Any explicit and implicit relation between r and theta will give a curve determined by the points whose co-ordinates satisfy that relation.
Thus the equations
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
is symmetrical about the initial line , for ;
It may be noted that ;
r=a represents a circle with its center at the pole and radius a
theta=b represents the line through the pole obtained by revolving the initial line through the angle b.