TikZ — joining a plot and a line segment

Question

Is there a way to join a plot and a line segment in TikZ?

Here is a shape I'm trying to draw:

I almost happy with the result, but there is a gap between the plotted segment and the rest of the drawing:

\documentclass[12pt, border=0.5mm]{standalone}
\usepackage[ngerman]{babel}
\usepackage[T1]{fontenc}
\usepackage[sc]{mathpazo}    
\usepackage{tikz}
\usepackage{pgfplots}

\begin{document}    
\begin{tikzpicture}[x=1mm, y=1mm]
    \def\r{10.5}
    \def\w{18}  

    \coordinate (a) at ({\r*cos(180-\w)}, {\r*sin(180-\w)});
    \coordinate (b) at ({\r*cos(\w)}, {\r*sin(\w)});

    \draw[domain=\w:{180-\w}, variable=\x, samples=250] 
        (a) to[out=270, in=180] 
        (-8, 1.25) --
        (-8, 0.75) -- 
        (-1, 0.75) -- 
        (-1, 8) to[out=20, in=250] 
        (0, 9) to[out=290, in=160] 
        (1, 8) -- 
        (1, 0.75) -- 
        (8, 0.75) -- 
        (8, 1.25) to[out=0, in=270] 
        (b) plot ({\r*cos(\x)}, {\r*sin(\x)}) -- (a);
\end{tikzpicture}
\end{document}

The percusse's solution works well:

\documentclass[12pt, border=0.5mm]{standalone}
\usepackage[ngerman]{babel}
\usepackage[T1]{fontenc}
\usepackage[sc]{mathpazo}    
\usepackage{tikz}
\usepackage{pgfplots}

\begin{document}    
\begin{tikzpicture}[x=1mm, y=1mm]
    \def\r{10.5}
    \def\w{18}  

    \coordinate (b) at ({\r*cos(180-\w)}, {\r*sin(180-\w)});
    \coordinate (a) at ({\r*cos(\w)}, {\r*sin(\w)});
    \coordinate (c) at ({\r*cos(-\w)}, {\r*sin(-\w)});
    \coordinate (d) at ({\r*cos(-180+\w)}, {\r*sin(-180+\w)});

    \pgfmathsetmacro\t{90-\w}
    \pgfmathsetmacro\z{90+\w}

    \draw[line width=0.2mm, domain={180-\w}:\w, variable=\x, samples=250]  plot ({\r*cos(\x)}, {\r*sin(\x)}) -- 
        (a) to[out=-\t, in=0] 
        (8, 1.25) --
        (8, 0.75) -- 
        (1, 0.75) -- 
        (1, 8) to[out=160, in=290] 
        (0, 9) to[out=250, in=20]
        (-1, 8) --
        (-1, 0.75) -- 
        (-8, 0.75) -- 
        (-8, 1.25) to[out=180, in=-\z] 
        (b) -- cycle;
\end{tikzpicture}
\end{document}

I've adjusted the angles and plotted the round segment first. There is no gap even at 6400%:

If you wonder, why do I need a shape like this, it's a part of the Artobolevskii's tool for drawing of logarithmic spirals:

Answer 1

If --cycle does not work for you, one option would be to adjust the out and in angle so that the lines are at the same angle as what they are connecting to. This slightly changes the shape, so if that is not required to be that precise you can get (at 6400% zoom on the left hand side):

Notes:

• On the left hand side I changed 270 to 250, but on the right hand side the required change was from 270 to 288. Not sure why the right hand side required a slightly different angle delta. For more precision you could compute the actual angle, but trial and error works pretty well.

Code:

\documentclass[12pt, border=0.5mm]{standalone}
\usepackage[ngerman]{babel}
\usepackage[T1]{fontenc}
\usepackage[sc]{mathpazo}    
\usepackage{tikz}
\usepackage{pgfplots}

\begin{document}    
\begin{tikzpicture}[x=1mm, y=1mm, join=cap]
    \def\r{10.5}
    \def\w{18}  

    \coordinate (a) at ({\r*cos(180-\w)}, {\r*sin(180-\w)});
    \coordinate (b) at ({\r*cos(\w)}, {\r*sin(\w)});

    \draw[domain=\w:{180-\w}, variable=\x, samples=250] 
        (a) to[out=250, in=180] %% <--- Adjusted out angle here
        (-8, 1.25) --
        (-8, 0.75) -- 
        (-1, 0.75) -- 
        (-1, 8) to[out=20, in=250] 
        (0, 9) to[out=290, in=160] 
        (1, 8) -- 
        (1, 0.75) -- 
        (8, 0.75) -- 
        (8, 1.25) to[out=0, in=288] <--- Adjusted in angle here
        (b) plot ({\r*cos(\x)}, {\r*sin(\x)}) -- (a)
        ;
\end{tikzpicture}
\end{document}