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Transcript of Plane Curves and ParametriPlane Curves and Parametric Equationsc Equations
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8/13/2019 Plane Curves and ParametriPlane Curves and Parametric Equationsc Equations
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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. Ast 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 2
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 2
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EXAMPLE: Sketch and identify the curve defined by the parametric equations
x= sin 2t, y= cos 2t 0 t 2
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= 5cos t, y= 2 sin t 0 t 2
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 =
x5
2
+
y2
2
= cos2 t+ sin2 t= 1
Thus the point (x, y) moves on an ellipse x2
25+
y2
4 = 1.
4 2 2 4
2
1
1
2
x5cost, y2sint, 0t12Pi6
EXAMPLE: Sketch and identify the curve defined by the parametric equations
x= t cos t, y= t sin t t >0
2
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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
yx
Substituting this into x2 +y2 =t2,we get x2 +y2 = arctan2y
x
.
10 5 5 10
10
5
5
10
xt cost, yt sint, 0t12Pi4
EXAMPLE: Sketch and identify the curve defined by the parametric equations
x= t2 +t, y= 2t 1
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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 .
0.51.01.52.02.5
5
4
3
2
1
1
2
xt^2t, y2t1, 2t2124
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
xsin^2t, y2cost, 0t12Pi6
4
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The Cycloid
EXAMPLE: The curve traced out by a point Pon 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 ofPis the origin, find parametric equations for the cycloid.
Solution: We choose as parameter the angle of rotation of the circle ( = 0 when P is at the origin)Suppose the circle has rotated through 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 P T =r
Therefore, the center of the circle is C(r,r). Let the coordinates ofP be (x, y). Then from the Figureabove we see that
x= |OT| |P Q| =r r sin = r( sin )
y= |T C| |QC| =r r cos = r(1 cos )
Therefore, parametric equations of the cycloid are
x= r( sin ), y= r(1 cos ) R (1)
One arch of the cycloid comes from one rotation of the circle and so is described by 0 2. AlthoughEquations 1 were derived from the Figure above, which illustrates the case where 0 < < /2, it can beseen that these equations are still valid for other values of.
Although it is possible to eliminate the parameter from Equations 1, the resulting Cartesian equation inx and y is very complicated and not as convenient to work with as the parametric equations:
xr + 2
1
2
x
r
1
= cos1
1 y
r
2
2
y
r
yr
2
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Unit Circle
Parametric equations:x= cos t, y= sin t
wheret= k/6, k= 0, . . . , 12.
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xcos t, ysin t, 0t1Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xcos t, ysin t, 0t2Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xcos t, ysin t, 0t3Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xcos t, ysin t, 0t4Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xcos t, ysin t, 0t5Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xcos t, ysin t, 0t6Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xcos t, ysin t, 0t7Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xcos t, ysin t, 0t8Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xcos t, ysin t, 0t9Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xcos t, ysin t, 0t10Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xcos t, ysin t, 0t11Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xcos t, ysin t, 0t12Pi6
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Unit Circle
Parametric equations:x= sin t, y= cos t
wheret= k/6, k= 0, . . . , 12.
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xsint, ycost, 0t1Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xsint, ycost, 0t2Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xsint, ycost, 0t3Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xsint, ycost, 0t4Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xsint, ycost, 0t5Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xsint, ycost, 0t6Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xsint, ycost, 0t7Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xsint, ycost, 0t8Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xsint, ycost, 0t9Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xsint, ycost, 0t10Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xsint, ycost, 0t11Pi6
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
xsint, ycost, 0t12Pi6
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Ellipse
Parametric equations:x= 5 cos t, y= 2 sin t
wheret= k/6, k= 0, . . . , 12.
