Copy COMAP Inc. Not for Resale

Transcription

1 Examination Copy COMAP Inc. Not for Resale CHAPTER 4 Animation/ Special Effects LESSON ONE Get Moving LESSON TWO Get to the Point LESSON THREE Escalating Motion LESSON FOUR Calculator Animation LESSON FIVE Fireworks Chapter 4 Review Examination Copy COMAP Inc. Not for Resale 306

2 ANIMATION/SPECIAL EFFECTS Animation has always entertained and fascinated people. The first television cartoons were made from artists plates. Now, animated scenes in many movies are computer-generated. Virtual reality, a type of animation, is a passport to the world. You can visit faraway and exotic places without leaving your room. As you watch animated movies or play video games, the central question must be, How do they do that? How do animators make objects appear to move and change on a picture screen? Geometry and algebra are fundamental to animation. Graphs, equations, and matrices drive many computer animation programs that simulate motion. Computer programmers use mathematics to write software that allows children, adults, and animators to create their own animated cartoons. In this chapter, you will use mathematics to create your own simple animation models. 307

3 PREPARATION READING: Simulating Motion LESSON ONE Get Moving Key Concepts Continuous and discrete representations Variables and constants Rates of change Reference systems Linear functions Closed-form equations In a photo finish, two racers arrive at the finish line at the same time. People watching cannot tell who was first. Fortunately, many races are now videotaped. The videotape can be replayed frame by frame until the exact moment the first racer reached the finish line. Each frame is a snapshot of what took place. Videotape captures life in a series of frames or snapshots. When the video is played again at the same speed at which it was taped, you are convinced you are seeing the entire event again. You are not. The videotape simulates motion by playing back the series of frames faster than your eye can see the individual frames. Traditional animation is like video. Hundreds of cartoon frames or drawings are created so that each is slightly different from the one before it. The frames when viewed rapidly create the illusion of motion. Advances in technology have changed traditional animation, bringing new tools and capabilities to the animator. To create the illusion of motion through animation, you begin with the essential elements of motion and create a simple model. The purpose of this first lesson is to identify the basic elements of animation. In the lessons that follow, you start with a simple model and add complexity. Eventually you will design your own simple animation. DISCUSSION/REFLECTION 1. Videos, motion pictures, and animations simulate motion. A motion picture may show you 30 frames in one second, but it cannot show you the movement between frames. How are the scales of a piano like animation and the string on a violin like actual motion? 2. Suppose each frame in a series of frames is the same. What do you see if the frames are viewed rapidly in succession? 3. Suppose you view a series of frames as they are played rapidly and it appears that things are moving. What is actually changing as you view this illusion of motion? 308 Preparation Reading

4 Examination Activity 4.1: Copy The Living COMAP Marquee Inc. Not for Resale In this activity you simulate the motion of letters traveling across an electronic billboard by creating a living marquee. Marquees with lights moving across their screens can be found on buildings in large cities, along highways, and even in front of some schools. They are used to convey information to the people passing by. The lights on the marquee look like a parade of dots all moving in the same direction. Each dot seems to move along in a line. In order to simulate the motion on a marquee, you begin with a simplified model of the problem. Rather than creating many points that move about the marquee, you will focus on the simplest type of movement. Your first challenge is to physically model the horizontal movement of a single point in one direction at a constant rate. PART I: THE MARQUEE 1. Form a group of about ten people standing shoulder-to-shoulder. (See Figure 4.1.) Each person should hold a motion card that is one color on one side and another color on the other side. Your task is to make a point appear to move from one end of the line to the other at a predetermined rate. Figure After practicing, demonstrate your living marquee for the rest of the class, or videotape it and play it back. Try doing the simulation with your eyes closed. 3. Prepare a table, graph, equation, or arrow diagram to represent your living marquee. (Hint: Let time be the explanatory variable and horizontal location in the line of people be the response variable.) PART II: CREATIVE MARQUEE 4. Now it s your turn to design your own living marquee. With your group, brainstorm some ideas. Be creative. a) Write precise directions for your marquee. b) Practice your marquee using the directions you wrote. Revise the directions if they are not precise enough. Have another group use your directions to demonstrate your creative marquee. Activity 4.1 TAKE NOTE Sample Design Ideas Simulate two points that cross. Simulate a line (made up of more than one point) that grows longer as it moves from left to right. Change the rate at which a point moves from one end of a marquee to the other. Lesson One 309

