TikZ — Intersection line of a cone and a helicoid, too many lines

Question

Here is the image I'm working on: it's an archimedean spiral, shown as an orthogonal projection of an intersection line of a cone and a helicoid.

\documentclass[12pt, border=0.5mm]{standalone}
\usepackage{etex}
\usepackage[ngerman]{babel}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}

\begin{document}

\begin{tikzpicture}[x=1mm, y=1mm, z=1mm]
    \begin{axis}[
        axis equal image,
        view       = {22}{12},
        axis lines = none,
        xmax       = 50,
        xmin       = -50,
        ymax       = 50,
        ymin       = -50,
        zmax       = 100,
        zmin       = 0,             
        xtick      = \empty,
        ytick      = \empty,
        ztick      = \empty
    ]

    % Ground
    \addplot3+[
        ytick      = \empty,
        yticklabel = \empty,
        domain     = 0:12*pi,
        samples    = 1000,
        samples y  = 0,
        mark       = none,
        draw       = none,
        fill       = black!10   
    ]
    ({12*pi*sin(deg(x))}, {12*pi*cos(deg(x)}, {0});

    % Shadow
    \addplot3+[
        ytick      = \empty,
        yticklabel = \empty,
        domain     = 0:12*pi,
        samples    = 1000,
        samples y  = 0,
        mark       = none,
        line width = 0.2mm,
        line cap   = round,
        black!40
    ]
    ({x*sin(0.1*pi*deg(x))}, {x*cos(0.1*pi*deg(x)}, {0});

    % Intersection line
    \addplot3+[
        ytick      = \empty,
        yticklabel = \empty,
        domain     = 0:12*pi,
        samples    = 1000,
        samples y  = 0,
        mark       = none,
        line width = 0.2mm,
        line cap   = round,
        red
    ]
    ({x*sin(0.1*pi*deg(x))}, {x*cos(0.1*pi*deg(x)}, {x});

    % Base of a cone
    \addplot3+[
        ytick      = \empty,
        yticklabel = \empty,
        domain     = 0:12*pi,
        samples    = 1000,
        samples y  = 0,
        mark       = none,
        line width = 0.1mm,
        black
    ]
    ({12*pi*sin(deg(x))}, {12*pi*cos(deg(x)}, {12*pi});

    % Helicoid outer line
    \addplot3+[
        ytick      = \empty,
        yticklabel = \empty,
        domain     = 0:12*pi,
        samples    = 1000,
        samples y  = 0,
        mark       = none,
        line cap   = round,
        line width = 0.1mm,
        black
    ]
    ({12*pi*sin(0.1*pi*deg(x))}, {12*pi*cos(0.1*pi*deg(x)}, {x});

    % Helicoid plane
    \foreach \a in {0,1,...,48} {
        \edef\temp{\noexpand \draw[line width=0.05mm] ({(1/4)*\a*pi*sin(0.1*pi*deg(\a*pi/4))}, {(1/4)*\a*pi*cos(0.1*pi*deg(\a*pi/4)}, {\a*pi/4}) -- ({12*pi*sin(0.1*pi*deg(\a*pi/4))}, {12*pi*cos(0.1*pi*deg(\a*pi/4)}, {\a*pi/4});}
        \temp
    }

    %Cone walls
    \foreach \a in {0,1,...,13} {
        \edef\temp{\noexpand\draw [line cap=round, line width=0.05mm] (0, 0, 0) -- ({12*pi*cos(deg(\a*pi/7))}, {12*pi*sin(deg(\a*pi/7))}, {12*pi});}
        \temp
    }
\end{axis}
\end{tikzpicture}
\end{document}

The output looks like this:

Now I'm looking for a way to add the helicoid.

The problem is: there are too many lines in the output image and you are hardly able to recognize the single parts of an image (it should be black and white by the way).

Any ideas how to solve this problem?

Answer 1

How about changing the line below % Helicoid plane to

\foreach \a in {0,2,...,48} {

And for making everything black and white, change the nearest red below % Intersection line to black.

The result is