Lesson 3. Investigation. Name: a. Use the matrix to find the rotation image of the point (-1, 5).

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1 Check Unit Your 5 Understanding Lesson 3 Investigation 2 Build a matrix that represents a 135 counterclockwise rotation about the origin. a. Use the matrix to find the rotation image of the point (-1, 5). b. Use the matrix to find the rotation image of HJK = c. Sketch HJK and its rotation image on a coordinate grid. Name: Investigation 2 Building and Using Size Transformation Matrices Consider again the sequence of images in the mock animation of a space shuttle. Note how the shuttle image increases in size from frame to frame as it rolls over and moves forward. This simulation was accomplished by using size transformations, translations, and rotations. In this investigation, all size transformations will be centered at the origin. As you work on problems in this investigation, look for answers to the following questions: How can size transformations be represented using matrices? How can you animate size change and translation of two-dimensional shapes? How can size transformations and translations be combined with rotations to create more complex animations? LESSON 3 Transformations, Matrices, and Animation 237

2 1 Begin by examining the space shuttle model shown below. a. What is the coordinate matrix for a similar shuttle model (in the y same position) whose sides are twice the length of those in the E(0, 8) given model? Half the length? F D C(2, 4) b. Write two coordinate rules (x, y) (, ) that would resize the shuttle model as described H G B A(8, 0) in Part a. c. What is the coordinate matrix for the image of the given shuttle model when translated: i. 5 units to the right? ii. 3 units down? iii. 5 units to the right and 3 units down? d. Write coordinate rules for these three transformations in the form (x, y) (, ). 2 Matrix representations for size transformations with center at the origin can be found using the same method you used to find rotation matrices. a. Determine the entries a, b, c, and d of the matrix for a size transformation with magnitude 2. a c b d x y = 2x 2y b. What should be the image of the point F(-2, 4) under this transformation? Multiply the transformation matrix by the one-column matrix for point F and check to see if you get the correct image point. c. Multiply the size transformation matrix you found in Part a by the matrix for AEH, where A, E, and H are the wing tips of the shuttle model. i. Compare the coordinates of the image A E H with those found using the appropriate coordinate rule for a size transformation with magnitude 2. ii. Compare the lengths of EH and E H. Why does that relationship make sense? d. Find the matrix for a size transformation with magnitude _ 1 2. With magnitude 5. 3 How could you use the idea of multiplying a matrix by a real number to find the image of a point or a polygon under a size transformation of magnitude 3 with center at the origin? Of magnitude 1 _ 4? Compare methods with others and resolve any differences. x 238 UNIT 3 Coordinate Methods

3 4 Use the Roll Over Algorithm on page 235 and the following questions to help you develop an algorithm for a program that will repeatedly scale the space shuttle model by a factor of 1.5 using a size transformation with center at the origin. What information would you need to input? What processing would the program need to complete? What information should the calculator or computer output? CPMP-Tools Resizing Algorithm Step 1. Set up the coordinate matrix representing the shuttle. (input) Step 2. ( ) Step 3. ( ) 5 The animation described in Problem 4 can be created using commands such as: Resizing Program let shuttle = [[8,0][2,0][2,4][0,4][0,8][0,4][ 2,4][ 2,0][ 8,0][8,0]] let sizematrix = [[1.5,0][0,1.5]] repeat 4 [draw [let shuttle = [sizematrix*shuttle]] pause 500] a. Test the program by entering it in the Command window of your software. b. Suppose the last three lines of the program were replaced by these two lines: repeat 4 [draw [let shuttle = [scale shuttle 1.5]] pause 500] What animation do you think will be produced by the modified program? Check your conjecture by running the modified program. c. Write a series of commands that could be used to create an animation that repeatedly scales the shuttle by a factor of 4 with center at the origin. d. Test your program by entering it in the Command window. Revise commands as necessary. 6 Unlike in the cases of rotations and size transformations, translations cannot be represented with 2 2 matrices. (See Extensions Task 18.) Instead of using matrix multiplication, the coordinate rule form is frequently used to describe translations. For example, you can translate the shuttle and display the image with the commands shown below. let shuttle = [[8,0][2,0][2,4][0,4][0,8][0,4][ 2,4][ 2,0][ 8,0][8,0]] let shuttle = [translate shuttle [5, 3]] LESSON 3 Transformations, Matrices, and Animation 239

4 CPMP-Tools a. Write a series of commands that could be used to create an animation that repeatedly translates the shuttle 2 units to the right and 3 units up. b. Test your program. Revise it as necessary. In the following problems, you will explore ways in which transformations can be combined to create more complex animations. 7 Suppose you want to create an animation that starts with the shuttle wingspan repositioned so that its center is at (-4, 5) and is rescaled to 75% the original size. a. Describe the horizontal and vertical components of the translation and scale factor of the size transformation that will produce this image when drawn. b. Will the order in which you choose to apply the translation and size transformation matter? Explain. c. Write a series of programming commands that could be used to create this initial image of the shuttle. d. Test your program and revise it as necessary. 240 UNIT 3 Coordinate Methods

5 8 By repeatedly combining a size transformation, rotation, and a translation, you can create an animation where the shuttle model appears to move from being far away to flying by the observer. a. Study the two algorithms for gradually growing the size of the shuttle model. CPMP-Tools Grow Algorithm 1 Step 1. Set up the coordinate matrix representing the shuttle. Step 2. Set up a 0.25 scale factor. Step 3. Compute the image of the original shuttle under the size transformation. Step 4. Draw the image. Step 5. Increase the scale factor by Step 6. Pause. Step 7. Repeat Steps 3 6 as needed. Grow Algorithm 2 Step 1. Set up the coordinate matrix representing the shuttle. Step 2. Set up a 0.25 scale factor. Step 3. Compute the image of the original shuttle under the size transformation. Step 4. Draw the image. Step 5. Multiply the scale factor by 1.2. Step 6. Pause. Step 7. Repeat Steps 3 6 as needed. i. How does the size of the shuttle grow for each algorithm? ii. For each algorithm, how many times must the steps be repeated before the shuttle image reaches its original size? b. Suppose to start an animation sequence, the center of the shuttle wingspan needs to be translated to (-20, 20) and the shuttle scaled to 25%. In which order should you perform the translation and size transformation? c. Study the program below that creates an animation of the space shuttle similar to that produced by the Animate Shuttle custom tool. Describe as precisely as you can the effect of each line. Shuttle Animation Program 1. pgm animateshuttle 2. gridstyle grid off axes off 3. let shuttle=[shape [[10,0][2.5,0][2.5,5][0,5][0,10][0,5][ 2.5,5][ 2.5,0][ 10,0][10,0]]] 4. style shuttle filled on fillcolor visible off label off 5. let currentcenter=[ 18,18] 6. let currentangle= let currentscale= let shuttleimage=[translate [scale [rotate shuttle currentangle] currentscale] currentcenter] 9. style shuttleimage filled on fillcolor label off visible on 10. pause repeat 18 [let currentangle=currentangle+10 let currentscale=currentscale+0.05 let currentcenter=currentcenter+[1, 1] let shuttleimage=[translate [scale [rotate shuttle currentangle] currentscale] currentcenter] Image pause 100] 12. clear currentcenter currentangle currentscale shuttleimage 13. gridstyle grid on axes on end d. Compare your description of the program with the execution of the Animate Shuttle custom tool. LESSON 3 Transformations, Matrices, and Animation 241