5 PART III: REFLECTING ON THE SIMULATION 5. When you created your marquees in Parts I and II, how did you know when to flip your card? 6. Suppose the equation L = 2t + 5 represents your living marquee with time t measured in seconds and location L in people. Assuming the motion begins when t = 0. How does the person at location 11 know when to flip his/her card? 7. What could you change to make the motion look smoother? In Chapter 3, Prediction, you worked with the concept of slope. In that chapter, slope is defined as the ratio of the change in the response variable to the corresponding change in the explanatory variable. Thus, slope is the rate of change of y with respect to x. In this activity, the rate of change of location with respect to time refers to how fast the dot appears to move along the line of people. It is a ratio of the change in location to the change in time. For example, if the dot moves from location 5 to location 13 in 4 seconds, then rate of change = = = 2 people per second. Thus, the rate of change is a fraction or ratio that compares the change in one quantity to the change in another. Rates of change are always calculated as: change in the response variable change in the explanatory variable Rates are expressed as units of the response variable per units of the explanatory variable. Every rate of change can be seen as a slope by graphing the first quantity as y on the vertical axis and the second quantity as x on the horizontal axis. 8. Calculate the rate of change of location with respect to time. a) Movement: location 2 to location 11 Elapsed time: 5 seconds 310 Chapter 4 Animation/Special Effects Activity 4.1

6 b) Movement: location 0 to location 15 Elapsed time: 3 seconds c) Recall your marquee from Question 1. Compute the rate of change of your moving point with respect to time over the entire duration of your design. Activity Summary In this activity, you: created a living marquee to physically simulate horizontal motion. used precise verbal directions to describe the movement of a single point horizontally. created a table, graph, equation, or arrow diagram to model the illusion of movement in your marquee. learned to calculate the rate of change of location of a point with respect to time as it moved along the line of people. Activity 4.1 Lesson One 311

7 Individual Work 4.1: Next in Line In this Individual Work, you review the concepts from Activity 1 including rates of change. In addition you are introduced to several new mathematical terms: reference system, reference point, closed-form equation, velocity, and displacement. FYI Some of the earliest cartoons were made using flip books or flip frames. Each page of the flipbook is slightly different from the previous page. So, when pages are viewed in rapid succession, the objects appear to move. 1. In the world of animation, a frame is one picture in a series of pictures. Examine the sequences of frames in Figures Describe how the object changes from one frame to the next. Draw the next frame in the sequence. a) b) Figure 4.2. c) Figure 4.3. Figure The five frames pictured in Figure 4.5 represent the word HEAR on a moving marquee. Figure 4.5. Moving marquee for HEAR. R AR EAR HEA HE a) Draw the sixth frame. b) Describe how to improve the marquee. 312 Chapter 4 Animation/Special Effects Individual Work 4.1

8 When you created directions for your marquee in Activity 1, you probably used a reference system and said something like, Begin at location 3 and move the dot 5 frames to the right. In a reference system, a starting location or reference point is chosen and a consistent measuring scale is used to indicate another point s distance and direction from that starting point. Numbers are used in reference systems to determine how far you have moved from a starting location. A ruler uses numbers to tell how many centimeters or inches you are from one end. Innings are used in baseball to tell how far you have come from the beginning of the game. Number lines are mathematical reference systems. In such number line systems, left is negative and right is positive. There is nothing special about these designations, but there is general agreement that these directions are standard. 3. Describe two situations in which numbers are used to determine how far you have come from a starting location. Identify the reference point. 4. Suppose the members of your class stand in a line and form a living marquee. To make references less confusing, consider the first person at the left end of the line to be location 1. The second person in line is location 2. Each change in the cards is considered a frame for the animation. a) Suppose at t = 0 seconds the moving dot is at location 0, which means no person in the line displays the moving dot. Assume that the speed of the dot is constant, that the dot does not skip people, and that frames change every half-second. So, at the end of 1 second, the dot will be at location 2. Where is the dot at t = 3 seconds? b) If frames change every half-second and the dot begins at location 0 for t = 0, when does the dot reach location 16? c) Suppose the dot begins at location 5 when t = 0 seconds and frames change 10 times each second. Where is the dot at t = 4 seconds? 5. a) Suppose the dot in a living marquee begins at location 12 when t = 0 seconds and changes 2 times each second. Use this relationship between location and time to complete the table in Figure 4.6. Individual Work 4.1 Lesson One 313

