Draw tangent of an arbitrary curve at intersections

Question

I have this picture:

and I want to draw a tangent line at each of the marked intersections. I have found some answers (and examples in the TikZ manual) using the decorations.markings library. The thing is that (from what I understood) this library only knows the tangent of a curve given the position, from 0 to 1, so I'd have to guess where, in the length of the curve, the intersection is, but I'd rather do that automagically :)

I thought about mapping the coordinate of the intersection to a normalised [0, 1] value, but I couldn't figure out how to do that either.

Code for the picture:

\documentclass[tikz]{standalone}
\usetikzlibrary{intersections}
\begin{document}
\begin{tikzpicture}
  \draw (0, 0.2) .. controls ++(0.5,0.75) and ++(-0.5,-0.75) .. (1,0.2) coordinate (LP1);
  \draw [xshift=1cm, blue, name path=eta]
    (0, 0.3) .. controls ++(0.5,0.75) and ++(-0.5,0) .. (1.0,-0.4)
             .. controls ++(0.3,0)    and ++(-0.3,0) .. (1.6, 0.3) coordinate (LP2);
  \draw [gray, name path=eta0] (LP1) -- (LP1-|LP2);
  \draw [red, name intersections={of=eta and eta0,total=\t}]
    \foreach \s in {1,...,\t}{ (intersection-\s) circle (1pt) };
\end{tikzpicture}
\end{document}

_{I think that this problem is fairly common, but I browsed plenty of intersections/tangents/markings questions but couldn't find what I'm trying to do, so sorry if it is a duplicate.}

Answer 1

This is a possible way. If you do not want to load pgfplots, I am afraid it will be considerably more effort.

How does this proposal work?

1. The pgplots (!) library fillbetween allows us to decompose paths into intersection segments. These are called A1 and A2 in the examples below, and can be reversed, if needed.
2. Once we have such an intersection segment, the problem boils down to attaching tangents at the ends of these segments, which is a long-solved problem, see this answer.

These codes illustrate this.

\documentclass[tikz]{standalone}
\usepackage{pgfplots}
\usepgfplotslibrary{fillbetween}
\usetikzlibrary{decorations.markings}
\begin{document}
\begin{tikzpicture}[tangent at/.style={% cf. https://tex.stackexchange.com/questions/25928/how-to-draw-tangent-line-of-an-arbitrary-point-on-a-path-in-tikz/25940#25940
    decoration={ markings,
      mark =at position #1 with {\draw[purple,-latex](0,0) -- (0.5,0);},
    }, decorate
  }]

  \draw (0, 0.2) .. controls ++(0.5,0.75) and ++(-0.5,-0.75) .. (1,0.2) coordinate (LP1);
  \draw [xshift=1cm, blue, name path=eta]
    (0, 0.3) .. controls ++(0.5,0.75) and ++(-0.5,0) .. (1.0,-0.4)
             .. controls ++(0.3,0)    and ++(-0.3,0) .. (1.6, 0.3) coordinate (LP2);
  \draw [gray, name path=eta0] (LP1) -- (LP1-|LP2);
  \draw [red, name intersections={of=eta and eta0,total=\t}]
    \foreach \s in {1,...,\t}{ (intersection-\s) circle (1pt) };
  \path [%draw,blue,
    name path=middle arc,
    intersection segments={
        of=eta and eta0,
        sequence={A1}
    },
    postaction={tangent at/.list={0,1}}];   
\end{tikzpicture}
\end{document}

You can do that with more intersections, too. You only need to add the tangents at the appropriate intersection segments.

\documentclass[tikz]{standalone}
\usepackage{pgfplots}
\usepgfplotslibrary{fillbetween}
\usetikzlibrary{decorations.markings}
\begin{document}
\begin{tikzpicture}[tangent at/.style={% cf. https://tex.stackexchange.com/questions/25928/how-to-draw-tangent-line-of-an-arbitrary-point-on-a-path-in-tikz/25940#25940
    decoration={ markings,
      mark =at position #1 with {\draw[purple,-latex](0,0) -- (0.5,0);},
    }, decorate
  }]

  \draw (0, 0.2) .. controls ++(0.5,0.75) and ++(-0.5,-0.75) .. (1,0.2)
  coordinate (LP1);
  \draw [xshift=1cm, blue, name path=eta]
    (0, 0.3) .. controls ++(0.5,0.75) and ++(-0.5,0) .. (1.0,-0.4)
             .. controls ++(0.3,0)    and ++(-0.3,0) .. (1.6, 0.3) 
   .. controls ++(0.6,0) and ++(-0.6,0) .. (2.8,-0.3) coordinate (LP2);
  \draw [gray, name path=eta0] (LP1) -- (LP1-|LP2);
  \draw [red, name intersections={of=eta and eta0,total=\t}]
    \foreach \s in {1,...,\t}{ (intersection-\s) circle (1pt) };
  \path [%draw,red,thick,
    intersection segments={
        of=eta and eta0,
        sequence={A1}
    },
    postaction={tangent at/.list={0,1}}];   
  \path [%draw,green,thick,
    intersection segments={
        of=eta and eta0,
        sequence={A2}
    },
    postaction={tangent at/.list={1}}]; 
\end{tikzpicture}
\end{document}

Of course, this can be automatized.

\documentclass[tikz]{standalone}
\usepackage{pgfplots}
\usepgfplotslibrary{fillbetween}
\usetikzlibrary{decorations.markings}
\begin{document}
\begin{tikzpicture}[tangent at/.style={% cf. https://tex.stackexchange.com/questions/25928/how-to-draw-tangent-line-of-an-arbitrary-point-on-a-path-in-tikz/25940#25940
    decoration={ markings,
      mark =at position #1 with {\draw[purple,-latex](0,0) -- (0.5,0);},
    }, decorate
  }]

  \draw (0, 0.2) .. controls ++(0.5,0.75) and ++(-0.5,-0.75) .. (1,0.2)
  coordinate (LP1);
  \draw [xshift=1cm, blue, name path=eta]
    (0, 0.3) .. controls ++(0.5,0.75) and ++(-0.5,0) .. (1.0,-0.4)
             .. controls ++(0.3,0)    and ++(-0.3,0) .. (1.6, 0.3) 
   .. controls ++(0.6,0) and ++(-0.6,0) .. (2.8,-0.3) coordinate (LP2);
  \draw [gray, name path=eta0] (LP1) -- (LP1-|LP2);
  \draw [red, name intersections={of=eta and eta0,total=\t},
  /utils/exec={\xdef\NumIntersection{\t}}]
    \foreach \s  in {1,...,\t}{ 
    (intersection-\s) circle (1pt) };
  \foreach \X [count=\Y starting from 0] in {1,...,\NumIntersection}    
  {
  \path [%draw,red,thick,
    intersection segments={
        of=eta and eta0,
        sequence={A\Y}
    },
    postaction={tangent at=1}];   
  } 
\end{tikzpicture}
\end{document}

(Same result as above.)