# How can I plot a Chebyshev spiral?

The Chebyshev polynomials T_n of the first kind are a certain set of orthogonal polynomials. They can be defined by T_n(cos(x))=cos(nx), the first of them are

• T_0(x) = 1
• T_1(x) = x
• T_2(x) = 2x^2 - 1
• T_3(x) = 4x^3 - 3x
• T_4(x) = 8x^4 - 8x^2 + 1
• T_5(x) = 16x^5 - 20x^3 + 5x
• T_6(x) = 32x^6 - 48x^4 + 18x^2 - 1
• T_7(x) = 64x^7 − 112x^5 + 56x^3 − 7x

A plot in the cartesian plane looks like this, seen on Wolfram Mathworld: On the same page, I found a beautiful polar plot of those polynomials: There they wrote, referring to "Graphica 1" by Michael Trott:

A beautiful plot can be obtained by plotting T_n(x) radially, increasing the radius for each value of n, and filling in the areas between the curves (Trott 1999, pp. 10 and 84).

A version without filling, posted by Paul Kemper: I would like to understand the way that plot is generated. The filling doesn't matter to me, I just would like to know how I could plot such a polynomial radially. As the domain ranges from -1 to 1, I should take pi as a factor. Does anybody know a formula to display those functions in polar coordinates in such a way with increasing radius?

– eldo
Jul 18, 2014 at 11:52
• PolarPlot[Evaluate@Table[n+ChebyshevT[n,t/Pi-1],{n,0,20,2}],{t,0,2Pi}] Jul 18, 2014 at 12:01
• @eldo Thank you! I did not see this link. Jul 18, 2014 at 12:14
• @SimonWoods Thank you very much! That's all I needed as help. If you like, post as an answer with some short words to let future readers easily understand, I'd gladly upvote and accept it. Jul 18, 2014 at 12:16

You can use PolarPlot to plot the curves. As noted in the question you need to map the polar angle onto the -1 to 1 domain of the polynomials. You should also note that only the even polynomials are plotted.

PolarPlot[Evaluate @ Table[n + ChebyshevT[n, t/Pi - 1], {n, 0, 40, 2}], {t, 0, 2 Pi}] To get the filling effect you can used FilledCurve:

Graphics @ FilledCurve @ Cases[%, _Line, -1] • this so nice and the code so terse...FilledCurve Jul 18, 2014 at 12:51