Mercyhurst UniversityMath DeptDr Williams Home

Polar Graphs

Function Options Function:
Interval: [ , ]

Graph Options Click and drag the graph to change the viewport
r = A + B sin(n θ)

Using the Applet

Choose a polar function family from the drop down menu. Then use the inputs to construct your graph. You can restrict the domain of your polar function. Use the slider to zoom in or out on the graph, and drag to reposition.

Polar Graphs

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

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.

About the Applet

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This applet was last updated February 2018.