of computer animation with which ou are familiar? In the investigations of this lesson, ou will learn how to use matrices to perform transformations of two-dimensional shapes and create simple animations. The tools that ou develop have straightforward etensions to work in three dimensions and the methods that are tpicall used in computer animation. Investigation 1 Building and Using Rotation Matrices For the purpose of this lesson, ou can simplif the space shuttle animation b representing the space shuttle and sequence of images with two-dimensional figures similar to the ones shown below. Such simple representations are used when a storboard, or outline, of an animation is developed. As a class, stud the animation created b the interactive geometr custom tool Animate Shuttle. In that animation, the space shuttle performs a rollover maneuver as if in preparation for re-entr. As ou work on the problems of this investigation, look for answers to the following questions: How can a rotation with center at the origin be represented b a matri? How can rotation matrices be used to animate the rotation of two-dimensional shapes? 232 UNIT 3 Coordinate Methods

2 1 One possible coordinate model of the shuttle is shown below. F E(0, 8) D C(2, 4) H G B A(8, 0) a. What are the coordinates for points F and H? For points B and G? b. Find the coordinates of the image of the shuttle model when rotated 180 about the origin. c. Write a smbolic rule (, ) (, ) that gives the coordinates of the image of an point P(, ) under a 180 rotation. d. What is a smbolic rule that gives the coordinates of the image of an point P(, ) under a 90 counterclockwise rotation about the origin? e. How would ou modif our rule in Part d so that it describes a 90 clockwise rotation about the origin? 2 Matri multiplication can be used to epress each of the rotations in Problem 1. To do this, coordinates of points (, ) need to be represented as one-column matrices,. For eample, the one-column or point matri for (-2, 4) is a. Look back at the smbolic rule for a 180 rotation that ou found in Problem 1 Part c. To build a 2 2 matri representation for the 180 rotation, find numbers a, b, c, and d that make this matri equation true. 180 Rotation General Image Matri Point Matri Point Matri a c b d = - - i. Test our rotation matri b using it to find the rotation images of points A(8, 0) and C(2, 4). Compare our image points with those found in Problem 1 Part b. ii. Wh is the general point matri placed to the right of the rotation matri? LESSON 3 Transformations, Matrices, and Animation 233

3 b. Determine the matri for a 90 counterclockwise rotation about the origin. a b c d = - i. Check our rotation matri b using it to find the images of points B(2, 0) and C(2, 4) in Problem 1. ii. Do these image points make sense? c. Geometricall, ou know that a 90 counterclockwise rotation followed b another 90 counterclockwise rotation gives a 180 rotation. See if multipling the matri for the 90 counterclockwise rotation b itself ields the matri for the 180 rotation. d. What do ou notice about the entries of the matrices used to epress these rotations? 3 One advantage of a matri representation of a transformation is that ou can use it quickl to transform an entire shape. Consider AEH = determined b the tips of the shuttle model a. Multipl the matri representation of AEH b the 90 counterclockwise rotation matri. Using the coordinate rule for the 90 counterclockwise rotation, verif that the result of our calculation is the image triangle, A E H. b. When transforming an n-sided polgon using matrices, wh should the coordinate matri of the polgon be the factor on the right? 4 Designing animations often requires use of rotations through man different angles, in addition to those that are multiples of 90. When building matri representations for rotations and other transformations, it is ver useful to know what happens to the points (1, 0) and (0, 1). Diagram I below shows the images of points P(1, 0) and Q(0, 1) under a counterclockwise rotation about the origin. Diagram I Diagram II Q (, ) Q(0, 1) P (, ) 2 2, 2 P' ( 2 ) O P(1, 0) O 1 M P(1, 0) a. Eplain wh the image of point P and the image of point Q will be on a circle of radius 1 with center at the origin. 234 UNIT 3 Coordinate Methods

4 b. Using Diagram II, eplain as precisel as ou can wh the image of point P under the rotation has coordinates ( 2 2, 2 2 ). c. Find the coordinates of point Q. d. Now find the entries of the counterclockwise rotation matri R = a b c d b solving the two matri equations below. Begin b entering the coordinates of point P and point Q in the appropriate column matrices. a b 1 i. c d 0 = CPMP-Tools ii. a c b d 0 1 = iii. So, R =. e. Check that multipling the counterclockwise rotation matri b itself (with entries epressed in radical form) gives the matri for a 90 counterclockwise rotation about the origin that ou found in Problem 2 Part b. 5 Look back at the entries for the counterclockwise rotation matri R and how the were calculated. a. How are the entries of matri R related to the rotation images of P(1, 0) and Q(0, 1)? b. Does the pattern hold for the 180 and 90 rotation matrices ou found in Problem 2? Eplain. 6 A computer or calculator program can be written that will rotate the space shuttle model counterclockwise about the origin using steps of. Stud the Roll Over Algorithm given below. Roll Over Algorithm Step 1. Set up the coordinate matri representing the space shuttle. Step 2. Set up the counterclockwise rotation matri. Step 3. Draw the shuttle. Step 4. Compute and store the coordinates of the shuttle rotated. Step 5. Clear the old shuttle and draw the rotated image. Step 6. Pause. Step 7. Repeat Steps 4 6 as needed. a. Identif the input, processing, and output parts of the Roll Over Algorithm. b. Step 7 is a control command. It controls the action of the algorithm. To make the shuttle rotate all the wa around once, how man times should Steps 4 6 be performed? LESSON 3 Transformations, Matrices, and Animation 235

5 CPMP-Tools 7 The roll over portion of the animation can be created using commands such as those below. Note how assigning names to the shuttle coordinate matri and the rotation matri simplifies the programming. Roll Over Program let shuttle = [[8,0][2,0][2,4][0,4][0,8][0,4][ 2,4][ 2,0][ 8,0][8,0]] let rotmatri = [[0.7071,0.7071][ ,0.7071]] draw shuttle repeat 8 [draw [let shuttle = [rotmatri*shuttle]] pause 500] a. Discuss with our classmates how the commands in this program match corresponding steps in the Roll Over Algorithm. b. Test the program b entering it in the Command window of our interactive geometr software. c. Predict the animation that will be produced b replacing the last three lines of the program b these two lines: draw shuttle repeat 8 [draw [let shuttle = [rotate shuttle 45]] pause 500] Run the program to test our prediction. Make notes of an misunderstandings of programming commands. Summarize the Mathematics In this investigation, ou eplored how to find matri representations for certain rotations and how matrices can be used to create an animation. a Eplain how to use the coordinate rule for a 270 counterclockwise rotation about the origin to find the matri representation for that rotation. Find the matri. b How would ou modif the Roll Over program so that it will rotate the space shuttle clockwise about the origin in steps of? c Describe a sstematic wa of determining the entries of a rotation matri. d The matri is the matri for a 60 clockwise rotation about the origin. Eplain how to use the matri to find a coordinate rule for the image of a point (, ) under this rotation. Be prepared to share our ideas and reasoning with the class. 236 UNIT 3 Coordinate Methods