Understanding Engineering Mathematics

7.2.8 Parametric representation of curves

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205 222➤

Sometimes, instead of writing an equation in the formy= f(x),i.e.intermsofx,y,itis
convenient to introduce someparametertin terms of whichxandyare jointly expressed
in the form

x=x(t) y=y(t)

For example the position of a projectile at timetmay be expressed as

x=Vtcosα=x(t)

y=Vtsinα−^12 gt^2 =y(t)

whereV,αare the initial projection velocity and angle respectively.
In principle, we can always return to thex,yform by eliminatingt. In the projectile
for example we have

t=

x

Vcosα

from the first equation, then substitution in the second gives the parabola:

y=V

x

Vcosα

sinα−

1

2

g

( x

Vcosα

) 2

=xtanα−

1

2

gx^2

V^2

sec^2 α

Solution to review question 7.1.8

(i) From the equationsx= 3 t+1,y=t+2wehavet=y−2, so

x= 3 t+ 1 = 3 (y− 2 )+ 1 = 3 y− 5

The Cartesian equation is thus

x− 3 y+ 5 = 0

which is a straight line.

(ii)x=6cos2t;y=6sin2t

This is a good example of the need to have the elementary trig iden-

tities at your fingertips. Otherwise, it might not immediately occur to

you to use the fact that cos^2 θ+sin^2 θ=1 (185

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) to eliminatet,

by squaring and adding

x^2 +y^2 =36 cos^22 t+36 sin^22 t= 36

which is a circle centre the origin and radius 6. This parametric form

of the circle is very useful in the theory of oscillating systems.

(iii) In general, the parametric equations

x=rcosθ+a, y=rsinθ+b