Normalized tangent vectors along a curve

Question

I am using the following code to add the tangent vector at various points along a curve. I would like to normalize each vector, thus resulting in tangent vectors of equal unit length. How might I achieve this?

\documentclass[11pt]{article}   
\usepackage{tikz}       
 \usetikzlibrary{arrows}      
\begin{document}

 \begin{tikzpicture}[allow upside down,x=1.5cm,y=1.4cm]
    \draw[red,thick] (0,0)
    .. controls +(right:6cm) and +(left:4cm) .. (2,6)
    \foreach \p in {0,10,...,100} {
      node[sloped,inner sep=0cm,above,pos=\p*0.01,
      anchor=south west,
      minimum height=(10+\p)*0.03cm,minimum width=(10+\p)*0.03cm]
      (N \p){}
    }
    .. controls +(right:2cm) and +(right:3cm) .. (6,2)
    \foreach \p in {0,10,...,100} {
      node[sloped,inner sep=0cm,above,pos=\p*0.01,
      anchor=south west,
      minimum height=(110-\p)*0.03cm,minimum width=(110-\p)*0.03cm]
      (N2 \p){}
    }
    ;
    \foreach \p in {0,10,...,100} {
      \draw[-latex,color=black] (N \p.south west) -- (N \p.south east);
    }
    \foreach \p in {0,10,...,100} {
      \draw[-latex,color=black] (N2 \p.south west) -- (N2 \p.south east);
    }
  \end{tikzpicture}
 \end{document}

score 8 · Accepted Answer · answered Dec 28 '14 at 15:58

With the calc tikz-library, the syntax

($A!1cm!B$)

means a point 1cm from A in the direction B. Using this we can easily modify your tangent vector expressions to

\foreach \p in {0,10,...,100} {
  \draw[-latex,color=black] (N \p.south west)
    -- ($(N \p.south west)!1cm!(N \p.south east)$);
  }
  \foreach \p in {0,10,...,100} {
  \draw[-latex,color=black] (N2 \p.south west)
    -- ($(N2 \p.south west)!1cm!(N2 \p.south east)$);
  }

which produces

Sample output

\documentclass[11pt]{article}   
\usepackage{tikz}       
\usetikzlibrary{arrows,calc}      

\begin{document}

\begin{tikzpicture}[allow upside down,x=1.5cm,y=1.4cm]
  \draw[red,thick] (0,0)
  .. controls +(right:6cm) and +(left:4cm) .. (2,6)
  \foreach \p in {0,10,...,100} {
    node[sloped,inner sep=0cm,above,pos=\p*0.01,
    anchor=south west,
    minimum height=(10+\p)*0.03cm,minimum width=(10+\p)*0.03cm]
    (N \p){}
  }
  .. controls +(right:2cm) and +(right:3cm) .. (6,2)
  \foreach \p in {0,10,...,100} {
    node[sloped,inner sep=0cm,above,pos=\p*0.01,
    anchor=south west,
    minimum height=(110-\p)*0.03cm,minimum width=(110-\p)*0.03cm]
    (N2 \p){}
  }
  ;
  \foreach \p in {0,10,...,100} {
  \draw[-latex,color=black] (N \p.south west)
    -- ($(N \p.south west)!1cm!(N \p.south east)$);
  }
  \foreach \p in {0,10,...,100} {
  \draw[-latex,color=black] (N2 \p.south west)
    -- ($(N2 \p.south west)!1cm!(N2 \p.south east)$);
  }
\end{tikzpicture}


\end{document}

score 5 · Answer 2 · edited Apr 13 '17 at 12:35

For comparison, here's a Metapost version, where defining the path, drawing it, and marking the tangents are done separately.

enter image description here

prologues := 3;
outputtemplate := "%j%c.eps";

beginfig(1);
u = 2cm;
% define the curve
path curve; 
curve = ( origin .. controls (6,0) and (-2,6)
        .. (2,6) .. controls (4,6) and (9,2) 
        .. (6,2) ) scaled u;

% draw it in pink
draw curve withpen pencircle scaled 1.44 withcolor .67[red,white];

% mark it with tangents at regular intervals
pair p, q;
A := arclength curve;
s := A/20;
for a = 0 step s until A+eps:
  t := arctime a of curve;
  p := point t of curve;
  q := p + 1/2u * unitvector(direction t of curve);
  drawarrow p -- q;
endfor

endfig;
end.

score 5 · Answer 3 · answered Dec 30 '14 at 00:21

Here is another method to get tangents via the decorations.markings library:

enter image description here

\documentclass[tikz]{standalone}   
\usetikzlibrary{arrows,calc,decorations.markings}      
\begin{document}
\begin{tikzpicture}[x=1.5cm,y=1.4cm]
  \tikzset{
    draw tangents/.style={decorate,decoration={
        markings,mark=between positions 0 and 1 step .04 with
        {\draw[-latex,black,thin] (0,0) -- (1cm,0);},
      }
    }
  }
  \draw[red,thick,postaction=draw tangents]
  (0,0) .. controls +(right:6cm) and +(left:4cm) .. (2,6)
  .. controls +(right:2cm) and +(right:3cm) .. (6,2);
\end{tikzpicture}
\end{document}

Normalized tangent vectors along a curve

3 Answers3

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