4 2 2 4
2
1
1
2
x5cost, y2sint, 0t1Pi6
4 2 2 4
2
1
1
2
x5cost, y2sint, 0t2Pi6
4 2 2 4
2
1
1
2
x5cost, y2sint, 0t3Pi6
4 2 2 4
2
1
1
2
x5cost, y2sint, 0t4Pi6
4 2 2 4
2
1
1
2
x5cost, y2sint, 0t5Pi6
4 2 2 4
2
1
1
2
x5cost, y2sint, 0t6Pi6
4 2 2 4
2
1
1
2
x5cost, y2sint, 0t7Pi6
4 2 2 4
2
1
1
2
x5cost, y2sint, 0t8Pi6
4 2 2 4
2
1
1
2
x5cost, y2sint, 0t9Pi6
4 2 2 4
2
1
1
2
x5cost, y2sint, 0t10Pi6
4 2 2 4
2
1
1
2
x5cost, y2sint, 0t11Pi6
4 2 2 4
2
1
1
2
x5cost, y2sint, 0t12Pi6
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Spiral
Parametric equations:x= t cos t, y= t sin t
wheret= k/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
xt cost, yt sint, 0t1Pi4
2 1 1 2
2
1
1
2
xt cost, yt sint, 0t2Pi4
3 2 1 1 2 3
3
2
1
1
2
3
xt cost, yt sint, 0t3Pi4
4 2 2 4
4
2
2
4
xt cost, yt sint, 0t4Pi4
4 2 2 4
4
2
2
4
xt cost, yt sint, 0t5Pi4
6 4 2 2 4 6
6
4
2
2
4
6
xt cost, yt sint, 0t6Pi4
6 4 2 2 4 6
6
4
2
2
4
6
xt cost, yt sint, 0t7Pi4
5 5
5
5
xt cost, yt sint, 0t8Pi4
5 5
5
5
xt cost, yt sint, 0t9Pi4
10 5 5 10
10
5
5
10
xt cost, yt sint, 0t10Pi4
10 5 5 10
10
5
5
10
xt cost, yt sint, 0t11Pi4
10 5 5 10
10
5
5
10
xt cost, yt sint, 0t12Pi4
9
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Parabola
Parametric equations:x= t2 +t, y= 2t 1
wheret= 2 +k/4, k= 0, . . . , 12.
0.51.01.52.02.5
5
4
3
2
1
1
2
xt^2t, y2t1, 2t214
0.51.01.52.02.5
5
4
3
2
1
1
2
xt^2t, y2t1, 2t224
0.51.01.52.02.5
5
4
3
2
1
1
2
xt^2t, y2t1, 2t234
0.51.01.52.02.5
5
4
3
2
1
1
2
xt^2t, y2t1, 2t244
0.51.01.52.02.5
5
4
3
2
1
1
2
xt^2t, y2t1, 2t254
0.51.01.52.02.5
5
4
3
2
1
1
2
xt^2t, y2t1, 2t264
0.51.01.52.02.5
5
4
3
2
1
1
2
xt^2t, y2t1, 2t274
0.51.01.52.02.5
5
4
3
2
1
1
2
xt^2t, y2t1, 2t284
0.51.01.52.02.5
5
4
3
2
1
1
2
xt^2t, y2t1, 2t294
0.51.01.52.02.5
5
4
3
2
1
1
2
xt^2t, y2t1, 2t2104
0.51.01.52.02.5
5
4
3
2
1
1
2
xt^2t, y2t1, 2t2114
0.51.01.52.02.5
5
4
3
2
1
1
2
xt^2t, y2t1, 2t2124
10
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Restricted Parabola
Parametric equations:x= sin2 t, y= 2 cos t
wheret= k/6, k= 0, . . . , 12.
0.40.60.81.01.2
2
1
1
2
xsin^2t, y2cost, 0t1Pi6
0.40.60.81.01.2
2
1
1
2
xsin^2t, y2cost, 0t2Pi6
0.40.60.81.01.2
2
1
1
2
xsin^2t, y2cost, 0t3Pi6
0.40.60.81.01.2
2
1
1
2
xsin^2t, y2cost, 0t4Pi6
0.40.60.81.01.2
2
1
1
2
xsin^2t, y2cost, 0t5Pi6
0.40.60.81.01.2
2
1
1
2
xsin^2t, y2cost, 0t6Pi6
0.40.60.81.01.2
2
1
1
2
xsin^2t, y2cost, 0t7Pi6
0.40.60.81.01.2
2
1
1
2
xsin^2t, y2cost, 0t8Pi6
0.40.60.81.01.2
2
1
1
2
xsin^2t, y2cost, 0t9Pi6
0.40.60.81.01.2
2
1
1
2
xsin^2t, y2cost, 0t10Pi6
0.40.60.81.01.2
2
1
1
2
xsin^2t, y2cost, 0t11Pi6
0.40.60.81.01.2
2
1
1
2
xsin^2t, y2cost, 0t12Pi6
11
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Butterfly Curve
Parametric equations:
x= sin t
ecos t 2 cos(4t) + sin5
t
12
, y= cos t
ecos t 2cos(4t) + sin5
t
12
wheret= k/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
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Butterfly Curve
Parametric equations:
x= sin t
ecos t 2 cos(4t) + sin5
t
12
, y= cos t
ecos t 2cos(4t) + sin5
t
12
wheret= k/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
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Parametric Curve
Parametric equations:x= t+ 2sin 2t, y= t+ 2cos 5t
wheret= 2+k/3, k= 0, . . . , 12.
