PARAMETRIC CURVES
Imagine that a particle moves along the curve C shown in Figure 1. It is impossible to describe C by an equation of the form because C fails the Vertical Line Test.
But the x- and y-coordinates of the particle are functions of time and so we can write and . Such a pair of equations is often a convenient way of describ- ing a curve and gives rise to the following definition.
Suppose that and are both given as functions of a third variable (called a parameter) by the equations
(called parametric equations). Each value of determines a point , which we can plot in a coordinate plane. As varies, the point varies and traces out a curve , which we call a parametric curve. The parameter t does not nec- essarily represent time and, in fact, we could use a letter other than t for the parame- ter. But in many applications of parametric curves, t does denote time and therefore we can interpret as the position of a particle at time t.
EXAMPLE 1 Sketch and identify the curve defined by the parametric equations
SOLUTION Each value of gives a point on the curve, as shown in the table. For instance, if , then , and so the corresponding point is . In Figure 2 we plot the points determined by several values of the parameter and we join them to produce a curve.
FIGURE 2 0 t=0 t=1
t=2
t=3
t=4
t=_1
t=_2 (0, 1)
y
x 8
x, yy 1 0, 1 t
x 0
t 0 t
y t 1 x t2 2t
x, y f t, tt
C
x, y f t, tt
t
x, y
t y tt
x f t
t y
x y tt
x f t
y f x
9.1
PARAMETRIC EQUATIONS AND POLAR COORDINATES
So far we have described plane curves by giving as a function of or as a function of or by giving a relation between and that defines implicitly as a function of
. In this chapter we discuss two new methods for describing curves.
Some curves, such as the cycloid, are best handled when both and are given in terms of a third variable called a parameter . Other curves, such as the cardioid, have their most convenient description when we use a new coordinate system, called the polar coordi- nate system.
x f t, y tt
t
y x
f x, y 0 y x ty x y y x
y f x x x
y
9
482
t x y
2 8 1
1 3 0
0 0 1
1 1 2
2 0 3
3 3 4
4 8 5
C
0
(x, y)={ f(t), g(t)}
FIGURE 1 y
x
Module 9.1A gives an anima- tion of the relationship between motion along a parametric curve , and motion along the graphs of and as functions of .t t f
y tt
x f t
A particle whose position is given by the parametric equations moves along the curve in the direction of the arrows as 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 increases.
It appears from Figure 2 that the curve traced out by the particle may be a parabola. This can be confirmed by eliminating the parameter as follows. We obtain from the second equation and substitute into the first equation.
This gives
and so the curve represented by the given parametric equations is the parabola
. ■
No restriction was placed on the parameter in Example 1, so we assumed that t could be any real number. But sometimes we restrict t to lie in a finite interval. For instance, the parametric curve
shown in Figure 3 is the part of the parabola in Example 1 that starts at the point and ends at the point . The arrowhead indicates the direction in which the curve is traced as increases from 0 to 4.
In general, the curve with parametric equations
has initial point and terminal point .
EXAMPLE 2 What curve is represented by the following parametric equations?
SOLUTION If we plot points, it appears that the curve is a circle. We can confirm this impression by eliminating Observe that
Thus the point moves on the unit circle . Notice that in this example the parameter can be interpreted as the angle (in radians) shown in Figure 4. As increases from 0 to , the point moves once around the circle in the counterclockwise direction starting from the point . ■ EXAMPLE 3 What curve is represented by the given parametric equations?
SOLUTION Again we have
so the parametric equations again represent the unit circle . But as increases from 0 to , the point starts at and moves twice around the circle in the clockwise direction as indicated in Figure 5.2 x, y sin 2t, cos 2t x2 y20, 1 1 t ■
x2 y2 sin22t cos22t 1 0 t 2
y cos 2t x sin 2t
1, 0
x, y cos t, sin t
2
t
t
x2 y2 1
x, y
x2 y2 cos2t sin2t 1 t.
