Conical and cylindircal spirals

Question

I want to draw something like the following in TikZ, but, unfortunately, I'm not sure how to get to the needed result. The figure shows the path of ions in a quadrupole mass spectrometer. Outside the quadrupole (those 4 rods) no electromagnetic field applies to the ions and thus they fly in a straight line. If they enter the quadrupole they can either come into resonance with the electromagnetic field and thus be on a cylindrical spiral path or not be in resonance and thus be on a conical spiral path and sooner or later exit the quadrupole at a side.

My take to this problem was to use pgfplots to draw the spirals using a 3D plot with the function {x*cos(deg(x))},{x*sin(deg(x)},{x} for the conical plot and {cos(deg(x))},{sin(deg(x)},{x} for the cylindrical one. Unfortunately, I find myself unable to solve the following issues:

• correctly position the spirals
• draw a straight line that transforms into a spiral and then back to a straight line after exiting the quadrupole (only for the cylindrical one)
• stop the conical helix shortly after the path has exited the quadrupole

I'm well aware that this is quite a lot of issues and thus I'm happy for any hints.

My current (miserable) attempt

\documentclass{standalone}
\usepackage{xparse}
\usepackage{ifthen}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.8}
\usetikzlibrary{calc}
\usetikzlibrary{decorations.markings}
\begin{document}
\begin{tikzpicture}
    % General constants
    % %%%%%%%%%%%%%%%%%
\coordinate (msOrigin) at (0,0);
\pgfmathsetmacro{\msY}{3}

\pgfmathsetmacro{\offsetX}{0.3}
\pgfmathsetmacro{\offsetY}{0.2}
\pgfmathsetmacro{\spacer}{0.75}
\pgfmathsetmacro{\arrowLength}{1}
\pgfmathsetmacro{\centerOffset}{0.3}


% Quadrupole constants
% %%%%%%%%%%%%%%%%%%%%

\pgfmathsetmacro{\quadrupoleRadiusHorizontal}{0.08}
\pgfmathsetmacro{\quadrupoleRadiusVertical}{0.2}
\pgfmathsetmacro{\quadrupoleLength}{3}
\pgfmathsetmacro{\quadrupolePathLength}{\quadrupoleLength - (2 * \quadrupoleRadiusHorizontal)}

\pgfmathsetmacro{\quadrupoleTopFrontY}{0.5 * \msY + \centerOffset + 2 * \quadrupoleRadiusVertical}
\pgfmathsetmacro{\quadrupoleTopBackY}{\quadrupoleTopFrontY + \offsetY}
\pgfmathsetmacro{\quadrupoleBottomBackY}{0.5 * \msY - \centerOffset}
\pgfmathsetmacro{\quadrupoleBottomFrontY}{\quadrupoleBottomBackY - \offsetY}

\NewDocumentCommand{\cylinder}{m m m m m m m m}{%  coordX, coordY, length, radiusX, radiusY, colorCylinder, colorEllipse, opacity
    \fill [#6, fill opacity = #8]
        ($ (msOrigin) + ({#1},{#2}) $)
        --
        ++({#3},0)
        arc
        (90:270:-{#4} and {#5})
        --
        ++(-{#3},0)
        arc
        (270:90:-{#4} and {#5});

    \draw [fill = #7, fill opacity = #8]
        ($ (msOrigin) + ({#1},{#2}) + (0,{-#5}) $)
        ellipse
        ({#4} and {#5});

    \draw
        ($ (msOrigin) + ({#1},{#2}) $)
        --
        ++({#3},0)
        arc
        (90:270:-{#4} and {#5})
        --
        ++(-{#3},0);
}

\NewDocumentCommand{\quadrupoleRod}{m m m}{% segment, top/bottom, front/back
    \ifthenelse{\equal{#2}{top} \AND \equal{#3}{front}}{%
        \pgfmathsetmacro{\coordX}{\quadrupoleRadiusHorizontal + \offsetX}
        \pgfmathsetmacro{\coordY}{\quadrupoleTopFrontY}
    }{}

