How to draw tangent lines at control points of a path in TikZ?

Question

I'm drawing smooth paths by using the

\draw plot[smooth, tension=.7] coordinates ...

command. Now I would like to draw tangents at the specified coordinates. I alreade tried Jake's answer, but I cannot figure out how to define the tangent points at certain coordinates instead of path positions like 0.4.

My current code is the following:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}

\begin{document}
\begin{tikzpicture}[scale=0.35, transform shape,
    tangent/.style={
        decoration={
            markings,% switch on markings
            mark=
                at position #1
                with
                {
                    \coordinate (tangent point-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (0pt,0pt);
                    \coordinate (tangent unit vector-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (1,0pt);
                    \coordinate (tangent orthogonal unit vector-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (0pt,1);
                }
        },
        postaction=decorate
    },
    use tangent/.style={
        shift=(tangent point-#1),
        x=(tangent unit vector-#1),
        y=(tangent orthogonal unit vector-#1)
    },
    use tangent/.default=1
]

\begin{scope}[every node/.style={fill,draw,blue!50,thick,circle}]
  \draw[thick, tangent=0.9] plot[smooth, tension=.7] coordinates {(-8,5) (-5.5,7) (-3,5.5)};
  \node (m1) at (-8,5)   {};
  \node (m2) at (-5.5,7) {};
  \node (m3) at (-3,5.5) {};
  \draw [blue, use tangent] (-0.5,0) -- (0.5,0);
\end{scope}

\begin{scope}[every node/.style={fill,draw,red!50,thick,circle}, rotate=8, shift={(-0.5,3.5)}]
  \draw[thick] plot[smooth, tension=.7] coordinates {(-8,5) (-5.5,7) (-3,5.5)};
  \node at (-8,5)   {} edge (m1);
  \node at (-5.5,7) {} edge (m2);
  \node at (-3,5.5) {} edge (m2);
\end{scope}

\end{tikzpicture}
\end{document}

Which leads to:

I would like to draw tangent lines at the blue nodes and then replace the three direct connection lines by lines which start at the red nodes and finish orthogonally at the tangent lines:

Thanks in advance!

Answer 1

It didn't turn out to be as easy-to-use as I would thought about it but here is one possibility.

You first name your curves with curve prefix key and then draw them with some custom mark type my mark which places marks. An advantage is that you don't put extra nodes but as a shortcoming you have to provide mark indices={1,...,n} to actually number the marks. Also the end points are excluded from this process but you can use Jake's style anyway since pos=0 or pos=1 is always the point you wish to draw tangent from.

The node names are always names as (tpt-<curve prefix>-<mark number>) and you have to use the get tangle=<mark number> of <curve prefix>. You then use it to rotate the drawing to draw the custom tangent line.

Just put the code between \makeatletter ... \makeatother somewhere in the preamble.

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{plotmarks}

\makeatletter
\tikzset{curve prefix/.code={
    \xdef\pgf@mark@prefix{#1}
    }
}
\tikzset{get tangle/.code args={#1 of #2}{% Looks at the previous and next marks and fakes a tangent
\pgfmathparse{int(#1-1)}
\pgfpointanchor{tpt-#2-\pgfmathresult}{center}
\pgf@xa=\pgf@x\pgf@ya=\pgf@y
\pgfmathparse{int(#1+1)}
\pgfpointanchor{tpt-#2-\pgfmathresult}{center}
\pgf@xb=\pgf@x\pgf@yb=\pgf@y
\pgfpointdiff{\pgfpoint{\pgf@xa}{\pgf@ya}}{\pgfpoint{\pgf@xb}{\pgf@yb}}
\pgfmathparse{atan2(\pgf@x,\pgf@y)}
\xdef\tangle{\pgfmathresult}
    }
}

\pgfdeclareplotmark{my mark}% Places coordinates on the marks to be used above
{\pgfsetfillcolor{\pgf@mark@color}\pgfpathcircle{\pgfpoint{0cm}{0cm}}{1ex}\pgfusepathqfill
\pgfcoordinate{tpt-\pgf@mark@prefix-\the\pgf@plot@mark@count}{\pgfpointorigin}
}
\makeatother

\begin{document}
\begin{tikzpicture}
%First Curve
\draw[thick,curve prefix=a] plot[mark color=blue!50,mark=my mark,mark indices={1,...,5},smooth,tension=0.7] 
coordinates {(-8,5) (-5.5,2) (-3,5.5) (-1,-2) (1,3)};

% Second curve
\begin{scope}[every node/.style={fill,draw,red!50,thick,circle}, rotate=8, shift={(-0.5,3.5)}]
\draw[thick,curve prefix=b] plot[mark color=red,mark=my mark,mark indices={1,...,4},smooth,tension=0.7] 
coordinates {(-8,5) (-5.5,7) (-3,5.5) (0,3)};
\end{scope}


% First tangent drawing 
\draw[get tangle=2 of a,dashed,ultra thick,purple,rotate=\tangle] (tpt-a-2) -| (tpt-b-2) (tpt-a-2) -| (tpt-b-4);

% Second tangent drawing 
\draw[get tangle=4 of a,dashdotted,ultra thick,orange,rotate=\tangle] (tpt-a-4) -| (tpt-b-3) (tpt-a-4) -| (tpt-b-1);

\end{tikzpicture}
\end{document}