Section 8.4 Plane Curves and Parametric Equations
16
Section 8.4 Plane Curves and Parametric Equations Suppose that x and y are both given as functions of a third variable t (called a parameter) by the equations x = f (t), y = g(t) (called parametric equations). 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. EXAMPLE: Sketch and identify the curve defined by the parametric equations x = cos t, y = sin t 0 ≤ t ≤ 2π Solution: If we plot points, it appears that the curve is a circle (see the figure below and page 6). We can confirm this impression by eliminating t. If fact, we have x 2 + y 2 = cos 2 t + sin 2 t =1 Thus the point (x, y) moves on a unit circle x 2 + y 2 =1 . EXAMPLE: Sketch and identify the curve defined by the parametric equations x = sin 2t, y = cos 2t 0 ≤ t ≤ 2π 1
Transcript of Section 8.4 Plane Curves and Parametric Equations
Section 8.4 Plane Curves and Parametric Equations
Suppose that x and y are both given as functions of a third variable t (called a parameter) by theequations
x = f(t), y = g(t)
(called parametric equations). Each value of t determines a point (x, y), which we can plot in acoordinate plane. As t varies, the point (x, y) = (f(t), g(t)) varies and traces out a curve C, which we calla parametric curve.
EXAMPLE: Sketch and identify the curve defined by the parametric equations
x = cos t, y = sin t 0 ≤ t ≤ 2π
Solution: If we plot points, it appears that the curve is a circle (see the figure below and page 6). We canconfirm this impression by eliminating t. If fact, we have
x2 + y2 = cos2 t+ sin2 t = 1
Thus the point (x, y) moves on a unit circle x2 + y2 = 1 .
EXAMPLE: Sketch and identify the curve defined by the parametric equations
x = sin 2t, y = cos 2t 0 ≤ t ≤ 2π
1
EXAMPLE: Sketch and identify the curve defined by the parametric equations
x = sin 2t, y = cos 2t 0 ≤ t ≤ 2π
Solution: If we plot points, it appears that the curve is a circle (see the Figure below). We can confirmthis impression by eliminating t. If fact, we have
x2 + y2 = sin2 2t+ cos2 2t = 1
Thus the point (x, y) moves on a unit circle x2 + y2 = 1 .
EXAMPLE: Sketch and identify the curve defined by the parametric equations
x = 5 cos t, y = 2 sin t 0 ≤ t ≤ 2π
Solution: If we plot points, it appears that the curve is an ellipse (see page 8). We can confirm thisimpression by eliminating t. If fact, we have
x2
25+
y2
4=
(x
5
)2
+(y
2
)2
= cos2 t+ sin2 t = 1
Thus the point (x, y) moves on an ellipsex2
25+
y2
4= 1 .
-4 -2 2 4
-2
-1
1
2
x=5cosHtL, y=2sinHtL, 0<t<12Pi�6
EXAMPLE: Sketch and identify the curve defined by the parametric equations
x = t cos t, y = t sin t t > 0
2
EXAMPLE: Sketch and identify the curve defined by the parametric equations
x = t cos t, y = t sin t t > 0
Solution: If we plot points, it appears that the curve is a spiral (see page 9). We can confirm this impressionby the following algebraic manipulations:
x2 + y2 = (t cos t)2 + (t sin t)2 = t2 sin2 t+ t2 cos2 t = t2(sin2 t+ cos2 t) = t2 =⇒ x2 + y2 = t2
To eliminate t completely, we observe that
y
x=
t sin t
t cos t=
sin t
cos t= tan t =⇒ t = arctan
(y
x
)
Substituting this into x2 + y2 = t2, we get x2 + y2 = arctan2
(y
x
)
.