    \ifthenelse{\equal{#2}{top} \AND \equal{#3}{back}}{%
        \pgfmathsetmacro{\coordX}{\quadrupoleRadiusHorizontal}
        \pgfmathsetmacro{\coordY}{\quadrupoleTopBackY}
    }{}

    \ifthenelse{\equal{#2}{bottom} \AND \equal{#3}{front}}{%
        \pgfmathsetmacro{\coordX}{\quadrupoleRadiusHorizontal + \offsetX}
        \pgfmathsetmacro{\coordY}{\quadrupoleBottomFrontY}
    }{}

    \ifthenelse{\equal{#2}{bottom} \AND \equal{#3}{back}}{%
        \pgfmathsetmacro{\coordX}{\quadrupoleRadiusHorizontal}
        \pgfmathsetmacro{\coordY}{\quadrupoleBottomBackY}
    }{}

    \cylinder
        {\coordX}
        {\coordY}
        {\quadrupolePathLength}
        {\quadrupoleRadiusHorizontal}
        {\quadrupoleRadiusVertical}
        {gray}
        {white}
        {1}
}

\NewDocumentCommand{\quadrupolePair}{m m}{% segment, front/back
    \ifthenelse{\equal{#2}{front} \OR \equal{#2}{back}}{%
        \quadrupoleRod{#1}{top}{#2}
        \quadrupoleRod{#1}{bottom}{#2}
    }{}
}

\quadrupolePair{1}{back}
\begin{axis}[
    rotate around={-90:(current axis.origin)},
    view = {30}{20},
    axis line style = {draw = none},
    tick style = {draw = none},
    zmax = 60,
    xtick=\empty,
    ytick=\empty,
    ztick=\empty
]
    \addplot3+[
        mark = none,
        thick,
        red,
        domain = 0:50*pi,
        samples = 1000,
        samples y = 0,
    ]
    % ({x*cos(deg(x))},{x*sin(deg(x)},{x});
    ({cos(deg(x))},{sin(deg(x)},{x});
\end{axis}
\quadrupolePair{1}{front}

\end{tikzpicture}
\end{document}

Update 2020-11-26

I found this answer on TeX.SX helping to draw the cylindrical coil. By some modifications, I was able to get relatively far in the process. One remaining issue is the line connecting the horizontal path with the spiral as the code mark=at position #1 with \coordinate (#2); throws a Dimension too large. error, even if I don't understand why. The coils are small and definitely below 19 feet...

Another issue that remains is the conical spiral. I have a starting point, but unfortunately, it looks gross.

\documentclass{standalone}
\usepackage{xparse}
\usepackage{ifthen}
\usepackage{tikz}
\usetikzlibrary{calc}
\usetikzlibrary{decorations.markings}
\tikzset{
    mark position/.style args={#1(#2)}{
        postaction={
            decorate,
            decoration={
                markings,
                mark=at position #1 with \coordinate (#2);
            }
        }
    }
}
\NewDocumentCommand{\cylinder}{m m m m m m m m}{%  coordX, coordY, length, radiusX, radiusY, colorCylinder, colorEllipse, opacity
    \fill [#6, fill opacity = #8]
        ($ (msOrigin) + ({#1},{#2}) $)
        --
        ++({#3},0)
        arc
        (90:270:-{#4} and {#5})
        --
        ++(-{#3},0)
        arc
        (270:90:-{#4} and {#5});
\draw [fill = #7, fill opacity = #8]
    ($ (msOrigin) + ({#1},{#2}) + (0,{-#5}) $)
    ellipse
    ({#4} and {#5});

\draw
    ($ (msOrigin) + ({#1},{#2}) $)
    --
    ++({#3},0)
    arc
    (90:270:-{#4} and {#5})
    --
    ++(-{#3},0);

