9 ARAMETRIC EQUATIONS AND OLAR COORDINATES So far we have described plane curves b giving as a function of f or as a function of t or b giving a relation between and that defines implicitl as a function of f,. In this chapter we discuss two new methods for describing curves. Some curves, such as the ccloid, are best handled when both and are given in terms of a third variable t called a parameter f t, tt. Other curves, such as the cardioid, have their most convenient description when we use a new coordinate sstem, called the polar coordinate sstem. 9. ARAMETRIC CURVES FIGURE C (, )={ f(t), g(t)} Module 9.A gives an animation of the relationship between motion along a parametric curve f t, tt and motion along the graphs of f and t as functions of t. Imagine that a particle moves along the curve C shown in Figure. It is impossible to describe C b an equation of the form f because C fails the Vertical Line Test. But the - and -coordinates of the particle are functions of time and so we can write f t and tt. Such a pair of equations is often a convenient wa of describing a curve and gives rise to the following definition. Suppose that and are both given as functions of a third variable t (called a parameter) b the equations f t tt (called parametric equations). Each value of t determines a point,, which we can plot in a coordinate plane. As t varies, the point, f t, tt varies and traces out a curve C, which we call a parametric curve. The parameter t does not necessaril represent time and, in fact, we could use a letter other than t for the parameter. But in man applications of parametric curves, t does denote time and therefore we can interpret, f t, tt as the position of a particle at time t. EXAMLE Sketch and identif the curve defined b the parametric equations t t t SOLUTION Each value of t gives a point on the curve, as shown in the table. For instance, if t, then, and so the corresponding point is,. In Figure we plot the points, determined b several values of the parameter t and we join them to produce a curve. t 8 3 3 3 3 4 4 8 5 t= t= t= (, ) t=3 t=_ t=4 8 t=_ 48 FIGURE
SECTION 9. ARAMETRIC CURVES 483 This equation in and describes where the particle has been, but it doesn t tell us when the particle was at a particular point. The parametric equations have an advantage the tell us when the particle was at a point. The also indicate the direction of the motion. A particle whose position is given b the parametric equations moves along the curve in the direction of the arrows as t increases. Notice that the consecutive points marked on the curve appear at equal time intervals but not at equal distances. That is because the particle slows down and then speeds up as t increases. It appears from Figure that the curve traced out b the particle ma be a parabola. This can be confirmed b eliminating the parameter t as follows. We obtain t from the second equation and substitute into the first equation. This gives t t 4 3 and so the curve represented b the given parametric equations is the parabola 4 3. (8, 5) No restriction was placed on the parameter t in Eample, so we assumed that t could be an real number. But sometimes we restrict t to lie in a finite interval. For instance, the parametric curve t t t t 4 (, ) shown in Figure 3 is the part of the parabola in Eample that starts at the point, and ends at the point 8, 5. The arrowhead indicates the direction in which the curve is traced as t increases from to 4. In general, the curve with parametric equations FIGURE 3 f t tt a t b π t= (cos t, sin t) has initial point f a, ta and terminal point f b, tb. V EXAMLE What curve is represented b the following parametric equations? t=π t t= (, ) t=π cos t sin t t SOLUTION If we plot points, it appears that the curve is a circle. We can confirm this impression b eliminating t. Observe that 3π t= cos t sin t FIGURE 4 t=, π, π (, ) Thus the point, moves on the unit circle. Notice that in this eample the parameter t can be interpreted as the angle (in radians) shown in Figure 4. As t increases from to, the point, cos t, sin t moves once around the circle in the counterclockwise direction starting from the point,. EXAMLE 3 What curve is represented b the given parametric equations? sin t cos t t SOLUTION Again we have sin t cos t FIGURE 5 so the parametric equations again represent the unit circle. But as t increases from to, the point, sin t, cos t starts at, and moves twice around the circle in the clockwise direction as indicated in Figure 5.
