第十八單元

平面上之參數方程式

E- Learning

Parametric Equations

A plane curve is determined by a pair of parametric equations

( ) , ( )x f t y g t t I= = Î

t, Called the parameter, as measuring time

Example Eliminate the parameter in

2 2 , 3 2 3x t t y t t= + = - - £ £

3t y= +Q2( 3) 2( 3)x y y= + + +

2 8 15y y= + +

Solution

ExampleExample

Show 3cos , 2sin 0 2

represents a ellipse

x t y t t p= = £ £

Solution

cos , sin3 2

x yt t= =Q

2 2

13 2

x yæö æ ö÷ ÷ç ç\ + =÷ ÷ç ç÷ ÷ç çè ø è ø

ExampleFind the Cartesian equation of the curve

2 , 3 ; 0 3x t y t t= = £ £Solution :

3y t=Q2

2x

x tt= Þ =Q

23x

= ×

3

2y x\ =

Example

2 3

Find the Cartesian equation of the curve

, ; 1 2x t y t t= = - £ £Solution :

3y t=Q

2 1/ 2t xx t Þ ==Q

3/ 2x=

3y x\ =x

y

Differentiation for parametric curve

Theorem

Let and continuously differentiable with

( ) 0 on , then

f g

f t ta b¢ ¹ £ £

( ), ( )x f t y g t= =

/

/

dy dy dt

dx dx dt=

Example

1

Find , 0 withoutt

dyt

dx =

¹2 3,x t y t= =eliminate the parameter .

Solution :

dy

dx

dydtdxdt

æ ö÷ç ÷ç ÷çè ø=

æ ö÷ç ÷ç ÷çè ø

23

2

t

t=

3

2

t=

let 1t=dy

dx3

2

t=

1t=

3

2=

3

2

dy

dx=

2 3,t t< >

Area

y

dx

1

0

x

xA y dx=ò

1

0

( )( )

t

t

dx ty t dt

dt= ×ò

ExampleFind the area of the following region

Solution :

4

1A ydx=ò

3y t=Q2x t=Q 2dx t dtÞ =

3t=ò 2tdt×1

2

24

12 t dt= ò

25

1

12

5t= ×

5 522 1

5é ù= -ê úë û

62

5=

Some Famous Curves

Cycloid

{ ( sin )x r q q= -

(1 cos )y r q= -

www.chit.edu.tw/mathmet

Some Famous Curves

Lissajous

{ cosx t=sin 2y t=

Lissajous(1822 1880)-

Some Famous Curves

{ 3cosx t=3siny t=

Four Cusps

Some Famous Curves

{ 3

3

1

tx

t=

+2

3

3

1

ty

t=

+

The Folium of Descartes

> with(plots):

> g1:=animatecurve([3*t/(1+t^3),3*t^2/(1+t^3),t=0..40],numpoints=400,frames=40, scaling=unconstrained):g2:=animatecurve([3*t/(1+t^3),3*t^2/(1+t^3),t=-0.6..-0.1],color=black,numpoints=400,frames=40, > scaling=unconstrained):g3:=animatecurve([3*t/(1+t^3),3*t^2/(1+t^3),t=-100..-1.6],color=blue,numpoints=400,frames=40,scaling=unconstrained):display(g1,g2,g3);

: 1t -- ¥ ® -

: 1 0t +- ® 0:t ® ¥

(1596 1650)-

Swallowtail Catastrophe Curve

3

2 4

2 4

3

x ct t

y ct t

ìï = -ïíï =- +ïîfrom -10 to 10t

1000c=-

0c=

30c=

40c=

graph 60c=from 1000 to 1000c -

80c=

200c=

Ovals of Cassini

(Cassini)(1625 1712)-

4 2 2 4 42 cos 2 0r c r c aq- + - =

1c= 2c=3a=

Ovals of Cassini

4c=

3c=

2.5c=

2.8c=

3, from 1 to 5a c=

Witch of Agnesi

(1718 1799)-2 2 3Eq : ( )y x a a+ =

ìïïíïïî

x at=

21

ay

t=

+www.chit.edu.tw/mathmet

Newton’s Parabola

2 2( 2 )y x x ax b= - + 10b=

6a=- 0a= 2.5a= 3a=

4a= 5a= 6a= 8a=

Newton’s Parabola (animator)

2 2( 2 )y x x ax b= - +

10

from 8 to 8

b

a

=

-

單元結語

曲線以參數式來表示，在計算上及表示上更加明顯和方便。

還有許多曲線的例子，在此無法一一介紹，同學可進入本校網站www.chit.edu.tw/mathmet 進入函數圖形，那裡有更動畫圖形和微積分相關的分析。