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*,*t*^{2}).

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

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(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):

- parametric plot (1/cosh(
*t*),*t*– tanh(*t*)) - parametric plot (cos(
*t*) – sin^2*t*/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!

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.