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Vector-Valued Functions CHAPTER 11 P1: OSO/OVY P2: OSO/OVY QC: OSO/OVY T1: OSO GTBL001-11 GTBL001-Smith-v16.cls November 13, 2005 7:25 Vector-Valued Functions CHAPTER 11 RoboCup is the international championship of robot soccer. Unlike the remote-controlled destructive robots that you may have seen on television, the robots in this competition are engineered and programmed to respond automatically to the positions of the ball, goal and other players. Once play starts, the robots are completely on their own to analyze the field of play and use teamwork to outmaneuver their opponents and score goals. RoboCup is a chal- lenge for robotics engineers and artificial intelligence researchers. The competitive aspect of RoboCup focuses teams of researchers, while the annual tournament provides invaluable opportunities for feedback and exchange of information. The competition is divided into several categories, each with its own unique challenges. Overall, one of the greatest difficulties has been providing the robots with adequate vision. In the small size category, “vision” is provided by overhead cameras, with information relayed wirelessly to the robots. With the formidable sight problem removed, the focus is on effective movement of the robots and on providing artificial intelligence for teamwork. The remarkable abilities of these 853 P1: OSO/OVY P2: OSO/OVY QC: OSO/OVY T1: OSO GTBL001-11 GTBL001-Smith-v16.cls November 13, 2005 7:25 854 CHAPTER 11 .. Vector-Valued Functions 11-2 robots are demonstrated by the sequence of frames shown here, where a robot on the Cornell Big Red team of 2001 hits a wide-open teammate with a perfect pass that leads to a goal. There is a considerable amount of mathematics behind this play. To tell whether a teammate is truly open or not, a robot needs to take into account the positions and velocities of each robot, since opponents and teammates could move into the way by the time a pass is executed. In the sequence of photos, notice that all the robots are in motion. Both position and velocity can be described using vectors, but a different vector may be required for each time. In this chapter, we introduce vector-valued functions, which assign a vector to each value of the time variable. The calculus introduced in this chapter is essential background knowledge for the programmers of RoboCup. 11.1 VECTOR-VALUED FUNCTIONS To describe the location of the airplane following the circuitous path indicated in Figure 11.1a, you might consider using a point (x, y, z)inthree dimensions. However, it turns out to be more convenient to describe its location at any given time by the endpoint of a vector whose initial point is located at the origin (a position vector). (See Figure 11.1b for vectors indicating the location of the plane at a number of times.) Notice that a function that gives us avector in V3 for each time t would do the job nicely. This is the concept of a vector-valued function, which we define more precisely in Definition 1.1. DEFINITION 1.1 A vector-valued function r(t)isamapping from its domain D ⊂ R to its range R ⊂ V3,sothat for each t in D, r(t) = v for exactly one vector v ∈ V3.Wecan always write a vector-valued function as r(t) = f (t)i + g(t)j + h(t)k, (1.1) for some scalar functions f, g and h (called the component functions of r). For each t,weregard r(t)asaposition vector. The endpoint of r(t) then can be viewed as tracing out a curve, as illustrated in Figure 11.1b. Observe that for r(t)asdefined z t3 t4 t1 t2 O x y FIGURE 11.1a FIGURE 11.1b Airplane’s flight path Vectors indicating plane’s position at several times P1: OSO/OVY P2: OSO/OVY QC: OSO/OVY T1: OSO GTBL001-11 GTBL001-Smith-v16.cls November 13, 2005 7:25 11-3 SECTION 11.1 .. Vector-Valued Functions 855 in (1.1), this curve is the same as that described by the parametric equations x = f (t), y = g(t) and z = h(t). In three dimensions, such a curve is referred to as a space curve. We can likewise define a vector-valued function r(t)inV2 by r(t) = f (t)i + g(t)j, for some scalar functions f and g. y REMARK 1.1 ͗Ϫ1, 2͘ ͗3, 2͘ 2 r(2) We routinely use the variable t to represent the independent variable for vector-valued functions, since in many applications