# Polar Graphs

Function:

Some typical polar graphs (that can be made with this applet) include:

• Circles. There are a few ways of drawing a circle in polar coordinates. The simplest is the function $$r(\theta) = A$$ for some constant $$A$$. This will result in a circle with radius $$A$$ centered at the origin. Use any of the functions available in the applet and set $$B$$ to zero to see this. Alternately, the function $$r(\theta) = B\sin(\theta)$$ is a circle with center $$(\frac{B}{2},\frac{\pi}{2})$$ with radius $$B/2$$, and $$r(\theta) = B\cos(\theta)$$ is a circle with center $$(\frac{B}{2},0)$$ with radius $$B/2$$. The entire graph in any case can be seen as $$theta$$ ranges from 0 to $$2\pi$$.
• Limacons and Cardioids. The functions will have the form $$r(\theta) = A \pm B\sin(\theta)$$ or $$r(\theta) = A \pm B\cos(\theta)$$, with $$A$$ and $$B$$ nonzero. If $$|A/B|$$ < 1, the graph will appear heart shaped with an inner loop. If $$|A/B|$$ = 1, the graph is a cardioid. If $$|A/B|$$ is between 1 and 2, the graph appears as a dimpled circle. If $$|A/B|$$ > 2, the limacon is convex, and will appear almost circular. The entire graph in any case can be seen as $$theta$$ ranges from 0 to $$2\pi$$.
• Rose Curves. An equation of the form $$r(\theta) = B\sin(n\theta)$$ or $$r(\theta) = B\cos(n\theta)$$ will produce a flower shaped graph with $$n$$ petals. The entire graph in any case can be seen as $$\theta$$ ranges from 0 to $$2\pi$$.
• Archimedean Spirals. Polar functions of the form $$r(\theta) = A \pm B\theta$$ will produce spirals where the distance between turnings is constant. The value of $$A$$ adjusts the starting point of the spiral, while $$B$$ controls how tightly it coils.
• Logarithmic Spirals. Polar functions of the form $$r(\theta) = B e^{n\theta}$$ will also produce spirals, but the distance between turnings will increase in geometric procession. These are the famous curves that are approximated with surprising frequency in nature (galaxies, nautilus shells, hurricanes, etc). The golden spiral is a logarithmic spiral where $$|n|$$ is approximately 0.30635.

## Using the Applet

Choose a polar function from the drop down menu. Then use the sliders to adjust the values of A, B, and n. You can also restrict the plot to any subinterval of [0, 20π].