Examination Copy © COMAP Inc. Not for ResaleAs you watch animated movies or play video games, the central question must be, How do they do that? How do animators make ... Videos,

306

C HA P T E R

Animation/Special EffectsLESSON ONE

Get Moving

LESSON TWO

Get to the Point

LESSON THREE

Escalating Motion

LESSON FOUR

Calculator Animation

LESSON FIVE

Fireworks

Chapter 4 Review

44Examination Copy © COMAP Inc. Not for Resale

Examination Copy © COMAP Inc. Not for Resale

307

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.



PREPARATION READING:

Simulating Motion

I n a photo finish, two racers arrive at the finish line at thesame time. People watching cannot tell who was first.Fortunately, many races are now videotaped. The

videotape can be replayed frame by frame until the exactmoment the first racer reached the finish line. Each frame is asnapshot 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 itwas taped, you are convinced you are seeing the entire eventagain. You are not. The videotape simulates motion by playingback the series of frames faster than your eye can see theindividual frames.

Traditional animation is like video. Hundreds of cartoonframes or drawings are created so that each is slightly differentfrom the one before it. The frames when viewed rapidly createthe illusion of motion. Advances in technology have changedtraditional animation, bringing new tools and capabilities tothe animator.

To create the illusion of motion through animation, you beginwith the essential elements of motion and create a simplemodel. The purpose of this first lesson is to identify the basicelements of animation. In the lessons that follow, you startwith a simple model and add complexity. Eventually you willdesign 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 stringon a violin like actual motion?

2. Suppose each frame in a series of frames is the same. Whatdo you see if the frames are viewed rapidly in succession?

3. Suppose you view a series of frames as they are playedrapidly and it appears that things are moving. What isactually changing as you view this illusion of motion?

308 Preparation Reading

LESSON ONE

Get Moving

Key Concepts

Continuous and discreterepresentations

Variables and constants

Rates of change

Reference systems

Linear functions

Closed-form equations



309

In this activity you simulate the motion of letters traveling across anelectronic billboard by creating a “living marquee.”

Marquees with lights moving across their screens can be found onbuildings in large cities, along highways, and even in front of someschools. 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 thesame direction. Each dot seems to move along in a line.

In order to simulate the motion on a marquee, you begin with asimplified model of the problem. Rather than creating many points thatmove about the marquee, you will focus on the simplest type ofmovement. Your first challenge is to physically model the horizontalmovement 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 onecolor 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 lineto the other at a predetermined rate.

2. After practicing, demonstrate your living marquee for therest of the class, or videotape it and play it back. Try doingthe simulation with your eyes closed.

3. Prepare a table, graph, equation, or arrow diagram torepresent your living marquee. (Hint: Let time be theexplanatory variable and horizontal location in the line ofpeople 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 thedirections if they are not precise enough. Have another group useyour directions to demonstrate your creative marquee.

Activity 4.1 Lesson One

Activity 4.1: The Living Marquee

Figure 4.1.

Sample DesignIdeas

• Simulate two points that cross.

• Simulate a line (made up of morethan one point) that grows longeras it moves from left to right.

• Change the rate at which a pointmoves from one end of amarquee to the other.

TAKE NOTE



310 Chapter 4 Animation/Special Effects Activity 4.1

PART III: REFLECTING ON THE SIMULATION

5. When you created your marquees in Parts I and II, how did youknow when to flip your card?

6. Suppose the equation L = 2t + 5 represents your living marquee withtime t measured in seconds and location L in people. Assuming themotion begins when t = 0. How does the person at location 11 knowwhen 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 thatchapter, slope is defined as the ratio of the change in the responsevariable 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 respectto time refers to how fast the dot appears to move alongthe line of people. It is a ratio of the change in location tothe 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 inone quantity to the change in another.

• Rates of change are always calculated as:

• Rates are expressed as units of the response variable per units of theexplanatory variable.

• Every rate of change can be seen as a slope by graphing the firstquantity as y on the vertical axis and the second quantity as x on thehorizontal axis.

8. Calculate the rate of change of location with respect to time.

a) Movement: location 2 to location 11

Elapsed time: 5 seconds

change in the response variablechange in thee explanatory variable

8

4

13 54–



Activity Summary

311Activity 4.1 Lesson One

b) Movement: location 0 to location 15

Elapsed time: 3 seconds

c) Recall your marquee from Question 1. Compute the rate of changeof your moving point with respect to time over the entire durationof your design.

