Curve of Parametric Equation

7/25/2019 Curve of Parametric Equation

1/15

Curve of Parametric Equation

The parametric equations of a plane curve are dened by x=et and y=t2-1 at

the range -2 t 2.

The folloing is the tabulation of t! x! and y.

t x y-2."" ".1# $.""

-1.%" ".1& 2.2#

-1.'" ".2" 1.('

-1.#" ".2( ".)'

-1.2" ".$" ".##

-1."" ".$& ".""

-".%" ".#( -".$'

-".'" ".(( -".'#

-".#" ".'& -".%#

-".2" ".%2 -".)'"."" 1."" -1.""

".2" 1.22 -".)'

".#" 1.#) -".%#

".'" 1.%2 -".'#

".%" 2.2$ -".$'

1."" 2.&2 ".""

1.2" $.$2 ".##

1.#" #."' ".)'

1.'" #.)( 1.('

1.%" '."( 2.2#2."" &.$) $.""


2/15

The folloing is the plotted plane curve

*oted that x-axis plotted ith the scale of 1." unit and y-axis plotted ith the

scale of ".( unit

" 1 2 $ # ( ' & %

-1.(

-1

-".(

"

".(

1

1.(

2

2.(

$

$.(

Curve of parametric equationx=et , y=t2-1 , -2 t 2


3/15

The position of two particles

+et x1 and y1represent the position of rst particle and x2and y2 represent

the position of second particle.

The position of rst particle at time tis x1=2 sin tand y1=$ cos thile theposition of second particle at time tis x2=cos t-2 and y2=1, sin there

both particles are move ithin " t 2

The folloing is the tabulation of x1 !y1! x2!y2 at certain t

t x1 y1 x2 2

"."" "."" $."" -1."" 1.""

".2" ".#" 2.)# -1."2 1.2"

".#" ".&% 2.&' -1."% 1.$)

".'" 1.1$ 2.#% -1.1& 1.('

".%" 1.#$ 2.") -1.$" 1.&2

1."" 1.'% 1.'2 -1.#' 1.%#

1.2" 1.%' 1.") -1.'# 1.)$

1.#" 1.)& ".(1 -1.%$ 1.))

1.'" 2."" -".1" -2."$ 2.""

1.%" 1.)( -".&" -2.2$ 1.)&

2."" 1.%2 -1.2" -2.#2 1.)1

2.2" 1.'2 -1.%" -2.() 1.%1

2.#" 1.$( -2.2" -2. 1.'%

2.'" 1."$ -2.'" -2.%' 1.(2

2.%" ".'& -2.%" -2.)# 1.$$

$."" ".2% -$."" -2.)) 1.1#

$.2" -".1" -$."" -$."" ".)#

$.#" -".(" -2.)" -2.)& ".

$.'" -".)" -2.&" -2.)" ".('

$.%" -1.2" -2.#" -2.&) ".$)

#."" -1.(" -2."" -2.'( ".2#

#.2" -1.&" -1.(" -2.#) ".1$

#.#" -1.)" -".)" -2.$1 "."(

#.'" -2."" -".$" -2.11 "."1

#.%" -2."" ".2' -1.)1 ".""

(."" -1.)" ".%( -1.&2 "."#

(.2" -1.%" 1.#1 -1.($ ".12

(.#" -1.(" 1.)" -1.$& ".2$

(.'" -1.$" 2.$$ -1.22 ".$&

(.%" -".)" 2.'' -1.11 ".(#

'."" -".'" 2.%% -1."# ".&2

'.2" -".2" 2.)) -1."" ".)2


4/15

'.#" ".2$ 2.)% -1."1 1.12

The folloing sho the paths of the particles on the same coordinate axes

according the tabulation above

The curve in blue is x1=2 sin tand y1=$ cos t and the curve in red is x2=cos

t-2 and y2=1, sin t.

