Polar Graphs

Notes (see the Notes page for further details of the various Notes collections).

Polar coordinates are simply a way of defining the position of a point in 2 dimensions. The distance from the origin (r) and the angle made with the x axis (measure in an anti-clockwise direction) define the position of the point. The point in the above diagram is a distance 3 from the origin and the angle made with the x axis is 30°.

Desmos - polar coordinates
Try this page on Desmos to experiment with plotting points. 

Try a join the dots exercisePolar Curves Join the dots
Polar join the dots

For some excellent resources on polar curves see these from the mathcentre and for some polar graph paper, scroll down the page on Mathsbits.

You can very easily experiment with families of polar curves using the excellent Desmos graphing calculator. Click on the image below and experiment with the sliders.

Desmos domain and sliders
Desmos polar curve, click on the image to experiment

It is possible to see how polar curves are traced out by using a slider in the domain on the Desmos graphing calculator. Try selecting the image to see r=acoskθ.
Further examples: r=acos2θ,   r=a(1-cosθ)  r=ae-kθ  r2=a2cos2θ

Alternatively, try this Polar Grapher; use the slider to change the angle and you will see how the curve is traced out. Note the value of R is displayed so you can easily see if it is positive or negative.
Polar Grapher

Polar & Cartesian PlotterFor a really clear plotter showing the connection between the Cartesian graph of r=f(θ) and the graph in polar coordinates try this Polar Curves and Cartesian Graphs applet. Watch the display carefully as you move the slider; you can easily see when r is negative for example.

See also The Polar Gallery from mathsdemos.org. If you scroll down the page you will see that you can download a set of 14 Excel files which will allow you to experiment with several families of polar curves.

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