Section 10.1 Curves Defined by Parametric Equations
Ruipeng Shen
March 5
Parametric Equations 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 y = f (x) because C fails the vertical
line test. But the x and y-coordinates of the particle are both functions of the time t. Thus we
can write
x = f (t)
, t ∈ [0, T ].
y = g(t)
In general, if both x and y are given as functions of a third variable t (called a parameter),
the equation above is called a parametric equation. Each value of t determines a point (x, y),
which we can plot in a coordinate plane. As t varies, the point (x, y) = (f (t), g(t)) varies and
traces out a curve C, which we call a parametric curve.
y
C
(x,y)=(f(t),g(t))
x
Figure 1: Trajectory of a Particle
Example 1. Sketch and identity the curve defined by the parametric equation
x = t2 − 2t
y =t+1
Solution By plugging in a few values of t, we can plot a few points on the plane and then
connect them with a smooth curve.
1
t
-2
-1
0
1
2
3
4
(x,y)
(8,-1)
(3,0)
(0,1)
(-1,2)
(0,3)
(3,4)
(8,5)
7.5
t=4
5
!
t=3
!
t=2
!
2.5
t=1
!
t=0
-2.5
!
0
2.5
!
t=-1
5
7.5
10
12.5
15
!
t=-2
-2.5
!!
We can identify this curve by eliminating the parameter t:
y = t + 1 =⇒ t = y − 1 =⇒ x = t2 − 2t = (y − 1)2 − 2(y − 1) = y 2 − 4y + 3.
This is a parabola.
Example 2. What curve is represented by the parametric equation
x = cos t
, 0 ≤ t ≤ 2π.
y = sin t
Solution Since x2 + y 2 = cos2 t + sin2 t = 1, the curve is the unit circle centered at the origin,
as shown in Figure 2.
Example 3. What curve is represented by the parametric equation
x = cos 2t
, 0 ≤ t ≤ 2π.
y = sin 2t
Solution Since x2 + y 2 = cos2 2t + sin2 2t = 1, the curve is still the unit circle centered at the
origin, but wrapped by two times, as shown in Figure 3.
Example 4. Find parametric equations for the circle with (x0 , y0 ) and radius r.
2
y
t=𝛑/2
(cos t, sin t)
t
t=𝛑
t=0, 2𝛑
x
t=3𝛑/2
Figure 2: The unit circle, Example 2
y
t=𝛑/4, 5𝛑/4
(cos 2t, sin 2t)
2t
t=𝛑/2, 3𝛑/2
t=0,𝛑,2𝛑
t=3𝛑/4, 7𝛑/4
Figure 3: The unit circle, Example 3
Solution
Using basic geometry, we have
x = x0 + r cos t
,
y = y0 + r sin t
0 ≤ t ≤ 2π;
as shown in the Figure 4
Example 5. Sketch the curve with parametric equation
x = sin t
y = sin2 t
3
x
y
(x0+r cos t, y0 + r sin t)
r
r sin t
t
(x0,y0) r cos t
x
Figure 4: A general circle
Solution It is clear that y = sin2 t = x2 . In addition, x moves back and forth between −1
and 1. Therefore the curve is a piece of parabola y = x2 , −1 ≤ x ≤ 1. As t increases, the point
(x, y) moves back and forth along this curve, as shown in Figure 5
y
(-1,1)
(1,1)
y=x2
x
Figure 5: A general circle
Example 6 (The Cycloid). The curve traced by a point P on the circumference of a circle with
radius r as the circle rolls along a straight line is called a cycloid. Find a parametric equation
4
for the cycloid.
O(rt,r)
t r cos t
P r sin t Q
A
rt
B
Figure 6: Cycloid
Solution Let us choose the parameter t to be angle from the vertical line OB to OP . This
implies the arc BP has a length rt. Since the circle rolls along the x-axis, the line segment AB
has the same length rt. Thus the point O has the coordinates (rt, r). Now let us consider the
right triangle OP Q. Using the conditions that |OP | = r and the angle ∠P OQ = t, we have
|OQ| = r cos t and |P Q| = r sin t. By these dimensions we have
x = rt − r sin t
y = r − r cos t
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