Helix on a cylinder

Question

I want to plot geodesics on a cylinder. These are straight lines, circles or helices. I can plot the first two, but I don't know how to draw the helices.

To be precise, I would like to know how to draw an helix on this cylinder (passing thru two points A and B, but the location of the points is not relevant), using TikZ exclusively (maybe with pgfplots, but without using pstricks):

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}
\begin{document}
\begin{tikzpicture}
\node [cylinder,rotate=90,draw,aspect=2,minimum width=2cm,minimum height=3.5cm](C){};
\draw[fill] (-0.5,-0.5) circle [radius=0.045]node[below]{$A$};
\draw[fill] (0.5,0.75) circle [radius=0.045]node[below]{$B$};
\end{tikzpicture}
\end{document}

Answer 1

I like @Gromolatto 's idea. However I believe that drawing hundreds of segments is an overkill. Here is his/her code using "\pgfplotfunction" which simplifies the code and increases precisicion since all points are ploted as they are without defining hunderes of sub-segments.

I left the original code in place and commented the lines that were replaced.

\documentclass{standalone}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usepackage{pgfplots}
\usetikzlibrary{shapes}
\tdplotsetmaincoords{60}{110}


\begin{document}
\begin{tikzpicture}[tdplot_main_coords]
  \node [cylinder,rotate=90,draw,aspect=2,minimum width=2cm,minimum height=3.5cm](C){};

  %\foreach \t in {-90,-75,...,0}{%
  %  \draw ({cos(\t)},{sin(\t)},{-0.25+\t/360})--({cos(\t+7)},{sin(\t +7)},{-0.23+\t/360});
  %}

  \begin{scope}[color=black, dashed]
  \pgfplothandlerlineto
  \pgfplotfunction{\t}{-90,-89,...,15}
       {\pgfpointxyz {cos(\t)}{sin(\t)}{-0.25+\t/360}} 
       \pgfusepath{stroke}
  \end{scope}

%  \foreach \t in {15,16,...,98}{%
%    \draw[line width=1.5pt,color=red] ({cos(\t)},{sin(\t)},{-0.25+\t/360})--({cos(\t+1)},{sin(\t +1)},{-0.22+\t/360});
%  }

  \begin{scope}[color=red]
  \pgfplothandlerlineto
  \pgfplotfunction{\t}{15,16,...,110}
       {\pgfpointxyz {cos(\t)}{sin(\t)}{-0.25+\t/360}} 
       \pgfusepath{stroke}
  \end{scope}



  %\foreach \t in {110,125,...,280}{%
  %  \draw[line width=1pt,color=red] ({cos(\t)},{sin(\t)},{-0.25+\t/360})--({cos(\t+7)},{sin(\t +7)},{-0.22+\t/360});
  %}


  \begin{scope}[color=red, dashed]
  \pgfplothandlerlineto
  \pgfplotfunction{\t}{110,111,...,303}
       {\pgfpointxyz {cos(\t)}{sin(\t)}{-0.25+\t/360}} 
       \pgfusepath{stroke}
  \end{scope}

 % \foreach \t in {303,304,...,340}{%
 %   \draw[line width=1.6pt,color=red] ({cos(\t)},{sin(\t)},{-0.25+\t/360})--({cos(\t+1)},{sin(\t +1)},{-0.19+\t/360});
 % }


  \begin{scope}[color=red]
  \pgfplothandlerlineto
  \pgfplotfunction{\t}{303,304,...,340}
       {\pgfpointxyz {cos(\t)}{sin(\t)}{-0.25+\t/360}} 
       \pgfusepath{stroke}
  \end{scope}

  %\foreach \t in {355,370}{%
  %  \draw ({cos(\t)},{sin(\t)},{-0.25+\t/360})--({cos(\t+7)},{sin(\t +7)},{-0.23+\t/360});
  %}


  \begin{scope}[color=black, dashed]
  \pgfplothandlerlineto
  \pgfplotfunction{\t}{340,341,...,370}
       {\pgfpointxyz {cos(\t)}{sin(\t)}{-0.25+\t/360}} 
       \pgfusepath{stroke}
  \end{scope}

  \def\ang{340}
  \pgfmathsetmacro\bx{cos(\ang)}
  \pgfmathsetmacro\by{sin(\ang)}
  \pgfmathsetmacro\bz{-0.24+ \ang/360}

  \coordinate (B) at (\bx,\by,\bz);



  \draw[fill] (0.9922,0.25,-0.2) circle [x=1cm,y=1cm,radius=0.045]node[below]{$A$};
  \draw[fill] (B) circle [x=1cm,y=1cm,radius=0.045]node[below]{$B$};
\end{tikzpicture}
\end{document}

Here is the figure obtained:

If you make a "zoom" you will see how this figure icreases precision.

Answer 2

Here is a tikz-3dplot solution. In case people are interested in drawing a more general helix (not just a geodesic) between two points of the cylinder, I've included a macro called \n to specify the number of additional turns around the cylinder.

Edit: thanks to Qrrbrbirlbel for his very helpful comment.