*oted that x-axis plotted ith the scale of 1." unit and y-axis plotted ith thescale of 1." unit

-# -$ -2 -1 " 1 2 $

-#

-$

-2

-1

"

1

2

$

#

x1=2sint ! y1=$cost x2=cost-2 ! y2=1,sint

/ased on the graph above! noted that the paths of the to particles have

to intersection points hich are

0ntersection point 1 x=-2.""! y=".""

0ntersection point 2 x=-1.#'! y=1.%#


5/15

/ased on rst tabulation on above! the particles may collide beteen t=#.'

and t=#.% because the data is very close at that time. ence ! an

investigation ere done .

3fter investigation !notice that the particles ill collide at the point t=$42.

The folloing table sho the data of investigation so that the collide point

can be found

0n the range " t 2 here =224& or can be said is in the range " t

'.2%'

tin term of t x1 y1 x2 2

" ".""" "."" $."" -1."" 1.""

4# ".&%' 1.#1 2.12 -1.2) 1.&1

4$ 1."#% 1.&$ 1.(" -1.(" 1.%&

42 1.(&1 2."" -"."" -2."" 2.""

24$ 2.")( 1.&$ -1.(" -2.(" 1.%&

$4# 2.$(& 1.#1 -2.12 -2.&1 1.&1

$.1#$ -"."" -$."" -$."" 1.""

#4$ #.1)" -1.&$ -1.(" -2.(" ".1$

$42 #.&1# -2."" "."" -2."" ".""

2 '.2%' "."1 $."" -1."" 1.""

51is same ith 52 and 1 is same ith 2 at t=$42 .

ence particles collide at that time .

The folloing sho the detail of the complete tabulation data of both

particles

t x1 y1 x2 2

"."" "."" $."" -1."" 1.""

".2" ".#" 2.)# -1."2 1.2"

".#" ".&% 2.&' -1."% 1.$)

".'" 1.1$ 2.#% -1.1& 1.('

".%" 1.#$ 2.") -1.$" 1.&2

1."" 1.'% 1.'2 -1.#' 1.%#

1.2" 1.%' 1.") -1.'# 1.)$

1.#" 1.)& ".(1 -1.%$ 1.))

1.'" 2."" -".") -2."$ 2.""

1.%" 1.)( -".'% -2.2$ 1.)&

2."" 1.%2 -1.2( -2.#2 1.)1

2.2" 1.'2 -1.&& -2.() 1.%1


6/15

2.#" 1.$( -2.21 -2. 1.'%

2.'" 1."$ -2.(& -2.%' 1.(2

2.%" ".'& -2.%$ -2.)# 1.$$

$."" ".2% -2.)& -2.)) 1.1#

$.2" -".1" -2.)) -$."" ".)#

$.#" -".(" -2.)" -2.)& ".

$.'" -".)" -2.') -2.)" ".('

$.%" -1.2" -2.$& -2.&) ".$)

#."" -1.(" -1.)' -2.'( ".2#

#.2" -1.&" -1.#& -2.#) ".1$

#.#" -1.)" -".)2 -2.$1 "."(

#.'" -2."" -".$# -2.11 "."1

#.'1 -2."" -".$1 -2.1" "."1

#.'2 -2."" -".2% -2.") ".""

#.'$ -2."" -".2( -2."% ".""

#.'# -2."" -".22 -2."& ".""

#.'( -2."" -".1) -2."' ".""

#.'& -2."" -".1$ -2."# ".""#.'% -2."" -".1" -2."$ ".""

#.') -2."" -"."& -2."2 ".""

#.&" -2."" -"."# -2."1 ".""

#.&1 -2."" -"."1 -2."" ".""

#.&1#2%(&1# -2."" "."" -2."" ".""

#.&2 -2."" "."2 -1.)) ".""

#.&$ -2."" "."( -1.)% ".""

#. -2."" "."% -1.)& ".""

#.&( -2."" ".11 -1.)' ".""

#.&' -2."" ".1# -1.)( ".""#.&& -2."" ".1& -1.)# ".""

#.&% -2."" ".2" -1.)$ ".""

#.&) -2."" ".2$ -1.)2 ".""

