Circumscribed polygons and circles

The figure is a unit circle, circumscribed by an equilateral triangle, circumscribed by another circle, circumscribed by a square, circumscribed by a circle, etc. (Interestingly enough, the radius of the circle sequence converges, so it makes for a pretty picture.) A Matlab script computes the circle radii and the polygons' vertices and creates a set of TikZ lines in a separate file for the rendering of the image. Then a separate tex file is used to include the created TikZ file.

The Matlab script src.m used to generate the figure is:

%  This Matlab script creates a set of LaTeX lines to generate a picture in
%  TikZ.  The picture is a unit circle, circumscribed by an equilateral
%  triangle, circumscribed by a circle, circumscribed by a square,
%  circumscribed by a circle, etc.  Here Matlab is used to generate the
%  sizes of the figures and the locations of the vertices of the polygons,
%  while TikZ is used to render a beautiful image.
fo = fopen('yl.tikz','w');
%  The initial circle's radius:
r1 = 1;
%  These commands are used to render a plot in Matlab, just to check that
%  things are working correctly.  Uncomment if desired:
%
% t = linspace(0,2*pi,100);
% x1 = r1*cos(t); y1 = r1*sin(t);
% plot(x1,y1,'linewidth',1.5);
% hold on
% axis square; axis equal;
fprintf(fo,'\\begin{tikzpicture}[scale=0.5]\n');
fprintf(fo,'\\draw (0,0) circle (1pt);\n');
fprintf(fo,'\\draw[blue] (0,0) circle (1);\n');
for n = 3:50
    %  Radius of the next circle:
    r2 = r1*sec(pi/n);
    %  Angle between successive vertices of the polygon:
    dtheta1 = 2*pi/n;
    %  (x,y) coordinates of the vertices of the polygon:
    px2 = r2*cos((0:n)*dtheta1);
    py2 = r2*sin((0:n)*dtheta1);
    %  Issue the plotting commands in TikZ:
    fprintf(fo,'\\draw[red] ');
    for j = 1:n
        fprintf(fo,'(%f,%f) -- ',px2(j),py2(j));
    end
    fprintf(fo,'cycle;\n');
    %  More Matlab plotting commands, if desired:
    %    x1 = r2*cos(t); y1 = r2*sin(t);
    %    plot(x1,y1,'linewidth',1.5);
    %  Draw the next circle, and then iterate to the next polygon:
    fprintf(fo,'\\draw[blue] (0,0) circle (%f);\n',r2);
    r1 = r2;
end
fprintf(fo,'\\end{tikzpicture}\n');
fclose(fo);

To compile the example you will need the files:

• yl.tikz
• src.m

Edit and compile if you like:

\documentclass{article}
%  Author:  Brian S. Marks
%  Title:  scr.tex
%  This latex file should be paired with the Matlab script scr.m, which 
%  creates the included yl.tex file.  The Matlab script creates a set of 
%  TikZ lines that create the figure of a circle circumscribed by a 
%  regular triangle, circumscribed by a circle, circumscribed by a 
%  square, etc.  

\usepackage{tikz}


\begin{document}

\begin{center}
	\input{yl.tikz}
\end{center}

The radius of the innermost circle is 1.  It is circumscribed by an 
equilateral triangle, which is circumscribed by a circle, and so forth.  
The radius of the circle as the number of figures approaches $\infty$ is 
\[
  \prod_{n=3}^\infty \sec\left(\frac{\pi}{n}\right) \approx 8.7
\]


\end{document}