\documentclass{article}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\begin{document}
\tdplotsetmaincoords{70}{15}
\tikzset{every circle/.append style={x=1cm, y=1cm}}
\begin{tikzpicture}[tdplot_main_coords]

% --- Independent parameters ---
\def\h{3}                          % cylinder height
\pgfmathtruncatemacro\tA{350}      % A angle
\def\zA{1}                         % A applicate
\pgfmathtruncatemacro\tB{150}      % B angle
\def\zB{2}                         % B applicate
\pgfmathtruncatemacro\n{0}         % number of additional turns
\pgfmathtruncatemacro\NbPt{51}     % number of dots for drawing the helix portion
\def\rhelixdots{0.02}              % radius of dots forming helix
\def\rAB{0.05}                     % radius of A and B dots

% --- Draw cylinder ---
% peripheral spokes
\foreach \t in {20,40,...,360} 
{ 
    \draw[gray,very thin,dashed] ({cos(\t)},{sin(\t)},0)
        --({cos(\t)},{sin(\t)},\h);
}

% lower circle
\draw[black,very thin] (1,0,0) 
    \foreach \t in {2,3,...,360}
    {
        --({cos(\t)},{sin(\t)},0)
    }
    --cycle;

% upper circle
\draw[black,very thin] (1,0,\h) 
    \foreach \t in {2,4,...,360}
    {
        --({cos(\t)},{sin(\t)},\h)
    }
    --cycle;


% --- Draw helix ---
\pgfmathsetmacro\tone{\tA}
\pgfmathsetmacro\tlast{\tB+\n*360}
\pgfmathsetmacro\ttwo{\tone+(\tlast-\tone)/(\NbPt-1)}
\pgfmathsetmacro\p{360*(\zB-\zA)/(\tB-\tA+360*\n)}
\foreach \t in {\tone,\ttwo,...,\tlast}{%
    \fill[red] ({cos(\t)},{sin(\t)},{\p*(\t-\tA)/360+\zA}) circle[radius=\rhelixdots];
}

% --- Draw A and B ---
\fill[blue] ({cos(\tA)},{sin(\tA)},\zA) circle [radius=\rAB]node[right]{$A$};
\fill[blue] ({cos(\tB)},{sin(\tB)},\zB) circle [radius=\rAB]node[left]{$B$};

\end{tikzpicture}
\end{document}

Answer 3

Here is the solution exactly as I wanted it (it is based on the replies by Jubobs and Qrrbrbirlbel, but in a more simplified form). There are 5 different sectors depending on the color and the type of line (dashed or not). The points coordinates are computed so they lie on the helix:

\documentclass{standalone}
\usepackage{tikz}
\usepackage{3dplot}
\usetikzlibrary{shapes.geometric}
\tdplotsetmaincoords{60}{110}
\begin{document}
\begin{tikzpicture}[tdplot_main_coords]
\node [cylinder,rotate=90,draw,aspect=2,minimum width=2cm,minimum height=3.5cm](C){};
\foreach \t in {-90,-75,...,0}{%
\draw ({cos(\t)},{sin(\t)},{-0.25+\t/360})--({cos(\t+7)},{sin(\t +7)},{-0.23+\t/360});
}
\foreach \t in {15,16,...,98}{%
\draw[line width=1.5pt,color=red] ({cos(\t)},{sin(\t)},{-0.25+\t/360})--({cos(\t+1)},{sin(\t +1)},{-0.22+\t/360});
}
\foreach \t in {110,125,...,280}{%
\draw[line width=1pt,color=red] ({cos(\t)},{sin(\t)},{-0.25+\t/360})--({cos(\t+7)},{sin(\t +7)},{-0.22+\t/360});
}
\foreach \t in {303,304,...,340}{%
\draw[line width=1.6pt,color=red] ({cos(\t)},{sin(\t)},{-0.25+\t/360})--({cos(\t+1)},{sin(\t +1)},{-0.19+\t/360});
}
\foreach \t in {355,370}{%
\draw ({cos(\t)},{sin(\t)},{-0.25+\t/360})--({cos(\t+7)},{sin(\t +7)},{-0.23+\t/360});
}
\draw[fill] (0.9922,0.25,-0.2) circle [x=1cm,y=1cm,radius=0.045]node[below]{$A$};
\draw[fill] (0.2739,-0.5,0.32) circle [x=1cm,y=1cm,radius=0.045]node[below]{$B$};
\end{tikzpicture}
\end{document}

There are some issues related to the density of lines near the turning points, but these can be easily adjusted by hand. When printed, this is OK:

Answer 4

Here is a solution with pstricks in answer to a similar question:

\documentclass[x11names]{standalone}%
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\pagestyle{empty}
\usepackage{pstricks-add, pst-3dplot}
\usepackage{auto-pst-pdf}
\def\localbasis{\psline{<->}(1,0)(0,0)(0,1)}
\def\\M{6*\pstPI1}
\begin{document}