-10 -5 5 10
-10
-5
5
10
x=t cosHtL, y=t sinHtL, 0<t<12Pi�4
EXAMPLE: Sketch and identify the curve defined by the parametric equations
x = t2 + t, y = 2t− 1
3
EXAMPLE: Sketch and identify the curve defined by the parametric equations
x = t2 + t, y = 2t− 1
Solution: If we plot points, it appears that the curve is a parabola (see page 10). We can confirm thisimpression by eliminating t. If fact, we have
y = 2t− 1 =⇒ t =y + 1
2=⇒ x = t2 + t =
(
y + 1
2
)2
+y + 1
2=
1
4y2 + y +
3
4
Thus the point (x, y) moves on a parabola x =1
4y2 + y +
3
4.
0.5 1.0 1.5 2.0 2.5
-5
-4
-3
-2
-1
1
2
x=t^2+t, y=2t-1, -2<t<-2+12�4
EXAMPLE: Sketch and identify the curve defined by the parametric equations
x = sin2 t, y = 2 cos t
Solution: If we plot points, it appears that the curve is a restricted parabola (see page 11). We can confirmthis impression by eliminating t. If fact, we have
y = 2 cos t =⇒ y2 = 4 cos2 t =⇒ 4x+ y2 = 4 sin2 t+ 4 cos2 t = 4 =⇒ x = 1−y2
4
We also note that 0 ≤ x ≤ 1 and −2 ≤ y ≤ 2. Thus the point (x, y) moves on the restricted parabola
x = 1−y2
4.
0.40.60.81.01.2
-2
-1
1
2
x=sin^2HtL, y=2cosHtL, 0<t<12Pi�6
4
The Cycloid
EXAMPLE: The curve traced out by a point P on the circumference of a circle as the circle rolls along astraight line is called a cycloid (see the Figure below). If the circle has radius r and rolls along the x-axisand if one position of P is the origin, find parametric equations for the cycloid.
Solution: We choose as parameter the angle of rotation θ of the circle (θ = 0 when P is at the origin).Suppose the circle has rotated through θ radians. Because the circle has been in contact with the line, wesee from the Figure below that the distance it has rolled from the origin is
|OT | = arc PT = rθ
Therefore, the center of the circle is C(rθ, r). Let the coordinates of P be (x, y). Then from the Figureabove we see that
x = |OT | − |PQ| = rθ − r sin θ = r(θ − sin θ)
y = |TC| − |QC| = r − r cos θ = r(1− cos θ)
Therefore, parametric equations of the cycloid are
x = r(θ − sin θ), y = r(1− cos θ) θ ∈ R (1)
One arch of the cycloid comes from one rotation of the circle and so is described by 0 ≤ θ ≤ 2π. AlthoughEquations 1 were derived from the Figure above, which illustrates the case where 0 < θ < π/2, it can beseen that these equations are still valid for other values of θ.
Although it is possible to eliminate the parameter θ from Equations 1, the resulting Cartesian equation inx and y is very complicated and not as convenient to work with as the parametric equations:
∣
∣
∣
∣
x
r+ 2π
⌈
1
2−
x
πr
⌉
− 1
∣
∣
∣
∣
= cos−1
(
1−y
r
)
− 2
√
2y
r−(y
r
)2
5
Unit Circle
Parametric equations:x = cos t, y = sin t
where t = kπ/6, k = 0, . . . , 12.
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=cosHtL, y=sin HtL, 0<t<1Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=cosHtL, y=sin HtL, 0<t<2Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=cosHtL, y=sin HtL, 0<t<3Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=cosHtL, y=sin HtL, 0<t<4Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=cosHtL, y=sin HtL, 0<t<5Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=cosHtL, y=sin HtL, 0<t<6Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=cosHtL, y=sin HtL, 0<t<7Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=cosHtL, y=sin HtL, 0<t<8Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=cosHtL, y=sin HtL, 0<t<9Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=cosHtL, y=sin HtL, 0<t<10Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=cosHtL, y=sin HtL, 0<t<11Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=cosHtL, y=sin HtL, 0<t<12Pi�6
6
Unit Circle
Parametric equations:x = sin t, y = cos t
where t = kπ/6, k = 0, . . . , 12.