484 CHATER 9 ARAMETRIC EQUATIONS AND OLAR COORDINATES r (h, k) FIGURE 6 =h+r cos t, =k+r sin t Eamples and 3 show that different sets of parametric equations can represent the same curve. Thus we distinguish between a curve, which is a set of points, and a parametric curve, in which the points are traced in a particular wa. EXAMLE 4 Find parametric equations for the circle with center h, k and radius r. SOLUTION If we take the equations of the unit circle in Eample and multipl the epressions for and b r, we get r cos t, r sin t. You can verif that these equations represent a circle with radius r and center the origin traced counterclockwise. We now shift h units in the -direction and k units in the -direction and obtain parametric equations of the circle (Figure 6) with center h, k and radius r: h r cos t k r sin t t (_, ) (, ) V EXAMLE 5 Sketch the curve with parametric equations sin t, sin t. FIGURE 7 SOLUTION Observe that sin t and so the point, moves on the parabola. But note also that, since sin t, we have, so the parametric equations represent onl the part of the parabola for which. Since sin t is periodic, the point, sin t, sin t moves back and forth infinitel often along the parabola from, to,. (See Figure 7.) GRAHING DEVICES Most graphing calculators and computer graphing programs can be used to graph curves defined b parametric equations. In fact, it s instructive to watch a parametric curve being drawn b a graphing calculator because the points are plotted in order as the corresponding parameter values increase. 3 EXAMLE 6 Use a graphing device to graph the curve 4 3. SOLUTION If we let the parameter be t, then we have the equations _3 3 t 4 3t t FIGURE 8 _3 Using these parametric equations to graph the curve, we obtain Figure 8. It would be possible to solve the given equation 4 3 for as four functions of and graph them individuall, but the parametric equations provide a much easier method. In general, if we need to graph an equation of the form t, we can use the parametric equations tt t Notice also that curves with equations f (the ones we are most familiar with graphs of functions) can also be regarded as curves with parametric equations t f t
SECTION 9. ARAMETRIC CURVES 485 8 Graphing devices are particularl useful for sketching complicated curves. For instance, the curves shown in Figures 9,, and would be virtuall impossible to produce b hand..5 _6.5 6.5 _.5.5 8 _.5 _ FIGURE 9 =t+ sin t =t+ cos 5t FIGURE =.5 cos t-cos 3t =.5 sin t-sin 3t FIGURE =sin(t+cos t) =cos(t+sin t) THE CYCLOID An animation in Module 9.B shows how the ccloid is formed as the circle moves. EXAMLE 7 The curve traced out b a point on the circumference of a circle as the circle rolls along a straight line is called a ccloid (see Figure ). If the circle has radius r and rolls along the -ais and if one position of is the origin, find parametric equations for the ccloid. FIGURE SOLUTION We choose as parameter the angle of rotation of the circle when is at the origin). Suppose the circle has rotated through radians. Because the circle has been in contact with the line, we see from Figure 3 that the distance it has rolled from the origin is O r r C(r, r) Q T OT arc T r Therefore, the center of the circle is Cr, r. Let the coordinates of be,. Then from Figure 3 we see that OT Q r r sin r sin FIGURE 3 TC QC r r cos r cos Therefore, parametric equations of the ccloid are r sin r cos One arch of the ccloid comes from one rotation of the circle and so is described b. Although Equations were derived from Figure 3, which illustrates the case, it can be seen that these equations are still valid for other values of (see Eercise 33).
486 CHATER 9 ARAMETRIC EQUATIONS AND OLAR COORDINATES A Although it is possible to eliminate the parameter from Equations, the resulting Cartesian equation in and is ver complicated and not as convenient to work with as the parametric equations. ccloid FIGURE 4 FIGURE 5 B One of the first people to stud the ccloid was Galileo, who proposed that bridges be built in the shape of ccloids and who tried to find the area under one arch of a ccloid. Later this curve arose in connection with the brachistochrone problem: Find the curve along which a particle will slide in the shortest time (under the influence of gravit) from a point A to a lower point B not directl beneath A. The Swiss mathematician John Bernoulli, who posed this problem in 696, showed that among all possible curves that join A to B, as in Figure 4, the particle will take the least time sliding from A to B if the curve is part of an inverted arch of a ccloid. The Dutch phsicist Hugens had alread shown that the ccloid is also the solution to the tautochrone problem; that is, no matter where a particle is placed on an inverted ccloid, it takes the same time to slide to the bottom (see Figure 5). Hugens proposed that pendulum clocks (which he invented) should swing in ccloidal arcs because then the pendulum takes the same time to make a complete oscillation whether it swings through a wide or a small arc. 9. EXERCISES 4 Sketch the curve b using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases.. st, t 4t,. cos t, t cos t, 3. 5 sin t, t, 4. e t t, e t t, t 5 t t t 5 8 (a) Sketch the curve b using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. (b) Eliminate the parameter to find a Cartesian equation of the curve. 5. 3t 5, t 6. 3t, 7. st, t 8. t, t 3 t. sec, tan, 3. e t, t 4. cos, cos 5 8 Describe the motion of a particle with position, as t varies in the given interval. 5. 3 cos t, sin t, t 3 6. sin t, 4 cos t, t 3 7. 5 sin t, cos t, 8. sin t, cos t, t 5 t 9 Use the graphs of f t and tt to sketch the parametric curve f t, tt. Indicate with arrows the direction in which the curve is traced as t increases. 9. 9 4 (a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. 9. sin, cos,. 4 cos, 5 sin,. sin t, csc t, t. _ t t t t
