Generating Polar and Parametric Plots in Wolfram|Alpha

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.

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.

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

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.

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:

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).

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

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):

• parametric plot (1/cosh(t), t – tanh(t))
• parametric plot (cos(t) – sin^2t/sqrt(2), cos(t) sin(t))
• parametric plot (1 – 3 t^2, t (3 – t^2))
• parametric plot (sin(8 t) (-csc(2 t)), -2 sin(3 t) sin(5 t) csc(2 t))

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!