}
\NewDocumentCommand{\quadrupoleRod}{m m m}{% segment, top/bottom, front/back
    \ifthenelse{\equal{#2}{top} \AND \equal{#3}{front}}{%
        \pgfmathsetmacro{\coordX}{\quadrupoleRadiusHorizontal + \offsetX}
        \pgfmathsetmacro{\coordY}{\quadrupoleTopFrontY}
    }{}
\ifthenelse{\equal{#2}{top} \AND \equal{#3}{back}}{%
    \pgfmathsetmacro{\coordX}{\quadrupoleRadiusHorizontal}
    \pgfmathsetmacro{\coordY}{\quadrupoleTopBackY}
}{}

\ifthenelse{\equal{#2}{bottom} \AND \equal{#3}{front}}{%
    \pgfmathsetmacro{\coordX}{\quadrupoleRadiusHorizontal + \offsetX}
    \pgfmathsetmacro{\coordY}{\quadrupoleBottomFrontY}
}{}

\ifthenelse{\equal{#2}{bottom} \AND \equal{#3}{back}}{%
    \pgfmathsetmacro{\coordX}{\quadrupoleRadiusHorizontal}
    \pgfmathsetmacro{\coordY}{\quadrupoleBottomBackY}
}{}

\cylinder
    {\coordX}
    {\coordY}
    {\quadrupolePathLength}
    {\quadrupoleRadiusHorizontal}
    {\quadrupoleRadiusVertical}
    {gray}
    {white}
    {1}

}
\NewDocumentCommand{\quadrupolePair}{m m}{% segment, front/back
    \ifthenelse{\equal{#2}{front} \OR \equal{#2}{back}}{%
        \quadrupoleRod{#1}{top}{#2}
        \quadrupoleRod{#1}{bottom}{#2}
    }{}
}
\begin{document}
% General constants
% %%%%%%%%%%%%%%%%%
\pgfmathsetmacro{\offsetX}{0.5}
\pgfmathsetmacro{\offsetY}{0.6}
\pgfmathsetmacro{\spacer}{0.75}
\pgfmathsetmacro{\centerOffset}{0.3}
% Quadrupole constants
% %%%%%%%%%%%%%%%%%%%%
\pgfmathsetmacro{\quadrupoleRadiusHorizontal}{0.08}
\pgfmathsetmacro{\quadrupoleRadiusVertical}{0.2}
\pgfmathsetmacro{\quadrupoleLength}{4}
\pgfmathsetmacro{\quadrupolePathLength}{\quadrupoleLength - (2 * \quadrupoleRadiusHorizontal)}
\pgfmathsetmacro{\quadrupoleTopFrontY}{\centerOffset + 2 * \quadrupoleRadiusVertical}
\pgfmathsetmacro{\quadrupoleTopBackY}{\quadrupoleTopFrontY + \offsetY}
\pgfmathsetmacro{\quadrupoleBottomBackY}{-\centerOffset}
\pgfmathsetmacro{\quadrupoleBottomFrontY}{\quadrupoleBottomBackY - \offsetY}
\begin{tikzpicture}
    \coordinate (msOrigin) at (0,0);
% Define a formula for the coil.
% This is what the numbers mean:
% 0.25: the x offset
% 0.13: how far the rings are apart
% 0.30: how much from the side the rings are seen
% 0.75: radius of the rings
\def\coil#1{
    {0.25 + 0.13 * (2 * #1 + \t) + 0.30 * sin(- \t  *  pi r))},
    {0.75 * cos(-\t * pi r)}
}

% Draw the background-rods
\quadrupolePair{1}{back}

% Draw the part of the coil behind
\foreach \n in {1,...,14} {
    \draw[domain={0:1},smooth,variable=\t,samples=15]
        plot (\coil{\n}); 
}

