How can I draw a spiral that gets arbitrary close to a unit circle?

Question

I would like to draw something like this:

The circle in the center is not connected to the spiral, but the spiral gets arbitrary close to the circle.

A problem why I can't draw this is that I don't know how to describe this mathematically / formally.

I have found this one:

\documentclass[varwidth=true, border=2pt]{standalone}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
    \draw [domain=0:30,variable=\t,smooth,samples=100]
        plot ({\t r}: {0.005*\t*\t});
\end{tikzpicture}
\end{document}

but it seems not to get closer to a fixed size circle in the center.

This image looks similar to what I'm looking for, but I only need one spiral.

Background of my question

I'm currently studying geometry and topology. Sadly the professor does not provide a script, so I write one by my own to make studying easier for other students (see repository with source files and compiled pdf). The space that is described by the spiral and the circle is connected when you have a definition that makes use of $\varepsilon$-environments, but not connected when you make use of a definition that demands the existance of a path.

Answer 1

You need a function for the radius that approaches 1. For example, you could use 1+2*exp(-0.1*\t):

\documentclass[border=5pt]{standalone}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
    \draw [red] (0,0) circle [radius=1];
    \draw [domain=0:50,variable=\t,smooth,samples=500]
        plot ({\t r}: {1+2*exp(-0.1*\t)});
\end{tikzpicture}
\end{document}

Answer 2

My spiral starts from the correct point, does not it?

\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pst-plot}

\begin{document}
\begin{pspicture}(-5,-5)(5,5)
    \psplot[algebraic,polarplot,plotpoints=2000]{0}{TwoPi 8 mul}{1+4*3^(-0.1*x)}
\end{pspicture}
\end{document}

Animated version

\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pst-plot}

\begin{document}
\multido{\n=0.0+0.2}{41}{
\begin{pspicture}(-5,-5)(5,5)
    \psplot[algebraic,polarplot,plotpoints=2000]{0}{TwoPi \n\space mul}{1+4*3^(-0.1*x)}
\end{pspicture}}
\end{document}

With an attenuated sinusoidal multiplier.

\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pst-plot}

\begin{document}
\multido{\n=0.0+0.2}{41}{%
\begin{pspicture}(-5,-5)(5,5)
    \pscircle[linecolor=red,dimen=m]{1}
    \psplot[algebraic,polarplot,plotpoints=2000]{0}{TwoPi \n\space mul}{1+4*(1-sin(3*Pi*x)/8*2^(-x/10))*3^(-x/10)}
\end{pspicture}}
\end{document}

The last edit (I promise). The following might be difficult in TikZ.

\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pstricks-add}
\psset{unit=8}
\begin{document}
\multido{\n=-6.0+.2}{61}{%
\begin{pspicture}(-.5,-.5)(1.5,1.5)
\psplotDiffEqn[whichabs=0,whichord=1,linecolor=red,method=rk4,algebraic,plotpoints=1000]{-6}{\n}{0 0}{cos(Pi*x^2/2)|sin(Pi*x^2/2)}
\end{pspicture}}
\end{document}

Answer 3

This doesn't have, for health reasons, the animated effect... but it allows through another approach to more easily construct denser spirals (it is not essentially different though and belong to the same family of "exponential" spirals).

The number of complete turns done is on purpose a bit too much.

\documentclass[border=2pt,tikz]{standalone}
%\usepackage{tikz}
\usepackage{xintfrac}

% \ratio will get defined inside each tikzpicture

% The floating point macros of xintfrac are used, not the exact ones as we don't
% need the *exact* values with all digits!
\xintDigits := 4;
\def\Rescale #1#2{\xintTrunc {3}
                      {\xintFloatMul {\xintFloatPow {\ratio}{#1}}{#2}}}
% \Rescale multiplies its second argument #2 by \ratio to the power #1

\newcount\cnta

\begin{document}
\begin{tikzpicture}
   \cnta 0
   \def\ratio     {0.9}
   \def\quarters  {120}
   \def\couleur   {red}
   \loop
   \def\Quarter {(1+\Rescale{\cnta}{2},0) arc 
                     (0:90:1+\Rescale{\cnta}{2} and 1+\Rescale{\cnta+1}{2})}

   \draw [color=\couleur!\the\numexpr 100-100*\the\cnta/\quarters\relax] 
                    \Quarter; 

   \advance\cnta 1
   \draw [rotate=90,
          color=\couleur!\the\numexpr 100-100*\the\cnta/\quarters\relax] 
                    \Quarter;

   \advance\cnta 1
   \draw [rotate=180,
          color=\couleur!\the\numexpr 100-100*\the\cnta/\quarters\relax] 
                    \Quarter;

   \advance\cnta 1
   \draw [rotate=270,
          color=\couleur!\the\numexpr 100-100*\the\cnta/\quarters\relax] 
                    \Quarter;

   \advance\cnta 1
   \ifnum \cnta < \quarters 
   \repeat
   \draw [thick,color=\couleur] (0,0) circle (1);
\end{tikzpicture}

\begin{tikzpicture}
   \cnta 0
   \def\ratio     {0.95}
   \def\quarters  {180}
   \def\couleur   {blue}
   \loop
   \def\Quarter {(1+\Rescale{\cnta}{2},0) arc 
                     (0:90:1+\Rescale{\cnta}{2} and 1+\Rescale{\cnta+1}{2})}

   \draw [color=\couleur!\the\numexpr 100-100*\the\cnta/\quarters\relax] 
                    \Quarter; 

   \advance\cnta 1
   \draw [rotate=90,
          color=\couleur!\the\numexpr 100-100*\the\cnta/\quarters\relax] 
                    \Quarter;

   \advance\cnta 1
   \draw [rotate=180,
          color=\couleur!\the\numexpr 100-100*\the\cnta/\quarters\relax] 
                    \Quarter;

   \advance\cnta 1
   \draw [rotate=270,
          color=\couleur!\the\numexpr 100-100*\the\cnta/\quarters\relax] 
                    \Quarter;

   \advance\cnta 1
   \ifnum \cnta < \quarters 
   \repeat
   \draw [thick,color=\couleur] (0,0) circle (1);
\end{tikzpicture}

\begin{tikzpicture}
   \cnta 0
   \def\ratio     {0.97}
   \def\quarters  {240}
   \def\couleur   {green}
   \loop
   \def\Quarter {(1+\Rescale{\cnta}{2},0) arc 
                     (0:90:1+\Rescale{\cnta}{2} and 1+\Rescale{\cnta+1}{2})}

   \draw [color=\couleur!\the\numexpr 100-100*\cnta/\quarters\relax]  
                    \Quarter; 

   \advance\cnta 1
   \draw [rotate=90,
          color=\couleur!\the\numexpr 100-100*\the\cnta/\quarters\relax]  
                    \Quarter;

   \advance\cnta 1
   \draw [rotate=180,
          color=\couleur!\the\numexpr 100-100*\the\cnta/\quarters\relax]
                    \Quarter;

   \advance\cnta 1
   \draw [rotate=270,
          color=\couleur!\the\numexpr 100-100*\the\cnta/\quarters\relax] 
                    \Quarter;

   \advance\cnta 1
   \ifnum \cnta < \quarters 
   \repeat
   \draw [thick,color=\couleur] (0,0) circle (1);
\end{tikzpicture}

\end{document}