Polar Graphs

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

• Just the Maths – Polar Curves
• The MathCentre

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°.

Try this page on Desmos to experiment with plotting points. 

Try a join the dots exercise! Polar Curves 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 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

Cardioids a+bsinθ       and a + bcosθ
When do you get a dimple?
When do you get an inner loop?

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.

For 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.