t represents time. 1 r(Ϫ2) x Ϫ1 O 1 23 EXAMPLE 1.1 Sketching the Curve Defined Ϫ1 r(0) by a Vector-Valued Function Sketch a graph of the curve traced out by the endpoint of the two-dimensional Ϫ2 ͗1, Ϫ2͘ vector-valued function FIGURE 11.2a r(t) = (t + 1)i + (t2 − 2)j. Some values of r(t) = (t + 1)i + (t 2 − 2)j Solution Substituting some values for t,wehaver(0) = i − 2j =1, −2, r(2) = 3i + 2j =3, 2 and r(−2) =−1, 2.Weplot these in Figure 11.2a. The y endpoints of all position vectors r(t) lie on the curve C, described parametrically by ͗Ϫ1, 2͘ ͗3, 2͘ = + , = 2 − , ∈ R. 2 C: x t 1 y t 2 t Ϫ r(2) r( 2) We can eliminate the parameter by solving for t in terms of x : 1 t = x − 1. x The curve is then given by Ϫ1 O 1 23 y = t2 − 2 = (x − 1)2 − 2. Ϫ1 r(0) Notice that the graph of this is a parabola opening up, with vertex at the point (1, −2), Ϫ2 as seen in Figure 11.2b. The small arrows marked on the graph indicate the orientation, ͗1, Ϫ2͘ that is, the direction of increasing values of t.Ifthe curve describes the path of an FIGURE 11.2b object, then the orientation indicates the direction in which the object traverses the path. Curve defined by In this case, we can easily determine the orientation from the parametric representation r(t) = (t + 1)i + (t 2 − 2)j of the curve. Since x = t + 1, observe that x increases as t increases. You may recall from your experience with parametric equations in Chapter 9 that eliminating the parameter from the parametric representation of a curve is not always so easy as it was in example 1.1. We illustrate this in example 1.2. EXAMPLE 1.2 A Vector-Valued Function Defining an Ellipse Sketch a graph of the curve traced out by the endpoint of the vector-valued function r(t) = 4 cos ti − 3 sin tj, t ∈ R. Solution In this case, the curve can be written parametrically as x = 4 cos t, y =−3 sin t, t ∈ R. P1: OSO/OVY P2: OSO/OVY QC: OSO/OVY T1: OSO GTBL001-11 GTBL001-Smith-v16.cls November 13, 2005 7:25 856 CHAPTER 11 .. Vector-Valued Functions 11-4 y Instead of solving for the parameter t,itoften helps to look for some relationship between the variables. Here, 3 x 2 y 2 + = cos2 t + sin2 t = 1 4 3 x x 2 y 2 Ϫ or + = 1, 4 4 4 3 which is the equation of an ellipse (see Figure 11.3). To determine the orientation of the Ϫ3 curve here, you’ll need to look carefully at both parametric equations. First, fix a starting place on the curve, for convenience, say (4, 0). This corresponds to t = 0, ±2π, ±4π,....As t increases, notice that cos t (and hence, x) decreases initially, FIGURE 11.3 while sin t increases, so that y =−3 sin t decreases (initially). With both x and y Curve defined by decreasing initially, we get the clockwise orientation indicated in Figure 11.3. r(t) = 4 cos ti − 3 sin tj Just as the endpoint of a vector-valued function in two dimensions traces out a curve, if we were to plot the value of r(t) = f (t)i + g(t)j + h(t)k for every value of t, the endpoints of the vectors would trace out a curve in three dimensions. EXAMPLE 1.3 A Vector-Valued Function Defining an Elliptical Helix z Plot the curve traced out by the vector-valued function r(t) = sin ti − 3 cos tj + 2tk, t ≥ 0. Solution The curve is given parametrically by x = sin t, y =−3 cos t, z = 2t, t ≥ 0. While most curves in three dimensions are difficult to recognize, you should notice that there is a relationship between x and y here, namely, y 2 x2 + = sin2 t + cos2 t = 1. (1.2) 3 x y In two dimensions, this is the equation of an ellipse. In three dimensions, since the equation does not involve z, (1.2) is the equation of an elliptic cylinder whose axis is the FIGURE 11.4a z-axis. This says that every point on the curve defined by r(t) lies on this cylinder. From Elliptical helix: the parametric equations for x and y (in two dimensions), the ellipse is traversed in the r(t) = sin ti − 3 cos tj + 2tk counterclockwise direction. This says that the curve will wrap itself around the cylinder (counterclockwise, as you look down the positive z-axis toward the origin), as t increases. Finally, since z = 2t, z will increase as t increases and so, the curve will wind its way up the cylinder, as t increases. We show the curve and the elliptical cylinder in Figure 11.4a.
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