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8/9/2019 PARAMETRIC EQUATIONS & POLAR COORDINATES
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PARAMETRIC EQUATIONSPARAMETRIC EQUATIONSAND POLAR COORDINATESAND POLAR COORDINATES
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PARAMETRIC EQUATIONS & POLAR COORDINATES
So far, we have described plane curves
by giving:
yas a function ofx [y= f(x)] orxas a function of y
[x = g(y)]
relation betweenxand ythat defines yi!plicitly
as a function ofx [f(x, y) = 0]
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PARAMETRIC EQUATIONS & POLAR COORDINATES
"n this chapter, we discuss
two new !ethods for describing
curves#
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So!e curves$such as the cycloid$are
best handled when bothxand yare given in
ter!s of a third variable tcalled a para!eter
[x= f(t), y= g(t)]#
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%ther curves$such as the cardioid$have
their !ost convenient description when
we use a new coordinate syste!, called
the polar coordinate syste!#
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10.1Curves Defined by
Pr!e"ri# E$u"i%ns
"n this section, we will learn about:
&ara!etric e'uations and generating their curves#
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INTRODUCTION
"!agine that a particle !oves along
the curve C shown here#
"t is i!possible to describe C by an e'uation
of the for! y= f(x)#
his is because
C fails the ertical
*ine est#
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+owever, thex and ycoordinates
of the particle are functions of ti!e#
So, we can writex = f(t) and y = g(t)#
INTRODUCTION
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Such a pair of e'uations is often
a convenient way of describing a curve
and gives rise to the following definition#
INTRODUCTION
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Supposexand yare both given as functions
of a third variable t(called a para!eter) by
the e'uations
x= f(t) and y= g(t)
hese are called para!etric e'uations#
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-ach value of tdeter!ines a point (x, y),
which we can plot in a coordinate plane#
s tvaries, the point (x, y) = (f(t), g(t)) varies
and traces out a curve C.
his is called a para!etric curve#
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he para!eter t does not necessarily
represent ti!e#
"n fact, we could use a letter other than t
for the para!eter#
PARAMETER t
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PARAMETER t
+owever, in !any applications of
para!etric curves, t does denote ti!e#
hus, we can interpret (x, y) = (f(t), g(t))
as the position of a particle at ti!e t#
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S.etch and identify the curve defined
by the para!etric e'uations
x= t/ /t y= t 1
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-ach value of tgives a point on the curve,
as in the table#
2or instance, if t= 0,
thenx= 0, y= 1#
So, the corresponding
point is (0, 1)#
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3ow, we plot the points (x, y) deter!ined by
several values of the para!eter, and 4oin the!
to produce a curve#
PARAMETRIC CURES E'!()e 1
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particle whose position is given by
the para!etric e'uations !oves along
the curve in the direction of the arrows as t
increases#
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3otice that the consecutive points !ar.ed
on the curve appear at e'ual ti!e intervals,
but not at e'ual distances#
hat is becausethe particle slows downand then speeds upas tincreases#
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"t appears that the curve traced out by
the particle !ay be a parabola#
5e can confir! this
by eli!inatingthe para!eter t,as follows#
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5e obtain t= y 1 fro! the e'uation y= t 1#
5e then substitute it in the e'uationx= t/ /t#
his gives:x= t/ /t= (y 1)/ /(y 1)= y/ 6y 7
So, the curve represented by the given para!etrice'uations is the parabolax = y/ 6y 7
PARAMETRIC CURES E'!()e 1
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his e'uation inxand ydescribes
where the particle has been#
+owever, it doesn8t tell us when the particle
was at a particular point#
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he para!etric e'uations have an
advantagethey tell us when the particle
was at a point#
hey also indicate the direction of the !otion#
ADANTA*ES
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3o restriction was placed on the para!eter t
in -9a!ple 1#
So, we assu!ed t could be any real nu!ber#
So!eti!es, however, we restrict t
to lie in a finite interval#
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2or instance, the para!etric curve
x= t/ /t y = t 1 0 t 6
shown is a part of the parabola in -9a!ple 1#
"t starts at the point
(0, 1) and ends at
the point (;,
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he arrowhead indicates the direction
in which the curve is traced as tincreases
fro! 0 to 6#
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5hat curve is represented by the following
para!etric e'uations
x = cos t y= sin t 0 t /
PARAMETRIC CURES E'!()e +
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"f we plot points, it appears the curve
is a circle#
5e can confir! this by eli!inating t#