#.%" -2."" ".2' -1.)1 ".""

(."" -1.)" ".%( -1.&2 "."#

(.2" -1.%" 1.#1 -1.($ ".12

(.#" -1.(" 1.)" -1.$& ".2$

(.'" -1.$" 2.$$ -1.22 ".$&

(.%" -".)" 2.'' -1.11 ".(#

'."" -".'" 2.%% -1."# ".&2'.2" -".2" 2.)) -1."" ".)2

'.#" ".2$ 2.)% -1."1 1.12

6rom the calculation ! it is shon that the particles collide at t=$42.

The particles collide at only one point ! 7 x ! y 8 hich is 7 -2."" ! "."" 8 .


7/15

The set of Parametric Equation

There are three sets of parametric equation for the curve hose equation is 7

y - 1 8

2

= x 9 #)The folloing are the sets of parametric equation

:et 1 ;ompare ith cartesian equation and standard parametric equation

7 y 9 1 82 = x 9 #) --------

/y substitute value ! h and a into parametric equation

5 = h , a t2

= , 2a t2

ence the rst set of parametric equation is

5 = #) , > t2 ! = 1 , ? t

:et 2 +et y become tand substitute into equation

+et y = t

7 y 9 1 82 = x 9 #)

7 t9 1 82 = x 9 #)

x = 7 t-1 82 , #)

x = #) , 7 t-1 82

ence the second set of parametric equation is

= t ! x = #) , 7 t-1 82


8/15

:et $ +et x become tand substitute into equation

+et x = t

7 y 9 1 82 = x 9 #)

7 y 9 1 82 = t9 #)

7 y 9 1 8=

+

t49

y = 1

+

t49

since t9 #) @ "

t@ #)

ence the third set of parametric equation is

5 = t , y = 1

+

t49here t@ #)

x= -t2is impossible to be chosen as the parametric equation for x in 7 y 9 1 82 = x 9 #)

This is because 7 y -1 82@ " for every values of y . 0f x = -t2! the equation ill

not be accepted .

Aroven

7 y 9 1 82 = x 9 #)

+et x = -t2

7 y 9 1 82 = -t29 #)

The left hand side of the equation is alays positive values hile the right

hand side of the equation is alays negative values.

ence 7 y 9 1 82 is not equal to -t29 #) . x is not equal to -t2.


9/15

x can be start ith x @ #) for the parametric equation .

y also can be start ith any choice for the parametric equation for y because

there is no restriction of value for y .

x = #) , 7y-1 8

2

! hence y can be any number.0f y = t! hence till be the rst choice of parametric equation .

BCTDED+DF

3pplication of GC5;C+H is used to setch a curve generated by a

pair of parametric equations. The simplest ay is to evaluate x and y for

several values of t. Then plot the points x and y in the plane and through

them dra a smooth curve .This idea of plotting points is identical to the

elementary graphing technique of graphing functions . The parametric

equations are x=et and y=t2-1 at the range -2 t 2 here the diIerence

beteen to t is ".2 . 3fter determined the corresponding values

ofxandy!then plotted and dran these points. :ame technique hich is

evaluate x1 !y1 !x2 and y2 for several values of tis used for the parametric

equation x1=2 sin tand y1=$ cos t !x2=cos t-2 and y2=1, sin t at the range

" t 2 here the diIerence beteen consecutivet is ".2.

;onverted the ;artesian equation to the parametric equations by

eliminate the parameter from the equations. This ill result in an equation

involving onlyxoryhich e may recogniJe. :ubstitute x ith tinto the

equation fory to get y = 1

+

t49. 3nd then substitute y ith t into the

equation for x to get x = #) , 7 t-1 82 . :tandard parametric equation hich

are 5 = h , a t2and = , 2a t2is also used to compare ith the

;artesian equation

7 y 9 82 = #a 7 x 9 h 8 . oever! it have to tae into account any restrictions

on the value of the parameter. 0n this case ! added the condition here t@

#) so that it ould not be square root of negative values .


