TikZ — involute of a circle

Question

I need to combine an involute (https://en.wikipedia.org/wiki/Involute) of a circle and an archimedean spiral in one TikZ drawing, but have a problem drawing the involute.

I need something like this:

Here is my code:

\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}

\usepackage{fp}
\usetikzlibrary{fixedpointarithmetic}

\begin{tikzpicture}[x=1mm, y=1mm]

    \draw[] (0, 0) circle (7mm);

    \draw[rotate=-90, blue, domain=0:2*pi, variable=\a, samples=500]
            plot[fixed point arithmetic] ({\a r}:{7*sqrt(1+pow(\a, 2))});   

    \draw[rotate=-90, red, domain=0:2*pi, variable=\a, samples=500]
        plot[fixed point arithmetic] ({\a r}:{(7*\a)});
\end{tikzpicture}

And the output is:

Found my error, here is the code for involute and the circle:

    \def\r{7}

    % Circle
    \draw[line width=0.1mm, domain=0:2*pi, variable=\t, samples=500] 
        plot[fixed point arithmetic] ({\r*cos(deg(\t))}, {\r*sin(deg(\t))});

    % Involute
    \draw[blue, line width=0.1mm, domain=0:2*pi, variable=\t, samples=500] 
        plot[fixed point arithmetic] ({\r*(cos(deg(\t))+\t*sin(deg(\t)))}, {\r*(sin(deg(\t))-\t*cos(deg(\t)))});

    % Spiral
    \draw[red, rotate=-90, line width=0.1mm, domain=0:2*pi, variable=\t, samples=500]
        plot[fixed point arithmetic] ({\t r}:{(7*\t)});

And the fixed output is:

Answer 1

I would use PGFPlots for this:

\documentclass{article}    \usepackage{pgfplots}
\pgfplotsset{compat=1.12}

\begin{document}
\begin{tikzpicture}
\begin{axis}[
    trig format plots=rad,
    samples=300,
    axis equal image,
    xtick=\empty, ytick=\empty,
    axis lines=middle, enlargelimits=true
]
\addplot [
    no markers,
    black,
    thick,
    domain=0:6.4844*pi
] ( {cos(x) + x*sin(x)}, { sin(x) - x*cos(x)} );
\addplot [
    no markers,
    red,
    thick,
    domain=0:6*pi
] ( {x * cos(x)}, { x* sin(x) } );
\end{axis}
\end{tikzpicture}
\end{document}

Answer 2

Without PGFPlots :

\documentclass[tikz,border=7mm]{standalone}
\begin{document}
  \begin{tikzpicture}[scale=.2,samples=200,smooth,thick]
    \draw[blue,domain=0:3*360+87] plot ([rotate=\x]1,-rad \x);
    \draw[red,domain=0:3*360] plot (\x:rad \x);
  \end{tikzpicture}
\end{document}