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%bserve that:
x/ y/= cos/ t sin/ t= 1
hus, the point (x, y) !oves
on the unit circlex/ y/= 1
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3otice that, in this e9a!ple, the para!eter t
can be interpreted as the angle (in radians),
as shown#
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s tincreases fro! 0 to /, the point
(x, y) = (cos t, sin t) !oves once around
the circle in the countercloc.wise direction
starting fro!the point (1, 0)#
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5hat curve is represented by the given
para!etric e'uations
x= sin /t y= cos /t 0 t /
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gain, we have:
x/ y/ = sin/ /t cos/ /t= 1
So, the para!etric e'uations again
represent the unit circlex/ y/ = 1
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+owever, as tincreases fro! 0 to /,
the point (x, y) = (sin /t, cos /t) starts at (0, 1),
!oving twicearound the circle inthe cloc.wise direction#
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-9a!ples / and 7 show that different
sets of para!etric e'uations can represent
the sa!e curve#
So, we distinguish between:
curve, which is a set of points
para!etric curve, where the points
are traced in a particular way
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2ind para!etric e'uations
for the circle with center (h, k)
and radius r.
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5e ta.e the e'uations of the unit circle
in -9a!ple / and !ultiply the e9pressions
forxand yby r.
5e get:x= r cost y= r sin t
>ou can verify these e'uations represent a circle withradius rand center the origin traced countercloc.wise#
E'!()e -PARAMETRIC CURES
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3ow, we shift hunits in thexdirection
and kunits in the ydirection#
E'!()e -PARAMETRIC CURES
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hus, we obtain the para!etric e'uations
of the circle with center (h, k) and radius r :
x= h r cos t y = k r sin t 0 t /
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S.etch the curve with para!etric
e'uations
x = sin t y = sin/ t
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PARAMETRIC CURES
%bserve that y= (sin t)/ =x/#
hus, the point (x, y) !oves on
the parabola y=x/#
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+owever, note also that, as 1 sin t 1,
we have 1 x 1#
So, the para!etric e'uations represent only
the part of the parabola for which 1 x 1#
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Since sin tis periodic, the point
(x, y) = (sin t, sin/ t) !oves bac. and forth
infinitely often along the parabola fro!
(1, 1) to (1, 1)#
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*RAP/IN* DEICES
?ost graphing calculators and co!puter
graphing progra!s can be used to graph
curves defined by para!etric e'uations#
"n fact, it8s instructive to watch a para!etric
curve being drawn by a graphing calculator#
he points are plotted in order as the correspondingpara!eter values increase#
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@se a graphing device to graph
the curve
x= y6 7y/
"f we let the para!eter be t = y,
we have the e'uationsx= t6 7t/ y= t
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@sing those para!etric e'uations,
we obtain this curve#
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"t would be possible to solve the given
e'uation for y as four functions ofx and
graph the! individually#
+owever, the para!etric e'uations providea !uch easier !ethod#
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"n general, if we need to graph
an e'uation of the for!x= g(y),
we can use the para!etric e'uations
x = g(t) y = t
*RAP/IN* DEICES
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3otice also that curves with e'uations y= f(x)
(the ones we are !ost fa!iliar with$graphs
of functions) can also be regarded as curves
with para!etric e'uations
x = t y= f(t)
*RAP/IN* DEICES
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Araphing devices are particularly
useful when s.etching co!plicated
curves#
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2or instance,
these curves would
be virtually i!possible
to produce by hand#
COMPLE CURES
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%ne of the !ost i!portant uses of
para!etric curves is in co!puteraided
design (BC)#
CAD
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"n the *aboratory &ro4ect after Section 10#/,
we will investigate special para!etric curves
called DEFier curves#
hese are used e9tensively in !anufacturing,
especially in the auto!otive industry#
hey are also e!ployed in specifying the shapesof letters and other sy!bols in laser printers#
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C5CLOID
he curve traced out by a point Pon
the circu!ference of a circle as the circle rolls
along a straight line is called a cycloid#
E'!()e 6
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2ind para!etric e'uations for
the cycloid if:
he circle has radius rand rolls along thexa9is#
%ne position of Pis the origin#
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5e choose as para!eter the angle of rotation
of the circle (= 0 when Pis at the origin)#
Suppose the circle has rotated through radians#
E'!()e 6C5CLOIDS
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s the circle has been in contact with
the line, the distance it has rolled fro!