SECTION 9. ARAMETRIC CURVES 487.. Match the parametric equations with the graphs labeled I VI. Give reasons for our choices. (Do not use a graphing device.) (a) t 3 t, t t (b) t 3, t (c) sin 3t, sin 4t (d) t sin t, t sin 3t (e) sint sin t, cost cos t (f) cos t, sint sin 5t I IV t II V ; 3. Graph the curve 3 3 5. ; 4. Graph the curves 5 and and find their points of intersection correct to one decimal place. 5. (a) Show that the parametric equations t where t, describe the line segment that joins the points, and,. (b) Find parametric equations to represent the line segment from, 7 to 3,. ; 6. Use a graphing device and the result of Eercise 5(a) to draw the triangle with vertices A,, B4,, and C, 5. 7. Find parametric equations for the path of a particle that moves along the circle 4 in the manner described. (a) Once around clockwise, starting at, (b) Three times around counterclockwise, starting at, (c) Halfwa around counterclockwise, starting at, 3 ; 8. (a) Find parametric equations for the ellipse a b. [Hint: Modif the equations of the circle in Eample.] III VI t t (b) Use these parametric equations to graph the ellipse when a 3 and b,, 4, and 8. (c) How does the shape of the ellipse change as b varies? ; 9 3 Use a graphing calculator or computer to reproduce the picture. 9. 3. 3 3 Compare the curves represented b the parametric equations. How do the differ? 3. (a) t 3, t (b) t 6, t 4 (c) e 3t, e t 3. (a) t, t (b) cos t, sec t (c) e t, e t 33. Derive Equations for the case. 34. Let be a point at a distance d from the center of a circle of radius r. The curve traced out b as the circle rolls along a straight line is called a trochoid. (Think of the motion of a point on a spoke of a biccle wheel.) The ccloid is the special case of a trochoid with d r. Using the same parameter as for the ccloid and assuming the line is the -ais and when is at one of its lowest points, show that parametric equations of the trochoid are r d sin Sketch the trochoid for the cases d r and d r. 35. If a and b are fied numbers, find parametric equations for the curve that consists of all possible positions of the point in the figure, using the angle as the parameter. Then eliminate the parameter and identif the curve. a b O r d cos 4 3 8
488 CHATER 9 ARAMETRIC EQUATIONS AND OLAR COORDINATES 36. A curve, called a witch of Maria Agnesi, consists of all possible positions of the point in the figure. Show that parametric equations for this curve can be written as Sketch the curve. a cot =a ; 37. Suppose that the position of one particle at time t is given b 3 sin t cos t t and the position of a second particle is given b 3 cos t (a) Graph the paths of both particles. How man points of intersection are there? (b) Are an of these points of intersection collision points? In other words, are the particles ever at the same place at the same time? If so, find the collision points. (c) Describe what happens if the path of the second particle is given b 3 cos t sin t t 38. If a projectile is fired with an initial velocit of v meters per second at an angle above the horizontal and air resis- a O a sin A sin t C t tance is assumed to be negligible, then its position after t seconds is given b the parametric equations v cos t where t is the acceleration due to gravit ( 9.8 ms ). (a) If a gun is fired with and v 5 ms, when will the bullet hit the ground? How far from the gun will it hit the ground? What is the maimum height reached b the bullet? ; (b) Use a graphing device to check our answers to part (a). Then graph the path of the projectile for several other values of the angle to see where it hits the ground. Summarize our findings. (c) Show that the path is parabolic b eliminating the parameter. ; 39. Investigate the famil of curves defined b the parametric equations t, t 3 ct. How does the shape change as c increases? Illustrate b graphing several members of the famil. ; 4. The swallowtail catastrophe curves are defined b the parametric equations ct 4t 3, ct 3t 4. Graph several of these curves. What features do the curves have in common? How do the change when c increases? ; 4. The curves with equations a sin nt, b cos t are called Lissajous figures. Investigate how these curves var when a, b, and n var. (Take n to be a positive integer.) ; 4. Investigate the famil of curves defined b the parametric equations sin t c sin t 3 v sin t tt cos t c sin t How does the shape change as c changes? In particular, ou should identif the transitional values of c for which the basic shape of the curve changes. 9. CALCULUS WITH ARAMETRIC CURVES Having seen how to represent curves b parametric equations, we now appl the methods of calculus to these parametric curves. In particular, we solve problems involving tangents, areas, and arc length. TANGENTS Suppose f and t are differentiable functions and we want to find the tangent line at a point on the parametric curve f t, tt where is also a differentiable function of. Then the Chain Rule gives d dt d d d dt