9 Figure 4.6. Time (sec) Examination Table for time and location. Copy COMAP Inc. Not for Resale Location (person) TAKE NOTE Recall from Chapter 2 that locations in a digital image (such as a computer monitor or television screen) are called pixels, for picture elements. b) Express the relationship between location and time as described in part (a) with an arrow diagram that begins with time and concludes with the location. c) In the relationship between location and time, which is the explanatory variable and what is the response variable? d) Use a symbolic equation to express the relationship between location and time as described in part (a). Use L for location (person) and t for time (seconds). 6. a) A dot moves to the right 20 people in 5 seconds. Calculate the rate of change of the location of the dot with respect to time. b) A club had 153 members in April and 216 members in August. Calculate the average rate of increase in members per month. 7. a) Sometimes rates are negative. What does a negative rate of change mean in the context of animating a point along a horizontal line? b) What does it mean if a company is earning $1000 per month? c) What does it mean if the water level in a pool is changing at a rate of 3 inches per hour? In Questions 8 11, the animation involves objects that move from one location to another. Three different rates of change are commonly used in animation: You can change how far a dot moves (in pixels) from one frame to the next. change in pixels change in frames or pixels per frame You can increase or decrease the number of times the animation changes frames in a second. (Most movies are changing at a rate of 30 frames per second.) change in frames change in seconds or frames per second The combination of pixels per frame and frames per second results in pixels per second, or the speed at which the dot appears to move. change in pixels change in seconds or pixels per second The first rate measures how much the pixel location changes in one frame, the second rate measures how much the frames change in one second, and the third rate measure how much the pixel location changes in one second. 314 Chapter 4 Animation/Special Effects Individual Work 4.1

10 Thus, you have rates involving the change of the location of the object on the screen, how often the screen changes, and the combination of those two rates. 8. Each person in a living marquee is like a pixel on a computer screen. If a dot moves 2 people with each frame (skips over 1 person), that is the same as 2 pixels per frame. a) If the dot moves 2 pixels per frame and 30 frames per second, how fast is the object moving in pixels per second? b) How far (how many pixels) does the dot move in 3 seconds? c) How many frames would it take to travel 94 pixels? d) How many seconds would it take to travel 94 pixels? 9. Assume a dot moves 3 people each frame (skips over 2 people) and the frames change 2 times each second. The marquee simulation begins with person 9. a) Complete the table in Figure 4.7. Time (sec) Location (person) Figure 4.7. b) Write a symbolic equation to represent the location in terms of time, expressed in seconds. Let L represent location and t the number of seconds. 10. Figure 4.8 shows a dot that begins at pixel 5 in Frame 0 and moves to the right at a rate of 2 pixels per frame. Frame Pixel location Frame Pixel location Frame Pixel location a) Write an equation to represent the location L after f frames. b) Where is the object located at the 12th frame? Individual Work Figure 4.8. Three frames for moving dot. Lesson One 315