the origin is:
G OT G = arc PT= r
hus, the center of
the circle is C(r, r)#
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*et the coordinates of Pbe (x, y)#
hen, fro! the figure,
we see that: x= GOTG GPQG
= r rsin
= r( sin)
y = GTCG GQCG= r rcos
= r(1 cos )
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herefore, para!etric e'uations of
the cycloid are:
x= r( sin ) y= r(1 cos ) R
E. 7. 68E$u"i%n 1C5CLOIDS
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%ne arch of the cycloid co!es fro!
one rotation of the circle#
So, it is described by 0 /#
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-'uations 1 were derived fro! the figure,
which illustrates the case where 0 H H I/#
+owever, it can be seen
that the e'uations are
still valid for other values
of .
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"t is possible to eli!inate the para!eter
fro! -'uations 1#
+owever, the resulting Bartesian e'uationinxand yis:
ery co!plicated
3ot as convenient to wor. with
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%ne of the first people to study
the cycloid was Aalileo#
+e proposed that bridges be built in the shape#
+e tried to find the area under one arch of a cycloid#
C5CLOIDS
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*ater, this curve arose in connection with
the brachistochrone proble!$proposed by
the Swiss !athe!atician John Dernoulli
in 1KLK:
2ind the curve along which a particle will slide in
the shortest ti!e (under the influence of gravity) fro!
a pointAto a lower point Bnot directly beneathA#
2RAC/ISTOC/RONE PRO2LEM
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Dernoulli showed that, a!ong all possible
curves that 4oinAto B, the particle will ta.e
the least ti!e sliding fro!Ato B if the curve
is part of an inverted arch of a cycloid#
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he Cutch physicist +uygens had already
shown that the cycloid is also the solution tothe tautochrone proble!:
3o !atter where a particle is placed on an invertedcycloid, it ta.es the sa!e ti!e to slide to the botto!#
TAUTOC/RONE PRO2LEM
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+e proposed that pendulu! cloc.s
(which he invented) swing in cycloidal
arcs#
hen, the pendulu! ta.es the sa!e ti!e to !a.ea co!plete oscillation$whether it swings througha wide or a s!all arc#
C5CLOIDS & PENDULUMS
C C S E ) :
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PARAMETRIC CURE 9AMILIES
"nvestigate the fa!ily of curves with
para!etric e'uations
x= a cos t y= a tan t sin t
5hat do these curves have in co!!on
+ow does the shape change as a increases
E'!()e :
E ) :PARAMETRIC CURE 9AMILIES
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5e use a graphing device to produce
the graphs for the cases a=
/, 1, 0#
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PARAMETRIC CURE 9AMILIES
3otice that: ll the curves (e9cept for a= 0) have two branches# Doth branches approach the vertical asy!ptotex= a
asxapproaches afro! the left or right#
E'!()e :
E ) :LESS T/AN 1
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5hen aH 1, both branches are
s!ooth#
E'!()e :LESS T/AN ;1
E ) :REAC/ES 1
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+owever, when areaches 1, the right
branch ac'uires a sharp point, called
a cusp#
E'!()e :REAC/ES ;1
E ) :2ET
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2or abetween 1 and 0, the cusp turns
into a loop, which beco!es larger as aapproaches 0#
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5hen a= 0, both branches co!e
together and for! a circle#
E'!()e :EQUALS 0
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2or abetween 0 and 1, the left branch
has a loop#
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EQUALS 1
5hen a = 1, the loop shrin.s
to beco!e a cusp#
E'!()e :
E'!()e :*REATER T/AN 1
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2or aM 1, the branches beco!e s!ooth
again#
s aincreases further,
they beco!e lesscurved#
E'!()e :*REATER T/AN 1
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3otice that curves with apositive are
reflections about the ya9is of thecorresponding curves with anegative#
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CONC/OIDS O9 NICOMEDES E'!()e :
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+e called the! so because the shape
of their outer branches rese!bles that
of a conch shell or !ussel shell#
CONC/OIDS O9 NICOMEDES E'!()e :