0 t 2
y sin t x cos t
V
f b, tb
f a, ta
a t b y tt
x f t
t
8, 5 0, 1
0 t 4 y t 1
x t2 2t
t x y2 4y 3
x t2 2t y 12 2y 1 y2 4y 3
t y 1 t
t t
■ 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––they tell us when the particle was at a point. They also indicate the direction of the motion.
y x
FIGURE 3 0
(8, 5)
(0, 1) y
x
FIGURE 4
3π
t=2 π
t=2
0 t
t=0 (1, 0) (cos t, sin t)
t=2π t=π
x y
0 t=0, π, 2π
FIGURE 5
x y
(0, 1)
Examples 2 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 para- metric curve, in which the points are traced in a particular way.
EXAMPLE 4 Find parametric equations for the circle with center and radius . SOLUTION If we take the equations of the unit circle in Example 2 and multiply the expressions for and by , we get , . You can verify that these equations represent a circle with radius and center the origin traced counter- clockwise. We now shift units in the -direction and units in the -direction and obtain parametric equations of the circle (Figure 6) with center and radius :
■
EXAMPLE 5 Sketch the curve with parametric equations , .
SOLUTION Observe that and so the point moves on the
parabola . But note also that, since , we have ,
so the parametric equations represent only the part of the parabola for which . Since is periodic, the point moves back and forth infinitely often along the parabola from to . (See Figure 7.) ■
GRAPHING DEVICES
Most graphing calculators and computer graphing programs can be used to graph curves defined by parametric equations. In fact, it’s instructive to watch a parametric curve being drawn by a graphing calculator because the points are plotted in order as the corresponding parameter values increase.
EXAMPLE 6 Use a graphing device to graph the curve . SOLUTION If we let the parameter be , then we have the equations
Using these parametric equations to graph the curve, we obtain Figure 8. It would be possible to solve the given equation for y as four functions of x and graph them individually, but the parametric equations provide a much easier
method. ■
In general, if we need to graph an equation of the form , we can use the parametric equations
Notice also that curves with equations (the ones we are most familiar with—
graphs of functions) can also be regarded as curves with parametric equations y f t
x t y f x
y t x tt
x ty
x y4 3y2 y t x t4 3t2
t y
x y4 3y2
1, 1
1, 1x, y sin t, sin2t sin t
1 x 1
1 x 1
1 sin t 1
y x2 y sin t2 x2 x, y
y sin2t x sin t
V
0 t 2
y k r sin t x h r cos t
h, ky r k
x h
r
y r sin t x r cos t
r y x
h, k r
FIGURE 6 0
(h, k) r
x y
x=h+r cos t, y=k+r sin t
FIGURE 7 0
(1, 1) (_1, 1)
x y
3
_3
_3 3
FIGURE 8
Graphing devices are particularly useful for sketching complicated curves. For instance, the curves shown in Figures 9, 10, and 11 would be virtually impossible to produce by hand.
THE CYCLOID
EXAMPLE 7 The curve traced out by a point on the circumference of a circle as the circle rolls along a straight line is called a cycloid (see Figure 12). If the circle has radius and rolls along the -axis and if one position of is the origin, find parametric equations for the cycloid.
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 13 that the distance it has rolled from the origin is
Therefore, the center of the circle is . Let the coordinates of be . Then from Figure 13 we see that
Therefore, parametric equations of the cycloid are
One arch of the cycloid comes from one rotation of the circle and so is described by . Although Equations 1 were derived from Figure 13, which illustrates the case , it can be seen that these equations are still valid for other values of (see Exercise 33).0 2
0 2
⺢ y r1 cos
x r sin
1
y
TCQC r r cos r1 cosx
OTPQ r r sin r sinx, y
P Cr, r
OT arc PT rP
0
FIGURE 12 P
P
P
P x
r
P
8
_8
_6.5 6.5
FIGURE 9 x=t+2 sin 2t y=t+2 cos 5t
2.5
_2.5
2.5
FIGURE 10
x=1.5 cos t-cos 30t y=1.5 sin t-sin 30t _2.5
1
_1
1
FIGURE 11 x=sin(t+cos 100t) y=cos(t+sin 100t) _1
FIGURE 13
x O
y
T C (r¨, r )
r ¨
x y
r¨
P Q
An animation in Module 9.1B shows how the cycloid is formed as the circle moves.