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=sinHtL, y=cosHtL, 0<t<1Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=sinHtL, y=cosHtL, 0<t<2Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=sinHtL, y=cosHtL, 0<t<3Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=sinHtL, y=cosHtL, 0<t<4Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=sinHtL, y=cosHtL, 0<t<5Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=sinHtL, y=cosHtL, 0<t<6Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=sinHtL, y=cosHtL, 0<t<7Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=sinHtL, y=cosHtL, 0<t<8Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=sinHtL, y=cosHtL, 0<t<9Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=sinHtL, y=cosHtL, 0<t<10Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=sinHtL, y=cosHtL, 0<t<11Pi�6
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
x=sinHtL, y=cosHtL, 0<t<12Pi�6
7
Ellipse
Parametric equations:x = 5 cos t, y = 2 sin t
where t = kπ/6, k = 0, . . . , 12.
-4 -2 2 4
-2
-1
1
2
x=5cosHtL, y=2sinHtL, 0<t<1Pi�6
-4 -2 2 4
-2
-1
1
2
x=5cosHtL, y=2sinHtL, 0<t<2Pi�6
-4 -2 2 4
-2
-1
1
2
x=5cosHtL, y=2sinHtL, 0<t<3Pi�6
-4 -2 2 4
-2
-1
1
2
x=5cosHtL, y=2sinHtL, 0<t<4Pi�6
-4 -2 2 4
-2
-1
1
2
x=5cosHtL, y=2sinHtL, 0<t<5Pi�6
-4 -2 2 4
-2
-1
1
2
x=5cosHtL, y=2sinHtL, 0<t<6Pi�6
-4 -2 2 4
-2
-1
1
2
x=5cosHtL, y=2sinHtL, 0<t<7Pi�6
-4 -2 2 4
-2
-1
1
2
x=5cosHtL, y=2sinHtL, 0<t<8Pi�6
-4 -2 2 4
-2
-1
1
2
x=5cosHtL, y=2sinHtL, 0<t<9Pi�6
-4 -2 2 4
-2
-1
1
2
x=5cosHtL, y=2sinHtL, 0<t<10Pi�6
-4 -2 2 4
-2
-1
1
2
x=5cosHtL, y=2sinHtL, 0<t<11Pi�6
-4 -2 2 4
-2
-1
1
2
x=5cosHtL, y=2sinHtL, 0<t<12Pi�6
8
Spiral
Parametric equations:x = t cos t, y = t sin t
where t = kπ/4, k = 0, . . . , 12.
-1.5 -1.0 -0.5 0.5 1.0 1.5
-1.5
-1.0
-0.5
0.5
1.0
1.5
x=t cosHtL, y=t sinHtL, 0<t<1Pi�4
-2 -1 1 2
-2
-1
1
2
x=t cosHtL, y=t sinHtL, 0<t<2Pi�4
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
x=t cosHtL, y=t sinHtL, 0<t<3Pi�4
-4 -2 2 4
-4
-2
2
4
x=t cosHtL, y=t sinHtL, 0<t<4Pi�4
-4 -2 2 4
-4
-2
2
4
x=t cosHtL, y=t sinHtL, 0<t<5Pi�4
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
x=t cosHtL, y=t sinHtL, 0<t<6Pi�4
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
x=t cosHtL, y=t sinHtL, 0<t<7Pi�4
-5 5
-5
5
x=t cosHtL, y=t sinHtL, 0<t<8Pi�4
-5 5
-5
5
x=t cosHtL, y=t sinHtL, 0<t<9Pi�4
-10 -5 5 10
-10
-5
5
10
x=t cosHtL, y=t sinHtL, 0<t<10Pi�4
-10 -5 5 10
-10
-5
5
10
x=t cosHtL, y=t sinHtL, 0<t<11Pi�4
-10 -5 5 10
-10
-5
5
10
x=t cosHtL, y=t sinHtL, 0<t<12Pi�4
9
Parabola
Parametric equations:x = t2 + t, y = 2t− 1
where t = −2 + k/4, k = 0, . . . , 12.
