Is there an efficient way to draw torus knots and related ''complicated lines'' (simple shapes, but on nontrivial surfaces) in TikZ?

Question

In the process of typing up a set of lecture notes for a course, I found myself in need of a nice picture of a torus knot, which basically comes down to drawing a very complicated line. With some trickery, I was able to produce the simplest such knot, which is not even really a knot (but it's a start...). The code is as follows:

\documentclass{article}
\usepackage{tikz}
\begin{document} 


\begin{tikzpicture}[scale=2]
\draw (-3.5,0) .. controls (-3.5,1.2) and (-1.5,1.5) .. (0,1.5);
\draw[xscale=-1] (-3.5,0) .. controls (-3.5,1.2) and (-1.5,1.5) .. (0,1.5);
\draw[rotate=180] (-3.5,0) .. controls (-3.5,1.2) and (-1.5,1.5) .. (0,1.5);
\draw[yscale=-1] (-3.5,0) .. controls (-3.5,1.2) and (-1.5,1.5) .. (0,1.5);

\draw (-2,.2) .. controls (-1.5,-0.05) and (-1,-0.15) .. 
    (0,-.15) .. controls (1,-0.15) and (1.5,-0.05) .. (2,0.2);
\draw (-1.7,0.1) .. controls (-1.5,0.15) and (-1,0.25) .. 
    (0,.25) .. controls (1,0.25) and (1.5,0.15) .. (1.7,0.1);
\begin{scope}[rotate=10]
    \draw[draw=none] (0,-1.32) arc (270:630:3.06cm and 1.32cm)
    coordinate[pos=0.375] (a) coordinate[pos=0.875] (b);
    \draw[red] (0,-1.32) arc (270:405:3.06cm and 1.32cm);
    \draw[red,densely dashed] (a) arc (45:225:3.06cm and 1.32cm);
    \draw[red] (b) arc (225:270:3.06cm and 1.32cm);
\end{scope}
\end{tikzpicture}

\end{document}

and the output is:

However, I'm also interested in being able to produce a picture of nontrivial knots such as a loop that wraps around one way three times while "going around the hole" once, and my method does not generalize. My question, therefore, is: Is it possible to efficiently produce pictures such as these (one may also think of natural generalizations like lines on surfaces of arbitrary genus), using TikZ? I should add that I already have a way of efficiently producing a picture of any (compact) orientable surface (without boundary, up to homotopy equivalence---I'm typing notes for an algebraic topology class): It's just the lines on them that I don't know how to quickly draw.

This is a related, yet much simpler question (not a duplicate).

Answer 1

This approach checks if the normal vector is pointing to your eyes -- if so, it draws solid lines.

Edit

Please compile this by XeLaTeX! Or one can replace greek letters in csnames by "safe" latin letters.

\documentclass[border=9,tikz]{standalone}
    \usepackage{tikz-3dplot}
    \usetikzlibrary{decorations}

\begin{document}

\foreach\j in{10,20,...,360}{
    \tdplotsetmaincoords{30}{\j}
    \def\R{5}
    \def\r{2}
    \def\N{2}
    \def\n{3}
        \pgfmathsetmacro\S{30*\N*\n}
        \pgfmathsetmacro\X{sin(-\tdplotmaintheta)*sin(-\tdplotmainphi)}
        \pgfmathsetmacro\Y{sin(-\tdplotmaintheta)*cos(-\tdplotmainphi)}
        \pgfmathsetmacro\Z{cos(-\tdplotmaintheta)}