Although it is possible to eliminate the parameter from Equations 1, the result- ing Cartesian equation in and is very complicated and not as convenient to work
with as the parametric equations. ■
One of the first people to study the cycloid was Galileo, who proposed that bridges be built in the shape of cycloids and who tried to find the area under one arch of a cycloid. 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 gravity) from a point to a lower point not directly beneath . The Swiss math- ematician John Bernoulli, who posed this problem in 1696, showed that among all possible curves that join to , as in Figure 14, the particle will take the least time sliding from to if the curve is part of an inverted arch of a cycloid.
The Dutch physicist Huygens had already shown that the cycloid is also the sol- ution to the tautochrone problem; that is, no matter where a particle is placed on an inverted cycloid, it takes the same time to slide to the bottom (see Figure 15).
Huygens proposed that pendulum clocks (which he invented) should swing in cycloi- dal arcs because then the pendulum takes the same time to make a complete oscilla- tion whether it swings through a wide or a small arc.
P B
A
B A
A B
A
y x
12. , ,
13. ,
14. ,
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
15–18 ■ Describe the motion of a particle with position as varies in the given interval.
15. , ,
16. , ,
, ,
18. , ,
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
19–21 ■ Use the graphs of and to sketch the parametric curve , . Indicate with arrows the direction in which the curve is traced as increases.
19.
20.
t x
1
1 t
y 1
1 t
x
_1
1 t
y 1
1 t y tt
x f t
y tt
x f t
2 t 2
y cos2t x sin t
t 5
y 2 cos t x 5 sin t
17.
0 t 32 y 4 cos t
x 2 sin t
2 t 32 y 1 2 sin t
x 3 2 cos t t
x, y
y 2 cos 1 x 1 cos
y t 1 x e2t
2 2 y tan
x sec
1– 4 ■ Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as increases.
1. , ,
2. , ,
3. , ,
, ,
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
5– 8 ■
(a) Sketch the curve by 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. , 6. ,
, 8. ,
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
9–14 ■
(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. , ,
10. , ,
, y csc t, 0 t 2 x sin t
11.
2 2 y 5 sin
x 4 cos
0 y cos
x sin
y t3 x t2
y 1 t x st
7.
y 2 t2 x 1 3t
y 2t 1 x 3t 5
2 t 2 y et t
x et t 4.
t y t2
x 5 sin t
0 t 2
y t cos t x 2 cos t
0 t 5 y t2 4t
x 1 st t
EXERCISES
9.1
FIGURE 14
P P
P P
P
FIGURE 15 A
B cycloid
3
7
11
13
17
19
3, 7, 11, 13, 17, 19, 27, 31, 35
(b) Use these parametric equations to graph the ellipse when and b 1, 2, 4, and 8.
(c) How does the shape of the ellipse change as b varies?
;29–30 ■ Use a graphing calculator or computer to reproduce the picture.
29. 30.
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
31–32 ■ Compare the curves represented by the parametric equations. How do they differ?
31. (a) ,
(b) ,
(c) ,
32. (a) ,
(b) ,
(c) ,
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
33. Derive Equations 1 for the case .
34. Let be a point at a distance from the center of a circle of radius . The curve traced out by as the circle rolls along a straight line is called a trochoid. (Think of the motion of a point on a spoke of a bicycle wheel.) The cycloid is the special case of a trochoid with . Using the same parameter as for the cycloid and assuming the line is the -axis and when is at one of its lowest points, show that parametric equations of the trochoid are
Sketch the trochoid for the cases and . If and are fixed 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 identify the curve.
