[Tex/LaTex] 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}