In this activity, you:

� created a “living marquee” to physically simulate horizontal motion.

� used precise verbal directions to describe the movement of a singlepoint horizontally.

� created a table, graph, equation, or arrow diagram to model theillusion of movement in your marquee.

� learned to calculate the rate of change of location of a point withrespect to time as it moved along the line of people.



Individual Work 4.1: Next in Line

312

In this Individual Work, you review the concepts from Activity 1including rates of change. In addition you are introduced to several newmathematical terms: reference system, reference point, closed-formequation, velocity, and displacement.

1. In the world of animation, a frame is one picture in a series ofpictures. Examine the sequences of frames in Figures 4.2–4.4.Describe how the object changes from one frame to the next. Drawthe next frame in the sequence.

a)

b)

c)

2. The five frames pictured in Figure 4.5 represent the word HEAR on amoving marquee.

a) Draw the sixth frame.

b) Describe how to improve the marquee.

Chapter 4 Animation/Special Effects Individual Work 4.1

Figure 4.2.

Figure 4.3.

Figure 4.4.

R HEAAR EAR HEFigure 4.5.Moving marqueefor HEAR.

Some of the earliestcartoons were made using flipbooks or flip frames. Eachpage of the flipbook is slightlydifferent from the previouspage. So, when pages areviewed in rapid succession,the objects appear to move.

FYI



313Individual Work 4.1 Lesson One

When you created directions for your marquee in Activity 1, youprobably used a reference system and said something like, “Begin atlocation 3 and move the dot 5 frames to the right.”

In a reference system, a starting location or referencepoint is chosen and a consistent measuring scale is used toindicate another point’s distance and direction from thatstarting point.

Numbers are used in reference systems to determine how far you havemoved from a starting location. A ruler uses numbers to tell how manycentimeters or inches you are from one end. Innings are used in baseballto tell how far you have come from the beginning of the game.

Number lines are mathematical reference systems. In such number linesystems, left is negative and right is positive. There is nothing specialabout these designations, but there is general agreement that thesedirections are standard.

3. Describe two situations in which numbers are used to determinehow far you have come from a starting location. Identify thereference point.

4. Suppose the members of your class stand in a line and form a livingmarquee. To make references less confusing, consider the first personat the left end of the line to be location 1. The second person in line islocation 2. Each change in the cards is considered a frame for theanimation.

a) Suppose at t = 0 seconds the moving dot is at location 0, whichmeans no person in the line displays the moving dot. Assume thatthe 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 1second, the dot will be at location 2. Where is the dot at t = 3seconds?

b) If frames change every half-second and the dot begins at location0 for t = 0, when does the dot reach location 16?

c) Suppose the dot begins at location 5 when t = 0 seconds andframes change 10 times each second. Where is the dot at t = 4seconds?

5. a) Suppose the dot in a living marquee begins at location 12 when t = 0 seconds and changes 2 times each second. Use thisrelationship between location and time to complete the table in Figure 4.6.



314 Chapter 4 Animation/Special Effects Individual Work 4.1

b) Express the relationship between location and time as described in part (a) with an arrow diagram that begins with time andconcludes with the location.

c) In the relationship between location and time, which is theexplanatory variable and what is the response variable?

d) Use a symbolic equation to express the relationship betweenlocation 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 rateof 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 changemean 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 rateof –3 inches per hour?

In Questions 8–11, the animation involves objects that move from onelocation to another. Three different rates of change are commonly usedin animation:

• You can change how far a dot moves (in pixels) from one frame tothe next.

or pixels per frame

• You can increase or decrease the number of times the animationchanges frames in a second. (Most movies are changing at a rate of30 frames per second.)

or frames per second

• The combination of pixels per frame and frames per second results inpixels per second, or the speed at which the dot appears to move.

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, andthe third rate measure how much the pixel location changes in onesecond.

change in pixelschange in seconds

change in frameschange in seconds

change in pixelschange in frames

0 3 5 8 10 15Time (sec)

Location (person)

Figure 4.6.Table for time and location.

Recall from Chapter 2that locations in adigital image (such asa computer monitor ortelevision screen) arecalled pixels, for“picture elements.”

TAKE NOTE




Thus, you have rates involving the change of the location of the objecton the screen, how often the screen changes, and the combination ofthose 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 isthe 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) andthe frames change 2 times each second. The marquee simulationbegins with person 9.

a) Complete the table in Figure 4.7.

b) Write a symbolic equation to represent the location in terms oftime, expressed in seconds. Let L represent location and t thenumber of seconds.

