Important Graphs of Polar Equations

Before we move to specific graphs, we first talk about a important property that is symmetry.

For equation like ${\displaystyle r=R(\theta )}$ in polar coordinates, we can use listed diagrams to test it and find its symmetry:

• If ${\displaystyle R(\theta )=R(-\theta )}$ or ${\displaystyle R(\theta )=-R(\pi -\theta )}$, we say it is symmetric about the x-axis.
• If ${\displaystyle R(\theta )=-R(-\theta )}$ or ${\displaystyle R(\theta )=R(\pi -\theta )}$, we say it is symmetric about the y-axis.
• If ${\displaystyle R(\theta )=-R(\theta )}$ or ${\displaystyle R(\theta )=R(\pi +\theta )}$, we say it is symmetric about the pole.

The three properties above help a lot while testing equations for their symmetry.

Lines

For lines across the pole, we have ${\displaystyle \theta ={\theta }_0}$.
For generalization, we have:

${\displaystyle r=\frac{d}{\cos (\theta -{\theta }_0)}}$

This is derived from the geometric relationship and trig functions.

Circles

We only talk about circles that go through the pole, since the equations of circles at general position ${\displaystyle (r_0,{\theta }_0)}$ would be quite complex.

${\displaystyle r=2a\cos (\theta -{\theta }_0)}$

And for circles centered at the pole, we have ${\displaystyle r=a}$.

Conics

We different ${\displaystyle e}$ to differ from ellipse, parabola and hyperbola.

${\displaystyle r=\frac{ed}{1+e\cos (\theta -{\theta }_0)}}$

This is the common equation for conics.

Cardioids

For equations in the form:

• ${\displaystyle r=a\pm b\cos \theta }$
• ${\displaystyle r=a\pm b\sin \theta }$ With ${\displaystyle a}$ and ${\displaystyle b}$ positive, we say they are cardioids.

Lemniscates

For equations in the form:

• ${\displaystyle r^2=\pm a\cos 2\theta }$
• ${\displaystyle r^2=\pm a\sin 2\theta }$
  They are figure-eight-shaped curves.

Spirals

Equations in the form ${\displaystyle r=a\theta }$ are spirals. The logarithmic spiral is the spiral in the form ${\displaystyle r=a{e}^{b\theta }}$.

Roses

Equations in the form:

• ${\displaystyle r=a\cos n\theta }$
• ${\displaystyle r=a\sin n\theta }$
  Are called roses. The graph have ${\displaystyle n}$ leaves if ${\displaystyle n}$ is odd, and ${\displaystyle 2n}$ leaves if ${\displaystyle n}$ is even.