    \tikz[tdplot_main_coords]{
        \path(-10cm,-10cm)(10cm,10cm);
        \draw[dashed](\R+\r,0,0)foreach\i in{0,...,\S}{
            \pgfextra{
                \pgfmathsetmacro\t{360*\N*(\i/\S)}
                \pgfmathsetmacro    \τ{\n/\N*\t}   \pgfmathsetmacro   \ι{\N*\t}
                \pgfmathsetmacro\rsinτ{\r*sin(\τ)} \pgfmathsetmacro\sinι{sin(\ι)}
                \pgfmathsetmacro\rcosτ{\r*cos(\τ)} \pgfmathsetmacro\cosι{cos(\ι)}
            }
            --({(\rcosτ+\R)*\cosι},{(\rcosτ+\R)*\sinι},\rsinτ)
        };
        \draw(\R+\r,0,0)foreach\i in{0,...,\S}{
            \pgfextra{
                \pgfmathsetmacro\t{360*\N*(\i/\S)}
                \pgfmathsetmacro    \τ{\n/\N*\t}   \pgfmathsetmacro   \ι{\N*\t}
                \pgfmathsetmacro\rsinτ{\r*sin(\τ)} \pgfmathsetmacro\sinι{sin(\ι)}
                \pgfmathsetmacro\rcosτ{\r*cos(\τ)} \pgfmathsetmacro\cosι{cos(\ι)}
                \pgfmathparse{
                    \rcosτ*\cosι*\X + \rcosτ*\sinι*\Y + \rsinτ*\Z
                }
                \ifdim\pgfmathresult pt<0pt
                    \def\to{--}
                \else
                    \def\to{}
                \fi
            }
            \to({(\rcosτ+\R)*\cosι},{(\rcosτ+\R)*\sinι},\rsinτ)
        };
    }
}
\end{document}

ascii version:

\documentclass[border=9,tikz]{standalone}
    \usepackage{tikz-3dplot}
    \usetikzlibrary{decorations}

\begin{document}

\foreach\j in{10,20,...,360}{
    \tdplotsetmaincoords{30}{\j}
    \def\R{5}
    \def\r{2}
    \def\N{2}
    \def\n{3}
        \pgfmathsetmacro\S{30*\N*\n}
        \pgfmathsetmacro\X{sin(-\tdplotmaintheta)*sin(-\tdplotmainphi)}
        \pgfmathsetmacro\Y{sin(-\tdplotmaintheta)*cos(-\tdplotmainphi)}
        \pgfmathsetmacro\Z{cos(-\tdplotmaintheta)}

    \tikz[tdplot_main_coords]{
        \path(-10cm,-10cm)(10cm,10cm);
        \draw[dashed](\R+\r,0,0)foreach\i in{0,...,\S}{
            \pgfextra{
                \pgfmathsetmacro\t{360*\N*(\i/\S)}
                \pgfmathsetmacro    \TAU{\n/\N*\t}   \pgfmathsetmacro   \IOTA{\N*\t}
                \pgfmathsetmacro\rsinTAU{\r*sin(\TAU)} \pgfmathsetmacro\sinIOTA{sin(\IOTA)}
                \pgfmathsetmacro\rcosTAU{\r*cos(\TAU)} \pgfmathsetmacro\cosIOTA{cos(\IOTA)}
            }
            --({(\rcosTAU+\R)*\cosIOTA},{(\rcosTAU+\R)*\sinIOTA},\rsinTAU)
        };
        \draw(\R+\r,0,0)foreach\i in{0,...,\S}{
            \pgfextra{
                \pgfmathsetmacro\t{360*\N*(\i/\S)}
                \pgfmathsetmacro    \TAU{\n/\N*\t}   \pgfmathsetmacro   \IOTA{\N*\t}
                \pgfmathsetmacro\rsinTAU{\r*sin(\TAU)} \pgfmathsetmacro\sinIOTA{sin(\IOTA)}
                \pgfmathsetmacro\rcosTAU{\r*cos(\TAU)} \pgfmathsetmacro\cosIOTA{cos(\IOTA)}
                \pgfmathparse{
                    \rcosTAU*\cosIOTA*\X + \rcosTAU*\sinIOTA*\Y + \rsinTAU*\Z
                }
                \ifdim\pgfmathresult pt<0pt
                    \def\to{--}
                \else
                    \def\to{}
                \fi
            }
            \to({(\rcosTAU+\R)*\cosIOTA},{(\rcosTAU+\R)*\sinIOTA},\rsinTAU)
        };
    }
}

\end{document}