Saturday, 25 January 2014

Polar Equation of Curves

 


Any explicit or implicit relation between    r    and    theta      will give a curve determined by the points whose co-ordinates satisfies that relation .

Thus the equations ;

r=f(theta)   or   F(r , theta)=0

determine curves .

The co-ordinates of two points symmetrically situated about the initial line are of the form   (r,Theta)    and    (r,-theta)   so that their vertical angles differ in sign only .

Hence a curve will be symmetrical about the initial line if on changing    theta    to     -theta    its equation does not change . For instance the curve

                    r=a[1+cos(theta)]   

is symmetrical about the initial line , for

        r=a[1+cos(theta)]=a[1+cos(-theta)]

It may be noted that

r=a    represents a circle with its center at pole and radius   a ;  and

theta=b    represents a line through the pole obtained by revolving the initial line through the angle   b   .

A few important curves will not be traced . To trace polar curves , we generally consider the variations in     r    as    theta    varies .


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