6 4 2 2 4 6
5
5
xt2sin2t,yt2cos5t, 2Pit2Pi1Pi3
6 4 2 2 4 6
5
5
xt2sin2t,yt2cos5t, 2Pit2Pi2Pi3
6 4 2 2 4 6
5
5
xt2sin2t,yt2cos5t, 2Pit2Pi3Pi3
6 4 2 2 4 6
5
5
xt2sin2t,yt2cos5t, 2Pit2Pi4Pi3
6 4 2 2 4 6
5
5
xt2sin2t,yt2cos5t, 2Pit2Pi5Pi3
6 4 2 2 4 6
5
5
xt2sin2t,yt2cos5t, 2Pit2Pi6Pi3
6 4 2 2 4 6
5
5
xt2sin2t,yt2cos5t, 2Pit2Pi7Pi3
6 4 2 2 4 6
5
5
xt2sin2t,yt2cos5t, 2Pit2Pi8Pi3
6 4 2 2 4 6
5
5
xt2sin2t,yt2cos5t, 2Pit2Pi9Pi3
6 4 2 2 4 6
5
5
xt2sin2t,yt2cos5t, 2Pit2Pi10Pi3
6 4 2 2 4 6
5
5
xt2sin2t,yt2cos5t, 2Pit2Pi11Pi3
6 4 2 2 4 6
5
5
xt2sin2t,yt2cos5t, 2Pit2Pi12Pi3
14
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Parametric Curves
Parametric equations:x= 1.5cos t cos kt, y= 1.5sin t sin kt
wherek= 1, . . . , 20.
2 1 1 2
2
1
1
2
x1.5costcos1t, y1.5sintsin1t
2 1 1 2
2
1
1
2
x1.5costcos2t, y1.5sintsin2t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin3t
2 1 1 2
2
1
1
2
x1.5costcos4t, y1.5sintsin4t
2 1 1 2
2
1
1
2
x1.5costcos5t, y1.5sintsin5t
2 1 1 2
2
1
1
2
x1.5costcos6t, y1.5sintsin6t
2 1 1 2
2
1
1
2
x1.5costcos7t, y1.5sintsin7t
2 1 1 2
2
1
1
2
x1.5costcos8t, y1.5sintsin8t
2 1 1 2
2
1
1
2
x1.5costcos9t, y1.5sintsin9t
2 1 1 2
2
1
1
2
x1.5costcos10t, y1.5sintsin10t
2 1 1 2
2
1
1
2
x1.5costcos11t, y1.5sintsin11t
2 1 1 2
2
1
1
2
x1.5costcos12t, y1.5sintsin12t
2 1 1 2
2
1
1
2
x1.5costcos13t, y1.5sintsin13t
2 1 1 2
2
1
1
2
x1.5costcos14t, y1.5sintsin14t
2 1 1 2
2
1
1
2
x1.5costcos15t, y1.5sintsin15t
2 1 1 2
2
1
1
2
x1.5costcos16t, y1.5sintsin16t
2 1 1 2
2
1
1
2
x1.5costcos17t, y1.5sintsin17t
2 1 1 2
2
1
1
2
x1.5costcos18t, y1.5sintsin18t
2 1 1 2
2
1
1
2
x1.5costcos19t, y1.5sintsin19t
2 1 1 2
2
1
1
2
x1.5costcos20t, y1.5sintsin20t
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Parametric Curves
Parametric equations:x= 1.5cos t cos3t, y= 1.5sin t sin kt
wherek= 1, . . . , 20.
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin1t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin2t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin3t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin4t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin5t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin6t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin7t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin8t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin9t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin10t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin11t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin12t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin13t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin14t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin15t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin16t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin17t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin18t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin19t
2 1 1 2
2
1
1
2
x1.5costcos3t, y1.5sintsin20t