0.5 1.0 1.5 2.0 2.5
-5
-4
-3
-2
-1
1
2
x=t^2+t, y=2t-1, -2<t<-2+1�4
0.5 1.0 1.5 2.0 2.5
-5
-4
-3
-2
-1
1
2
x=t^2+t, y=2t-1, -2<t<-2+2�4
0.5 1.0 1.5 2.0 2.5
-5
-4
-3
-2
-1
1
2
x=t^2+t, y=2t-1, -2<t<-2+3�4
0.5 1.0 1.5 2.0 2.5
-5
-4
-3
-2
-1
1
2
x=t^2+t, y=2t-1, -2<t<-2+4�4
0.5 1.0 1.5 2.0 2.5
-5
-4
-3
-2
-1
1
2
x=t^2+t, y=2t-1, -2<t<-2+5�4
0.5 1.0 1.5 2.0 2.5
-5
-4
-3
-2
-1
1
2
x=t^2+t, y=2t-1, -2<t<-2+6�4
0.5 1.0 1.5 2.0 2.5
-5
-4
-3
-2
-1
1
2
x=t^2+t, y=2t-1, -2<t<-2+7�4
0.5 1.0 1.5 2.0 2.5
-5
-4
-3
-2
-1
1
2
x=t^2+t, y=2t-1, -2<t<-2+8�4
0.5 1.0 1.5 2.0 2.5
-5
-4
-3
-2
-1
1
2
x=t^2+t, y=2t-1, -2<t<-2+9�4
0.5 1.0 1.5 2.0 2.5
-5
-4
-3
-2
-1
1
2
x=t^2+t, y=2t-1, -2<t<-2+10�4
0.5 1.0 1.5 2.0 2.5
-5
-4
-3
-2
-1
1
2
x=t^2+t, y=2t-1, -2<t<-2+11�4
0.5 1.0 1.5 2.0 2.5
-5
-4
-3
-2
-1
1
2
x=t^2+t, y=2t-1, -2<t<-2+12�4
10
Restricted Parabola
Parametric equations:x = sin2 t, y = 2 cos t
where t = kπ/6, k = 0, . . . , 12.
0.40.60.81.01.2
-2
-1
1
2
x=sin^2HtL, y=2cosHtL, 0<t<1Pi�6
0.40.60.81.01.2
-2
-1
1
2
x=sin^2HtL, y=2cosHtL, 0<t<2Pi�6
0.40.60.81.01.2
-2
-1
1
2
x=sin^2HtL, y=2cosHtL, 0<t<3Pi�6
0.40.60.81.01.2
-2
-1
1
2
x=sin^2HtL, y=2cosHtL, 0<t<4Pi�6
0.40.60.81.01.2
-2
-1
1
2
x=sin^2HtL, y=2cosHtL, 0<t<5Pi�6
0.40.60.81.01.2
-2
-1
1
2
x=sin^2HtL, y=2cosHtL, 0<t<6Pi�6
0.40.60.81.01.2
-2
-1
1
2
x=sin^2HtL, y=2cosHtL, 0<t<7Pi�6
0.40.60.81.01.2
-2
-1
1
2
x=sin^2HtL, y=2cosHtL, 0<t<8Pi�6
0.40.60.81.01.2
-2
-1
1
2
x=sin^2HtL, y=2cosHtL, 0<t<9Pi�6
0.40.60.81.01.2
-2
-1
1
2
x=sin^2HtL, y=2cosHtL, 0<t<10Pi�6
0.40.60.81.01.2
-2
-1
1
2
x=sin^2HtL, y=2cosHtL, 0<t<11Pi�6
0.40.60.81.01.2
-2
-1
1
2
x=sin^2HtL, y=2cosHtL, 0<t<12Pi�6
11
Butterfly Curve
Parametric equations:
x = sin t
[
ecos t − 2 cos(4t) + sin5
(
t
12
)]
, y = cos t
[
ecos t − 2 cos(4t) + sin5
(
t
12
)]
where t = kπ/6, k = 0, . . . , 12.
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
12
Butterfly Curve
Parametric equations:
x = sin t
[
ecos t − 2 cos(4t) + sin5
(
t
12
)]
, y = cos t
[
ecos t − 2 cos(4t) + sin5
(
t
12
)]
where t = kπ/2, k = 0, . . . , 12.