\footnotesize
\psset{xPlotpoints = 500, plotstyle=curve, linecolor = DarkSeaGreen3, algebraic, arrowinset=0.2, labelsep=3pt, dash=5pt 4pt}
\sffamily
\def\radius{2 }
\def\height {9.8}
\def\thetaend {0}
\def\thetaini {-305}
\def\sp{1.6}
\def\phase{0.7}
\psset{viewpoint=30 40 15 rtp2xyz, IIIDshowgrid = false}
\begin{pspicture}(-6,-2.5)(4,12)
    \pnodes(-4,-0.08){L1}(-4, 8.23){L2}(-2.2,-0.08){C1}(-2.2,8.23){C2}(-3,-0.08){S1}(-3, 3.36){S2}(-2,3.36){C3}(0,-0.08){O}(2,-0.08){R}
    \psCylinder[increment=5, opacity=0.3]{\radius}{\height}
    \ncline[linecolor=Coral1,linewidth=0.5pt]{*->}{O}{R}\nbput{large radius}
    \psset{linewidth=1.2pt, dimen=inner}
    \parametricplotThreeD[arrows=c-](0, 3.10){%
        \radius*cos(t - \phase) | \radius * sin(t - \phase) | t/\sp}
    \parametricplotThreeD[linecolor=DarkSeaGreen3!50!, linestyle=dashed](3.15,6.10){%
        \radius*cos(t-\phase) | \radius* sin(t-\phase) | t/\sp}
    \parametricplotThreeD(6.18, 9.36){%68
        \radius*cos(t-\phase) | \radius * sin(t-\phase) | t/\sp}
    \parametricplotThreeD[linecolor=DarkSeaGreen3!50!, linestyle=dashed](9.40, 12.35){%
        \radius*cos(t-\phase) | \radius * sin(t-\phase) | t/\sp}
    \parametricplotThreeD[arrows=-c](12.45, 15.56){%
        \radius*cos(t-\phase) | \radius * sin(t-\phase) | t/\sp}
    \psset{linewidth=0.5pt, linecolor=Coral1}
    \ncline[offset=6pt]{<->}{L2}{L1}\ncput*{length}
    \ncline{L1}{C1}\ncline{L2}{C2}
    \ncline[offset=-6pt]{<->}{S1}{S2}\ncput*{spacing}
    \ncline{S2}{C3}
\end{pspicture}

\end{document} 

Answer 5

This is just to mention that the nice tricks of Fritz' great answer can be used here, too. The mathematics is even simpler for cylinders. This allows one to let pgfplots decide whether some stretch is on the front or in the back, and pgfplots also allows us to shade the thing as we want by just dialing an appropriate point meta.

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{pgfplots} 
\pgfplotsset{compat=1.16} 
\begin{document}
% very much like https://tex.stackexchange.com/a/199715/121799
\pgfplotsset{visible stretch/.style={restrict expr to
domain={sin(atan2(rawy,rawx)-\pgfkeysvalueof{/pgfplots/view/az})}{-1.1:0}},hidden
stretch/.style={restrict expr to
domain={sin(atan2(rawy,rawx)-\pgfkeysvalueof{/pgfplots/view/az})}{0:1.1}}}
\def\addFGBGplot[#1]#2;{
    \addplot3[#1,hidden stretch, opacity=0.25] #2;
    \addplot3[#1,visible stretch] #2;
}
\begin{tikzpicture}[declare function={R=1;H=3;}]
\begin{axis}[hide axis]
 \addplot3[surf,shader=interp,domain y=0:H,opacity=0.5,domain=
 \pgfkeysvalueof{/pgfplots/view/az}:\pgfkeysvalueof{/pgfplots/view/az}-180,
 colormap/blackwhite,point meta={sin(atan2(y,x)+40)}] 
 ({R*cos(x)},
 {R*sin(x)},{y});
 \addplot3[samples y=0,domain=0:360,smooth,fill=gray!30,draw=black] ({R*cos(x)},{R*sin(x)},{H});
 \addFGBGplot[samples y=0,color=blue,domain=0:1440,smooth,samples=251]
 ({R*cos(x)},{R*sin(x)},{0.75*x/360});
 \addplot3[samples y=0,domain=\pgfkeysvalueof{/pgfplots/view/az}:\pgfkeysvalueof{/pgfplots/view/az}-180,smooth,color=black] 
 ({R*cos(x)},{R*sin(x)},{0});
 \draw  ({R*cos(\pgfkeysvalueof{/pgfplots/view/az})},{R*sin(\pgfkeysvalueof{/pgfplots/view/az})
 },{0}) -- ({R*cos(\pgfkeysvalueof{/pgfplots/view/az})},{R*sin(\pgfkeysvalueof{/pgfplots/view/az})},{H})
  ({R*cos(\pgfkeysvalueof{/pgfplots/view/az}-180)},{R*sin(\pgfkeysvalueof{/pgfplots/view/az}-180)
 },{0}) -- ({R*cos(\pgfkeysvalueof{/pgfplots/view/az}-180)},{R*sin(\pgfkeysvalueof{/pgfplots/view/az}-180)},{H});
\end{axis}
\end{tikzpicture}
\end{document}