% Draw the part of the coil in front
\foreach \n in {0,1,...,13} {
    \ifthenelse{\equal{\n}{0} \OR \equal{\n}{13}}
    {%
        \ifthenelse{\equal{\n}{0}}{%
            \draw[
                domain = {1:2},
                smooth,
                variable = \t,
                samples = 15,
                % mark position = 0(start)
            ]
                plot (\coil{\n});
        }{%
        \draw[
                domain = {1:2},
                smooth,
                variable = \t,
                samples = 15,
                % mark position = 1(end)
            ]
                plot (\coil{\n});
        }
    }{
        \draw[
            domain = {1:2},
            smooth,
            variable = \t,
            samples = 15
        ]
            plot (\coil{\n});
    }
}

% Draw the foreground-rods
\quadrupolePair{1}{front}

\draw 
    % (start) % to join the mark position "start"
    (0.25, -0.75)
    to [out = 180, in = 0] 
    ++(-1, 0.75);
\draw 
    % (end) % to join the mark position "end"
    (4, -0.75) 
    to [out = 0, in = 180] 
    ++(1, 0.75);

\end{tikzpicture}
\hspace{1em}
\begin{tikzpicture}
    \coordinate (msOrigin) at (0,0);
% Define a formula for the coil.
% This is what the numbers mean:
% 0.25: the x offset
% 0.13: how far the rings are apart
% 0.30: how much from the side the rings are seen
% 0.75: radius of the rings
\def\coil#1{
    {0.25 + 0.13 * (2 * #1 + \t) + 0.30 * sin(- \t  *  pi r)},
    {0.75 * #1/10 * \t * cos(-\t * pi r)}
}

% Draw the background-rods
\quadrupolePair{1}{back}

% Draw the part of the coil behind
\foreach \n in {1,...,14} {
    \draw[domain={0:1},smooth,variable=\t,samples=15]
        plot (\coil{\n});
}

% Draw the part of the coil in front
\foreach \n in {0,1,...,13} {
    \ifthenelse{\equal{\n}{0} \OR \equal{\n}{13}}
    {%
        \ifthenelse{\equal{\n}{0}}{%
            \draw[
                domain = {1:2},
                smooth,
                variable = \t,
                samples = 15,
                % mark position = 0(start)
            ]
                plot (\coil{\n});
        }{%
        \draw[
                domain = {1:2},
                smooth,
                variable = \t,
                samples = 15,
                % mark position = 1(end)
            ]
                plot (\coil{\n});
        }
    }{
        \draw[
            domain = {1:2},
            smooth,
            variable = \t,
            samples = 15
        ]
            plot (\coil{\n});
    }
}

% Draw the foreground-rods
\quadrupolePair{1}{front}


\end{tikzpicture}
\end{document}

Answer 1

I see no reason to use PGF code - you are almost there just by noticing that the spiral can be plotted by {cos(deg(x))},{sin(deg(x)},{x}. I normally love PGFPlots, but this is not a plot(axis, scale, ticks, labels, ...). I believe that the plot function in TikZ is the right way.

To straighten the ends of the spiral I let the amplitude decay at the same with the pitch of the loops. I am not sure how you want the conical to end - a simple way is just to let the amplitude of the coil go up fast and adjust the domain.

\documentclass[tikz, border=1cm]{standalone}
\begin{document}
\begin{tikzpicture}[ultra thick]
\newcommand{\domA}{-pi}
\newcommand{\domB}{0}
\newcommand{\domC}{2*pi}
\newcommand{\domD}{4*pi}
\newcommand{\domE}{\domC+0.5}
\newcommand{\pitch}{10}
\newcommand{\ampA}{(1/(1+\domB-\x))}
\newcommand{\ampB}{(1/(1-\domC+\x))}
\newcommand{\ampC}{(0.1*(\x-\domB)+1)}
\draw[red, domain={\domA:\domB}, smooth, samples=100] plot (\x, {\ampAcos((\ampA\pitch\x+(1-\ampA)\pitch\domB) r)}, {\ampAsin((\ampA\pitch\x+(1-\ampA)\pitch\domB) r)}  );
\draw[green, domain={\domB:\domC}, smooth, samples=200] plot (\x, {cos(\pitch\x r)} , {sin(\pitch\x r)} );
\draw[blue, domain={\domC:\domD}, smooth, samples=100] plot (\x, {\ampBcos((\ampB\pitch\x+(1-\ampB)\pitch\domC) r)}, {\ampBsin((\ampB\pitch\x+(1-\ampB)\pitch\domC) r)}  );
\begin{scope}[yshift=-4cm]
\draw[teal, domain={\domA:\domB}, smooth, samples=100] plot (\x, {cos((\ampA\pitch\x+(1-\ampA)\pitch\domB) r)}, {sin((\ampA\pitch\x+(1-\ampA)\pitch\domB) r)}  );
\draw[orange, domain={\domB:\domC}, smooth, samples=200] plot (\x, {\ampCcos(\pitch\x r)} , {\ampCsin(\pitch\x r)} );
\draw[violet, domain={\domC:\domE}, smooth, samples=100] plot (\x, {\ampC1/\ampBcos(\pitch\x r)} , {\ampC1/\ampBsin(\pitch\x r)} );
\end{scope}
\end{tikzpicture}
\end{document}