11 c) When does the dot reach location 37? d) If you were to graph location versus time expressed in frames, would the graph be a continuous line or a group of discrete points that form a linear pattern? Explain your answer. 11. a) Suppose the dot moves to the right at a rate of 2 pixels per frame and frames change at a rate of 10 frames per second. The point is located at position 5 when t = 0 seconds. Write an equation to represent location L at time t in seconds. b) Where is the point located after 30 seconds? c) When does the point reach location 173? What equation do you solve? d) If you graph location versus time in seconds, would the graph be a continuous line or a group of discrete points that form a linear pattern? Explain your answer. 12. An object moves continuously in a horizontal direction from left to right at a rate of 5 feet per second. At t = 0 seconds the object is located 36 feet to the right of the origin or reference point. a) Write a symbolic equation for location in terms of time. b) If you graph location versus time in seconds, would the graph be a continuous line or a group of discrete points that form a linear pattern? Explain your answer. 13. Suppose the equation L = 3t + 27 represents a living marquee with t measured in seconds and L in people. a) How does the person at location 42 know when to flip her card? b) Some members of another living marquee take a different approach. They know to watch the person before them in the line and flip the card exactly 1 second after that person. Write a word equation to represent this approach. Questions focus on equations that are expressed in closed form. Recall equations of the form y = mx + b (from Chapter 3, Prediction) and c = 2p + 4 (from Chapter 1, Secret Codes and the Power of Algebra). They are examples of closedform equations. Each equation directly relates one varying quantity to another. When a closed-form equation is used, it takes just two steps to calculate the location of a point. An arrow diagram can be used to represent this two-step process (see Figure 4.9). 316 Chapter 4 Animation/Special Effects Individual Work 4.1

12 Time Multiply by velocity Displacement Add starting location Location Figure 4.9. Arrow diagram to find current location. Notice the roles played by velocity and starting location. Velocity is the speed at which an object moves in a specific direction. Multiplying the velocity by the elapsed time gives you displacement. Displacement is distance in a particular direction. displacement = velocity time Adding the starting location to the displacement gives the current location. current location = displacement + starting location Suppose you want to write a closed-form equation to represent the location of an object at a particular time. The starting location of the object is 15 miles from the reference point. The object moves in a horizontal direction at 30 miles per hour. A word equation representing displacement is: displacement (miles) = 30 (mph) time (hours) A closed-form word equation for location is: current location (miles) = 30 (mph) time (hours) + 15 (miles) If the letter L represents current location in miles and the letter t represents time in hours, then a symbolic equation for location is: L = 30t The equations in parts (a) (c) represent the horizontal position of an object at t seconds. Find the position of the point when t = 8 seconds. a) L = 4.5t + 10 b) L = 3t + 32 c) L = 1.5t Suppose L = 2.5t + 3 represents the horizontal position (measured in pixels) at t seconds for an object moving from left to right. Individual Work 4.1 FYI In describing motion in a particular direction, the term velocity is used instead of speed and the term displacement is used instead of distance. The essential difference in these terms is that speed and distance are generally thought of as always being positive. Velocity and displacement may be either positive or negative, with the signs indicating a direction. TAKE You are encouraged NOTE to use both word equations and symbolic equations. When you use symbolic equations make sure to specify the meaning of the letters used as variables. In the equation L = 30t + 15, you should be able to state that t represents the time in hours that the object has been moving. And L represents the location in a horizontal reference system that uses miles. Lesson One 317

13 a) Assume the motion begins at t = 0. What is the starting location of Examination Copy the point? COMAP Inc. Not for Resale FYI You can graph the closed-form equation L = 2.5t + 3 on a graphing calculator, but you must be careful. Most calculators require that you use x and y as the variables, where x is the explanatory variable and y is the response variable. In the equation L = 2.5t + 3, t is the explanatory variable instead of x and L is the response variable instead of y. b) What equation would you solve if you wanted to know when the object reached the location L = 53? Solve the equation to find the time. 16. a) In Chapter 3, Prediction, you studied the concept of slope. What is the slope of the line representing the equation L = 2.5t + 3? What is the meaning of the slope if L represents the horizontal pixel location of an object and t represents the time the object has been moving? b) Graph the equation L = 2.5t + 3 on the graphing calculator. c) The object is moving horizontally, but the graph of L = 2.5t + 3 is not horizontal. Why not? 17. How might equations be useful in the context of computer animation? 318 Chapter 4 Animation/Special Effects Individual Work 4.1