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
13
Parametric Curve
Parametric equations:x = t+ 2 sin 2t, y = t+ 2 cos 5t
where t = −2π + kπ/3, k = 0, . . . , 12.
-6 -4 -2 2 4 6
-5
5
x=t+2sinH2tL,y=t+2cosH5tL, -2Pi<t<-2Pi+1Pi�3
-6 -4 -2 2 4 6
-5
5
x=t+2sinH2tL,y=t+2cosH5tL, -2Pi<t<-2Pi+2Pi�3
-6 -4 -2 2 4 6
-5
5
x=t+2sinH2tL,y=t+2cosH5tL, -2Pi<t<-2Pi+3Pi�3
-6 -4 -2 2 4 6
-5
5
x=t+2sinH2tL,y=t+2cosH5tL, -2Pi<t<-2Pi+4Pi�3
-6 -4 -2 2 4 6
-5
5
x=t+2sinH2tL,y=t+2cosH5tL, -2Pi<t<-2Pi+5Pi�3
-6 -4 -2 2 4 6
-5
5
x=t+2sinH2tL,y=t+2cosH5tL, -2Pi<t<-2Pi+6Pi�3
-6 -4 -2 2 4 6
-5
5
x=t+2sinH2tL,y=t+2cosH5tL, -2Pi<t<-2Pi+7Pi�3
-6 -4 -2 2 4 6
-5
5
x=t+2sinH2tL,y=t+2cosH5tL, -2Pi<t<-2Pi+8Pi�3
-6 -4 -2 2 4 6
-5
5
x=t+2sinH2tL,y=t+2cosH5tL, -2Pi<t<-2Pi+9Pi�3
-6 -4 -2 2 4 6
-5
5
x=t+2sinH2tL,y=t+2cosH5tL, -2Pi<t<-2Pi+10Pi�3
-6 -4 -2 2 4 6
-5
5
x=t+2sinH2tL,y=t+2cosH5tL, -2Pi<t<-2Pi+11Pi�3
-6 -4 -2 2 4 6
-5
5
x=t+2sinH2tL,y=t+2cosH5tL, -2Pi<t<-2Pi+12Pi�3
14
Parametric Curves
Parametric equations:x = 1.5 cos t− cos kt, y = 1.5 sin t− sin kt
where k = 1, . . . , 20.
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH1tL, y=1.5sinHtL-sinH1tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH2tL, y=1.5sinHtL-sinH2tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH3tL, y=1.5sinHtL-sinH3tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH4tL, y=1.5sinHtL-sinH4tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sinHtL-sinH5tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH6tL, y=1.5sinHtL-sinH6tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH7tL, y=1.5sinHtL-sinH7tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH8tL, y=1.5sinHtL-sinH8tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH9tL, y=1.5sinHtL-sinH9tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH10tL, y=1.5sinHtL-sinH10tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH11tL, y=1.5sinHtL-sinH11tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH12tL, y=1.5sinHtL-sinH12tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH13tL, y=1.5sinHtL-sinH13tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH14tL, y=1.5sinHtL-sinH14tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH15tL, y=1.5sinHtL-sinH15tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH16tL, y=1.5sinHtL-sinH16tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH17tL, y=1.5sinHtL-sinH17tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH18tL, y=1.5sinHtL-sinH18tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH19tL, y=1.5sinHtL-sinH19tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH20tL, y=1.5sinHtL-sinH20tL
15
Parametric Curves
Parametric equations:x = 1.5 cos t− cos 5t, y = 1.5 sin t− sin kt
where k = 1, . . . , 20.
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H1tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H2tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H3tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H4tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H5tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H6tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H7tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H8tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H9tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H10tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H11tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H12tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H13tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H14tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H15tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H16tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H17tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H18tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H19tL
-2 -1 1 2
-2
-1
1
2
x=1.5cosHtL-cosH5tL, y=1.5sin HtL-sin H20tL
16