10. Figure 4.8 shows a dot that begins at pixel 5 in Frame 0 and moves tothe right at a rate of 2 pixels per frame.

Frame 0

Frame 1

Frame 2

a) Write an equation to represent the location L after f frames.

b) Where is the object located at the 12th frame?

1 3 7 11 15Time (sec)

Location (person)Figure 4.7.

3 6 9 12 15 18 21 24 27 30 33 36 39 42Pixel location

45 48 51 54 57 60 63 66

3 6 9 12 15 18 21 24 27 30 33 36 39 42Pixel location

45 48 51 54 57 60 63 66

3 6 9 12 15 18 21 24 27 30 33 36 39 42Pixel location

45 48 51 54 57 60 63 66

Figure 4.8. Three frames for moving dot.




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 pointsthat form a linear pattern? Explain your answer.

11. a) Suppose the dot moves to the right at a rate of 2 pixels per frameand frames change at a rate of 10 frames per second. The point islocated at position 5 when t = 0 seconds. Write an equation torepresent 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 yousolve?

d) If you graph location versus time in seconds, would the graph bea continuous line or a group of discrete points that form a linearpattern? Explain your answer.

12. An object moves continuously in a horizontal direction from left toright at a rate of 5 feet per second. At t = 0 seconds the object islocated 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 bea continuous line or a group of discrete points that form a linearpattern? Explain your answer.

13. Suppose the equation L = 3t + 27 represents a living marquee with tmeasured 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 differentapproach. They know to watch the person before them in the lineand flip the card exactly 1 second after that person. Write a wordequation to represent this approach.

Questions 14–16 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 Codesand the Power of Algebra). They are examples of closed-form equations. Each equation directly relates onevarying quantity to another.

When a closed-form equation is used, it takes just two steps to calculatethe location of a point. An arrow diagram can be used to represent thistwo-step process (see Figure 4.9).




Notice the roles played by velocity and starting location.

Velocity is the speed at which an object moves in aspecific direction. Multiplying the velocity by the elapsedtime gives you displacement. Displacement is distance in aparticular direction.

displacement = velocity × time

Adding the starting location to the displacement gives thecurrent location.

current location = displacement + starting location

Suppose you want to write a closed-form equation to representthe location of an object at a particular time. The startinglocation of the object is 15 miles from the reference point. Theobject 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 trepresents time in hours, then a symbolic equation for location is:L = 30t + 15.

14. The equations in parts (a)–(c) represent thehorizontal position of an object at t seconds. Findthe position of the point when t = 8 seconds.

a) L = 4.5t + 10

b) L = –3t + 32

c) L = 1.5t + 25

15. Suppose L = 2.5t + 3 represents the horizontalposition (measured in pixels) at t seconds for anobject moving from left to right.

Time Displacement

Multiply by velocity

Location

Add starting location Figure 4.9.

Arrow diagram tofind current location.

In describing motion in aparticular direction, the termvelocity is used instead ofspeed and the termdisplacement is used insteadof distance.The essential difference inthese terms is that speedand distance are generallythought of as always beingpositive. Velocity anddisplacement may be eitherpositive or negative, with thesigns indicating a direction.

FYI

You are encouragedto use both wordequations and

symbolic equations. When you usesymbolic equations make sure to specifythe meaning of the letters used asvariables. In the equation L = 30t + 15,you should be able to state that trepresents the time in hours that theobject has been moving. And Lrepresents the location in a horizontalreference system that uses miles.

TAKE NOTE




a) Assume the motion begins at t = 0. What is the starting location ofthe point?

b) What equation would you solve if you wanted to know when theobject reached the location L = 53? Solve the equation to find thetime.

16. a) In Chapter 3, Prediction, you studied the concept ofslope. What is the slope of the line representing theequation L = 2.5t + 3? What is the meaning of the slope ifL represents the horizontal pixel location of an object andt represents the time the object has been moving?

b) Graph the equation L = 2.5t + 3 on the graphingcalculator.

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 computeranimation?

You can graph the closed-formequation L = 2.5t + 3 on agraphing calculator, but youmust be careful. Mostcalculators require that you usex and y as the variables, where xis the explanatory variable and yis the response variable. In theequation L = 2.5t + 3, t is theexplanatory variable instead of xand L is the response variableinstead of y.

FYI



Documents

Examination Copy © COMAP Inc. Not for ResaleAs you watch animated movies or play video games, the central question must be, How do they do that? How do animators make ... Videos,