10/15

3n interpretation of parametric equations is to thin of the

parameter as time ! in seconds and x1! y1and x2 !y2as functions that

describe the position of to particles moving in a plane. The parametric

equations x1=2 sin t!y1=$ cos t here " t 2 describe an obKect moving

around the unit circle. 0f t="! thin of the obKect as starting its Kourney at the

point 7"!$8 in the plane and if t=2 thin of the obKect ending its Kourney at

7"!$8 in the plane . 0n this case! the obKect is moving around the circle in a

clocise direction. 0t taes the obKect 2 seconds to travel around the circle.

The parametric equations x2=cos t-2 and y2=1, sin there " t 2

describe an obKect moving around the unit circle. 0f t="! thin of the obKect

as starting its Kourney at the point 7-1!18 in the plane and if t=2 thin of the

obKect ending its Kourney at 7-1!18 in the plane . Lhen to graph dran in

same plane! the time 7t8 here x1 = x2 and y1 = y2 indicate that to particlesill collide at that point.

0*TMDEN;T0D*

0n mathematics7T8 courseor! e need to study about the

application of the parametric equation .

Aarametric equations of a curve express the ;artesian

coordinates of the points of the curve as functions of a variable! called aparameter. Aarametric representations are generally non-unique! so the

same quantities may be expressed by a number of diIerent

parameteriJations. 3 single parameter is usually represented ith the

parameter ! hile the symbols and are commonly used for parametric

equations in to parameters.

6irst! tabulated the values of t! x and y based on the

parametric equation of a plane curve at certain range of t. Then plotted the

plane curve . /oth action are using GC5;C+H to get a clear and neat curve .

:econd ! converted from ;artesian equation to parametric

equation can be more than one method. 0n this research! converted

;artesian equation to parametric equation by comparing the standard


11/15

parametric equation ith the standard ;artesian equation. Then! substitute x

ith parameter for the equation of y and lastly substitute y ith parameter

for the equation of x .:o! it get three set of parametric equation in this

research.

Third! used graph to represent as the particles .:ince it haveto set of parametric equation! there are to particles. Tabulated the values

of t ! x1 ! y1 !x2and y2at certain range of t . Nsing the parametric equation to

construct the graph to nd out the number of intersection points beteen

the path ay of to particles .Nsing another table to determined the time

here 51is same ith 52 and 1 is same ith 2! this has shon the collision

point beteen to particles.

ence! shon that the number of intersection point is not sameas the collision point. The conversion from ;artesian equation to parametric

equation ill give more than one set of parametric equation.

;D*;+N:0D*

6rom the result! it is found that there is more than one set of

parametric equation hen converted from ;artesian equation into it.

oever! some of the parametric equations have some restriction on the

parameter 7t8. /esides! there are diIerence beteen intersection points and

collision point. The intersection point represent the overlapping of the path

ay of the moving particles .The collision point represent the point hich to

particle impact at one time. The number of the intersection point is not

necessary the same as the number of collision point. Aarticles ill collide atthe point here x1 = x2 and y1 = y2 .0n this case ! there are to intersection

points and only one collision point.


12/15

MC6CMC*;C

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Titles :TAB Text 7 Are-N 8 B3TCB3T0;: 7T8 9 :CB 1

3uthor +ee oon Loh ! Tan 3h Feo ! Tey Oim :oon

Aublisher Aenerbitan Aelangi :dn . /hd

6irst Aublished 2"1$

"nline


13/15

Lebsite https44en.iipedia.org4ii4ParametricPequation

+ast edited 1 :eptember 2"1(

Lebsite

http44tutorial.Bath.lamar.edu4;lasses4;alcll4AarametricCqn.aspx

0n text reference 2""$ 9 2"1( Aaul Eains

People

*ame +ie 3i +in

Dccupation Bathematics 7T8 teacher at :BO Eato G Aenggaa

Teacher ! Basai ! Qohor

Declaration

This is to certify that the assignment report submitted is based on

my own work.

Signature:

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ate : 2%.").2"1(
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15/15

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Curve of Parametric Equation