PARAMETRIC EQUATIONS & POLAR COORDINATES

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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"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)]#

PARAMETRIC EQUATIONS


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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!#

POLAR COORDINATES


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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#

PARAMETRIC EQUATIONS


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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#

PARAMETRIC CURE


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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

E'!()e 1PARAMETRIC CURES


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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)#

PARAMETRIC CURES E'!()e 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#



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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



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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#

PARAMETRIC CURES


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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#

PARAMETRIC CURES


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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#

PARAMETRIC CURES


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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


x= sin /t y= cos /t 0 t /

E'!()e ,PARAMETRIC CURES


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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

PARAMETRIC CURES E'!()e ,


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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#

PARAMETRIC CURES E'!()e ,


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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

PARAMETRIC CURES


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2ind para!etric e'uations

for the circle with center (h, k)

and radius r.

E'!()e -PARAMETRIC CURES


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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#



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3ow, we shift hunits in thexdirection

and kunits in the ydirection#



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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

E'!()e PARAMETRIC CURES


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PARAMETRIC CURES

%bserve that y= (sin t)/ =x/#

hus, the point (x, y) !oves on

the parabola y=x/#

E'!()e


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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

E'!()e *RAP/IN* DEICES


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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#

*RAP/IN* DEICES


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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#

234IER CURES


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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#

E'!()e 6C5CLOIDS


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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)#

E'!()e 6C5CLOIDS


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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 )

E'!()e 6C5CLOIDS


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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 /#

E'!()e 6C5CLOIDS


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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 .

E'!()e 6C5CLOIDS


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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

E'!()e 6PARAMETRIC S. CARTESIAN


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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#

2RAC/ISTOC/RONE PRO2LEM


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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


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#

E'!()e :2ET


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5hen a= 0, both branches co!e

together and for! a circle#

E'!()e :EQUALS 0

E'!()e :2ET


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2or abetween 0 and 1, the left branch

has a loop#

E'!()e :2ET


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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

E'!()e :PARAMETRIC CURE 9AMILIES


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3otice that curves with apositive are

reflections about the ya9is of thecorresponding curves with anegative#

E'!()e :PARAMETRIC CURE 9AMILIES


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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 :

PARAMETRIC EQUATIONS & POLAR COORDINATES

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