Edit:

The default z-vector in TikZ points to (−3.85mm, −3.85mm). To change the perspective, you can use e.g. z={(-3.85mm, 3.85mm)} like this:

\documentclass[tikz, border=1cm]{standalone}
\begin{document}
\begin{tikzpicture}[z={(-3.85mm, 3.85mm)}]
\newcommand{\domA}{-pi}
\newcommand{\domB}{0}
\newcommand{\domC}{2*pi}
\newcommand{\domD}{4*pi}
\newcommand{\domE}{\domC+0.5}
\newcommand{\pitch}{10}
\newcommand{\ampA}{(1/(1+\domB-\x))}
\newcommand{\ampB}{(1/(1-\domC+\x))}
\newcommand{\ampC}{(0.1*(\x-\domB)+1)}
\draw[fill=gray] (-1,1.2,1) -- (7,1.2,1) arc[start angle=90, end angle=-90, x radius=0.1cm, y radius=0.2cm] -- (-1,0.8,1);
\drawfill=white circle[x radius=0.1cm, y radius=0.2cm];
\draw[fill=gray] (-1,-1.2,1) -- (7,-1.2,1) arc[start angle=-90, end angle=90, x radius=0.1cm, y radius=0.2cm] -- (-1,-0.8,1);
\drawfill=white circle[x radius=0.1cm, y radius=0.2cm];
\draw[red, thick, domain={\domA:\domB}, smooth, samples=100] plot (\x, {\ampAcos((\ampA\pitch\x+(1-\ampA)\pitch\domB) r)}, {\ampAsin((\ampA\pitch\x+(1-\ampA)\pitch\domB) r)}  );
\draw[red, thick, domain={\domB:\domC}, smooth, samples=200] plot (\x, {cos(\pitch\x r)} , {sin(\pitch\x r)} );
\draw[red, thick, domain={\domC:\domD}, smooth, samples=100] plot (\x, {\ampBcos((\ampB\pitch\x+(1-\ampB)\pitch\domC) r)}, {\ampBsin((\ampB\pitch\x+(1-\ampB)\pitch\domC) r)}  );
\draw[fill=gray] (-1,1.2,-1) -- (7,1.2,-1) arc[start angle=90, end angle=-90, x radius=0.1cm, y radius=0.2cm] -- (-1,0.8,-1);
\drawfill=white circle[x radius=0.1cm, y radius=0.2cm];
\draw[fill=gray] (-1,-1.2,-1) -- (7,-1.2,-1) arc[start angle=-90, end angle=90, x radius=0.1cm, y radius=0.2cm] -- (-1,-0.8,-1);
\drawfill=white circle[x radius=0.1cm, y radius=0.2cm];
\end{tikzpicture}
\end{document}

The kink in the red spiral is because the smooth does not work across different plots. I can see two ways to correct this: Either remove the smooth option and increase the samples a lot. -or better: Use TikZ declare function to declare a piecewise function and only do one plot.