Graphs of equations in polar coordinates




Example 1: A circle of radius 1


Equation of the graph: \begin{equation} \,\,\,\,\, r(\theta) = 1, \,\,\,\,\, 0\leq \theta \leq 2 \pi.
\end{equation}
This is the the graph of the equation $r(\theta)$


      
The variable $\theta$







Example 2: A circle of radius 1/2 centered at (1/2,0)

\begin{equation} r(\theta) = cos\, \theta, \,\,\,\,\, 0\leq \theta \leq \pi \end{equation}
The graph of $r(\theta) = cos (\theta)$




       The graph of $cos (\theta), \,\,\,\,\, 0\leq \theta \leq \,\pi $






Example 3: A flower with four petals.

\begin{equation} r(\theta) = cos\, (2 \,\theta), \,\,\,\,\, 0\leq \theta \leq 2\,\pi \end{equation}
The graph of $r(\theta) = cos (2 \theta)$



       The graph of $cos ( \theta),\,\,\,\,\,0\leq \theta \leq 2\,\pi $






Example 4: A flower with three petals.

\begin{equation} r(\theta) = cos\,( 3 \,\theta), \,\,\,\,\, 0\leq \theta \leq \,\pi \end{equation}
The graph of $r(\theta) = cos (3 \theta)$



       The graph of $cos ( \theta),\,\,\,\,\,0\leq \theta \leq \,\pi $






Example 5: An carcoid

\begin{equation} r(\theta) = 1- cos( \theta), \,\,\,\,\, 0\leq \theta \leq \, 2\,\pi \end{equation}
The graph of $r(\theta) = 1- cos (\theta)$



       The graph of $1-cos ( \theta),\,\,\,\,\,0\leq \theta \leq \,2\pi $






Example 6: Another carcoid

\begin{equation} r(\theta) = 1+ cos( \theta), \,\,\,\,\, 0\leq \theta \leq \, 2\,\pi \end{equation}
The graph of $r(\theta) = 1+ cos (\theta)$



       The graph of $1+cos ( \theta),\,\,\,\,\,0\leq \theta \leq \,2\pi $