Polar Lemniscate Bernoulli in Latex

Question

I would like to plot a lemniscate Bernoulli (2D) in Latex and in polar coordinates, but I haven’t the appropriate code. Probably it would be easier to plot that in Octave or gnuplot, but I have a pressing time. Any help ?

Yes, but it is the case of three-dimensional space. I would like to plot curves on the plane according to equation: r^2 = a^2,cos(2,\phi). — Spyridon Kanavos, Dec 03 '21 at 09:43
Don't forget to accept and upvote brad's answer (checkmark and up-arrow to the left of his answer). — gernot, Jan 13 '22 at 15:29

brad · Answer 1 · 2022-01-13T15:23:38.343

3

Here is the code you need

\documentclass{standalone}
\usepackage{pgfplots}
\usepgfplotslibrary{polar}
\usepgfplotslibrary{fillbetween}
\pgfplotsset{compat=1.10}
\pgfdeclarelayer{bg}    
\pgfsetlayers{bg,main}
\begin{document}
\begin{tikzpicture}
  \begin{polaraxis}[style=black!10,grid style=black!10,ticklabel style=black!50,xticklabel=$\pgfmathprintnumber{\tick}^\circ$    ]
    \addplot [name path=A,thick, blue, domain=0:45, samples=50] {2sqrt(cos(2x))};
    \addplot [thick, red, domain=-45:0, samples=50] {2sqrt(cos(2x))};
    \addplot [thick, red, domain=135:225, samples=100] {2sqrt(cos(2x))};
    \draw [name path=B,blue,thick,opacity=.5,dotted] (0:0) -- (0:201);
    \tikzfillbetween[on layer=bg,of=A and B]{gray!50}
  \end{polaraxis}
\end{tikzpicture}
\end{document}

which provides

edited Jan 13 '22 at 15:23

answered Jan 11 '22 at 12:18

brad

517

The solution is superb. It works perfect. Thank you a lot. Now, I have due to symmetry to shade a quarter of the plot in order to calculate the whole lemniscate. The shading in Latex should be according to this information: https://tex.stackexchange.com/questions/407148/shading-a-part-of-a-plot – Spyridon Kanavos Jan 13 '22 at 08:47
Try changing \addplot [thick, red, domain=-45:45, samples=100] {2*sqrt(cos(2*x))}; into \addplot [thick, red, domain=-45:0, samples=50] {2*sqrt(cos(2*x))}; \addplot [thick, blue, domain=0:45, samples=50] {2*sqrt(cos(2*x))};
Does this suit your needs? – brad Jan 13 '22 at 11:35
Or do you mean that you want the inside area shaded? – brad Jan 13 '22 at 11:46
I will shade inside the quarter of the whole area of the lemniscate, which is at the first quadrant of the circle with boarders y = 0, x: [0, 2] and the upper line of the function's curve. – Spyridon Kanavos Jan 13 '22 at 12:19
I have edited the code to suit your needs – brad Jan 13 '22 at 15:24
Excellent. It works perfect! Thank you so much for your help. Your help is invaluable. I write books in Mathematics and until now I used other command-line and GUI programs that can generate two- and three-dimensional plots of functions, like gnuplot or CAS like Sage etc. Lastly, I use PS Tricks Latex, for direct insertion of figures in the text. Thanks to the scientific community of this website, I have solved many problems. – Spyridon Kanavos Jan 14 '22 at 08:24
@SpyridonKanavos You might be interested in https://tex.stackexchange.com/help/accepted-answer – samcarter_is_at_topanswers.xyz Oct 13 '22 at 10:24

Polar Lemniscate Bernoulli in Latex

1 Answers1