Making the size of the arrow head proportional to the magnitude of a vector field in TikZ

Question

Consider a scaled vector field v = (-x, 2x+y-1) living on the triangle at (0,1), (1,0), and (0,0):

\documentclass{minimal}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}

\coordinate (A) at (0,0);
\coordinate (B) at (1,0);
\coordinate (C) at (0,1);

\draw[thick] ($(A)$) -- ($(B)$) -- ($(C)$)-- cycle;
\foreach \x in {1, 2, ..., 10}
\foreach \y in {0, 1, ..., \x}
{\draw [-stealth,blue] ($(B)-(0.1,0)+0.1*(1-\x,\y)$) 
-- ++(0.02*\x - 0.2, -0.04*\x + 0.4 +0.02*\y-0.2);} 
\end{tikzpicture}
\end{document} 

The output is as follows:

Comparing with the vector fields drawn by MATLAB's quiver:

The MATLAB's quiver command automatically scales the size of the arrow heads according to the magnitude of the vector.

So my question is: how do we scale the size of the arrow heads in TikZ? Preferably in MATLAB's solution, the vector field vanishes at (0,1) so there is no arrow there.

For example I googled a little bit, and wonder if pgfmath could do some calculation about the length variable \l. Then doing something like will work:

\draw [-stealth,blue, single arrow head extend=\l mm] ($(B)-(0.1,0)+0.1*(1-\x,\y)$) 
-- ++(0.02*\x - 0.2, -0.04*\x + 0.4 +0.02*\y-0.2);

Thanks.

Answer 1

You could use PGFPlots' quiver functionality together with a bit of coordinate filtering to achieve this:

\documentclass{article}
\usepackage{pgfplots}

\begin{document}
\begin{tikzpicture}
\begin{axis}[
    view={0}{90},
    axis equal image,
    hide axis, clip=false
]
\addplot3 [
    blue,
    point meta={sqrt(x^2+(2*x+y-1)^2)},
    quiver={
        u={-x}, v={2*x+y-1},
        scale arrows=0.1,
        every arrow/.append style={
            line width=2pt*\pgfplotspointmetatransformed/1000,
            -latex
        },
    },
    samples=10,
    domain=0:1, y domain=0:1,
    y filter/.append code={\pgfmathparse{(x+y)>1 ? nan : y}},
    x filter/.append code={\pgfmathparse{(sqrt(x^2+(2*x+y-1)^2))==0 ? nan : x}}
] (x,y,0);

\draw (axis cs:0,0) -- (axis cs:1,0) -- (axis cs:0,1) -- cycle;
\end{axis}
\end{tikzpicture}

\end{document} 

Answer 2

Based on Qrrbrbirlbel's comment I have come up with my own solution as well besides the nice answer using quiver by Jake. I am using the New arrows library in this question made by Luigi (and it is great!): Is it possible to change the size of an arrowhead in TikZ/PGF?

You can download the two .tex files in the sourceforge link Luigi gave, and put them in the same directory of your main file then you can compile it using pdflatex or xelatex.

\documentclass{minimal}
\usepackage{tikz}
\usetikzlibrary{arrows,arrows.new}
\begin{tikzpicture}[scale=4]
\coordinate (A) at (0,0);
\coordinate (B) at (1,0);
\coordinate (C) at (0,1);
\draw[thick] ($(A)$) -- ($(B)$) -- ($(C)$)-- cycle;

\foreach \x in {1, 2, ..., 10}
\foreach \y in {0, 1, ..., \x}
{\pgfmathsetmacro\arrowHeadsize{sqrt(0.1*(-0.2*\x + 0.1*\y+1)*(-0.2*\x + 0.1*\y+1)
                                 +0.1*(1-0.1*\x)*(1-0.1*\x))}
\draw [thick, -stealth new, blue, arrow head=\arrowHeadsize cm] 
($(B)-(0.1,0)+0.1*(1-\x,\y)$) 
 -- ++(0.02*\x - 0.2, -0.04*\x + 0.4 +0.02*\y-0.2);}
\end{tikzpicture}
\end{document}

The output is as follows:

A major difference between above solution with the solution using PGFplots's quiver is that, by default, quiver scales the thickness of the arrow's body as well, here using this approach the arrow thickness is unchanged, only the size of the arrow heads are changed.