O y
x
¨
a b P
P
b 35. a
d r d r
y r d cos x r d sin
0 P x
d r
P r
d P
2 y e2t
x et
y sec2t x cos t
y t2 x t
y e2t x e3t
y t4 x t6
y t2 x t3
0 0
2 y
x
y
x 2
3 8
4
2 a 3 21.
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
22. Match the parametric equations with the graphs labeled I–VI. Give reasons for your choices. (Do not use a graphing device.)
(a) ,
(b) ,
(c) ,
(d) ,
(e) ,
(f ) ,
;23. Graph the curve .
;24. Graph the curves and and find their points of intersection correct to one decimal place.
(a) Show that the parametric equations
where , describe the line segment that joins
the points and .
(b) Find parametric equations to represent the line segment from to .
;26. Use a graphing device and the result of Exercise 25(a) to draw the triangle with vertices , , and . Find parametric equations for the path of a particle that moves along the circle in the manner described.
(a) Once around clockwise, starting at
(b) Three times around counterclockwise, starting at (c) Halfway around counterclockwise, starting at
; (a) Find parametric equations for the ellipse
. [Hint: Modify the equations of the circle in Example 2.]
x2a2 y2b2 1 28.
0, 32, 1
2, 1
x2 y 12 4 27.
C1, 5
B4, 2
A1, 1
3, 1
2, 7
P2x2, y2 P1x1, y1
0 t 1
y y1 y2 y1t x x1 x2 x1t
25.
x yy 12 y x5
x y 3y3 y5 VI
0 x
y
0 x
I II y
0 x
IV y
0 x
y
0 x
V y
III
0 x
y
y sint sin 5t
x cos t
y cost cos t
x sint sin t
y t sin 3t x t sin 2t
y sin 4t x sin 3t
y 2 t2 x t3 1
y t2 t x t3 2t
t y
1
1 t
x 1
1
27
31
35
tance is assumed to be negligible, then its position after seconds is given by the parametric equations
where is the acceleration due to gravity ( ms ).
(a) If a gun is fired with and ms, when will the bullet hit the ground? How far from the gun will it hit the ground? What is the maximum height reached by the bullet?
; (b) Use a graphing device to check your 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 your findings.
(c) Show that the path is parabolic by eliminating the parameter.
; Investigate the family of curves defined by the parametric equations , . How does the shape change as increases? Illustrate by graphing several members of the family.
;40. The swallowtail catastrophe curves are defined by the
parametric equations , .
Graph several of these curves. What features do the curves have in common? How do they change when increases?
; The curves with equations , are
called Lissajous figures. Investigate how these curves vary when , , and vary. (Take to be a positive integer.)
;42. Investigate the family of curves defined by the parametric equations
How does the shape change as c changes? In particular, you should identify the transitional values of c for which the basic shape of the curve changes.
y cos t c sin t
x sin t c sin t
n n
b a
y b cos t x a sin nt
41.
c y ct2 3t4 x 2ct 4t3
c
y t3 ct x t2
39.
v0 500
30
9.8 2
t
y v0sint 12tt2 x v0cost
t 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.
;37. Suppose that the position of one particle at time is given by
and the position of a second particle is given by
(a) Graph the paths of both particles. How many points of intersection are there?
(b) Are any 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 by
38. If a projectile is fired with an initial velocity of meters per second at an angle above the horizontal and air resis-
v0
x2 3 cos t y2 1 sin t 0 t 2
0 t 2
y2 1 sin t x2 3 cos t
0 t 2
y1 2 cos t x1 3 sin t
t O
A P
y=2a C
y
x a
¨
y 2a sin2 x 2a cot
P
CALCULUS WITH PARAMETRIC CURVES
Having seen how to represent curves by parametric equations, we now apply the meth- ods of calculus to these parametric curves. In particular, we solve problems involving tangents, areas, and arc length.
TANGENTS
Suppose and are differentiable functions and we want to find the tangent line at a point on the parametric curve , where is also a differentiable func- tion of . Then the Chain Rule gives
dy dt dy
dx dx dt x
y y tt
x f t
t f
