Polar coordinates

We have considered various ways of describing curves in the $x-y$ plane: the graph of a function $y=f(x)$, implicitly via an equation involving $x$ and $y$, or through parametrisation. We can also use polar coordinates where curves can be described using equations involving $r$ and $\theta$. Sometimes, it is far easier to consider the polar coordinate over the Cartesian coordinate system. To convert from Cartesian to polar coordinates, we use the following

and from polar to Cartesian,

Examples of common graphs in the polar coordinate system are given below

Lines

1. $r \cos \theta=c$ gives the vertical line, $x=c$;
2. $r \sin \theta=c$ gives the horizontal line $y=c$;
3. $\theta=\beta$ is the line that goes through the origin with slope $\tan \beta$.

Circles

1. $r=c$ is a circle, centred at the origin with radius $c$;
2. $r=2 a \cos \theta+2 b \sin \theta$ is a circle of radius $\sqrt{a^{2}+b^{2}}$ centred at $(a, b)$.

Conic

Consider also the polar form of the equation of a conic describing the hyperbolas, the parabola, and the ellipse given as follows

where $l$ is the semi-latus rectum and $e$ is the eccentricity. The latter tells us how much the conic section described by the equation deviates from being circular. It follows that for $e=0$, the equation reduces to $r=l$ giving the equation of a circle with radius $l$, centred at the origin, as seen above. For $0<e<1$, the curve gives an ellipse, for $e=1$ we obtain a parabola and for $e>1$ a hyperbola. Note that for $e>1,(1+e \cos \theta)^{-1}$ is negative for some values of $\theta$. However, in polar coordinates $r$ is non-negative and therefore values of $\theta$ giving a negative $r$ should be excluded from a plot of the curve.