Generating Polar and Parametric Plots in Wolfram|Alpha

August 3, 2011
Sam Blake
Posted by

In my last blog post on plotting functionality in Wolfram|Alpha, we looked at 2D and 3D Cartesian plotting. In this post, we will look at 2D polar and parametric plotting.

For those of you unfamiliar with polar plots, a point on a plane in polar coordinates is located by determining an angle θ and a radius r. For example, the Cartesian point (x, y) = (1, 1) has the polar coordinates (r, θ) = (√2,π/4). The following diagram illustrates the relationship between Cartesian and polar plots.

Relationship between Cartesian and polar plots

To generate a polar plot, we need to specify a function that, given an angle θ, returns a radius r that is a function r(θ). Making a polar plot in Wolfram|Alpha is very easy; for example, we can plot Archimedes’ spiral.

plot r = theta

Or we can get a little fancier and plot a polar rose with eight petals.

polar plot 2cos(4*θ)

Want to know how to graph this in Mathematica? We can easily extract the Mathematica code for this plot right from Wolfram|Alpha. By clicking the dog-ear in the bottom left of the images and then “Copyable plaintext”, you can see the Mathematica code used to generate the plots.

PolarPlot[2cos4θ],{θ, -pi, pi}]

Polar rose with 8 petals

Here are some other awesome polar plots:

The plot of a lemniscate:


And a lituus:


And a cochleoid:


Wolfram|Alpha can also handle more complicated inputs, like r(θ) = exp(cos(θ) – 2 cos(4θ) + sin (θ/12)^5:

plot r(θ) = exp(cos(θ)) - 2cos(4θ) + sin(θ/12)^5

Now that you have seen some great examples of polar plots, let’s move on to parametric plots.

What is the difference between a polar and parametric plot? Parametric coordinates specify points (x,y) in 2D with two functions, (x,y) = (f(t), g(t)) for a parameter t. Here are some examples of 2D parametric plots to try in Wolfram|Alpha.

Try to make a parametric plot of (x(t), y(t)) = (1-t,t2).

parametric plot (1-t, t^2)

We can easily see that this is the same as the Cartesian equation y = 1 – 2x + x2. (x(t) = 1 – tt = 1 – x(t) so y(x(t)) = (1 – x)2 = 1 – 2x + x2)

In the above example, we didn’t even need to enter a plot range; Wolfram|Alpha picked the plot range that best suits the graph. Of course, it’s possible to specify a range for the parameter, in this case we plot (x(t), y(t)) = (sin(t), sin(3t)) for t from 0 to 100.

parametric plot (sint, sin(sqrt(3)t)) for t from 0 to 100

Now let’s look at some other cool plots that Wolfram|Alpha can create.

How about the parametric plot of the astroid:


Or a similar plot, the deltoid:


If we had just said “plot” instead of “parametric plot”, then Wolfram|Alpha would have returned a Cartesian plot of 4cos(φ) + 2cos(2φ) and 4sin(φ) + 2sin(2φ), as well as a parametric plot of the deltoid.

Want some more examples? Check out these classic examples of parametric plots (the tractrix, fish curve, Tschirnhausen cubic, and Plateau curves, respectively):

In the previous plotting post, you had the opportunity to learn about 2D Cartesian plotting in Wolfram|Alpha, and now you are equipped with the ability to make 2D polar and parametric plots as well. Luckily, Wolfram|Alpha doesn’t stop there; start playing with some 3D graphs and it will not let you down!

1 Comment

Hello, can you tell me how to make plot with 2 function in it? so that I can see the intersection more clearly. Thank you very much.

Posted by Faldi Rianda May 15, 2012 at 9:32 pm Reply
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