Metapost 中的矢量场

Question 1

新的：相图有 3 种类型：斜率场、方向场、矢量场。在这种情况下，我选择矢量长度标准化的方向场（符合我的口味 ^^，如下所示.1*unit(...)）。

path vector(pair z) {return .1*unit((f(z.x,z.y),g(z.x,z.y)));}

更多：所选框中的剪辑允许根据 OP 的要求填充不变域。

// http://asymptote.ualberta.ca/
size(8cm);
import contour;
import graph;
real f(real x, real y) {return y+x*x-x*(y-1+2*x*x)/4;}
real g(real x, real y) {return -2x*(1+y);}
real p(real x) {return 1-2x^2;}
real a=3.5;
pair A=(-a,-a), B=(a,a);
guide[][] pf=contour(f,A,B, new real[] {0});
guide[][] pg=contour(g,A,B, new real[] {0});
path pp=graph(p,-2,2);

path invdomain=pp--(a,-a)--(a,-1)--(-a,-1)--(-a,-a)--cycle;
fill(invdomain,yellow);

// set up the vector of the vector field
path vector(pair z) {return .1*unit((f(z.x,z.y),g(z.x,z.y)));}
// set up the condition for the vector will be drawn
bool cond(pair z) {
if (g(z.x,z.y)==0) {return false;}   
else {return true;}  
}
add(vectorfield(vector,A,B,28,cond,paleblue,Arrow()));

draw(pf[0],blue);
draw(pg[0],magenta);
draw(pp,magenta);

clip(box(A,B));
xaxis("$x$",BottomTop,LeftTicks);
yaxis("$y$",LeftRight,RightTicks);

shipout(bbox(5mm,invisible));

老的：渐近线方法：隐函数的图形/轮廓/水平曲线，矢量场中的矢量（带条件）。

// http://asymptote.ualberta.ca/
size(8cm);
import contour;
import graph;
real f(real x, real y) {return y+x*x-x*(y-1+2*x*x)/4;}
real g(real x, real y) {return -2x*(1+y);}
real a=3.5;
pair A=(-a,-a), B=(a,a);

guide[][] pf=contour(f,A,B, new real[] {0});
guide[][] pg=contour(g,A,B, new real[] {0});

// set up the vector of the vector field
path vector(pair z) {return (f(z.x,z.y),g(z.x,z.y));}

// set up the condition for the vector will be drawn
bool cond(pair z) {
if (g(z.x,z.y)==0) {return false;}   
else {return true;}  
}

add(vectorfield(vector,A,B,20,cond,blue,Arrow()));

draw(pf[0],orange);
draw(pg[0],magenta);

//xaxis("$x$",BottomTop,LeftTicks);
//yaxis("$y$",LeftRight,RightTicks);
shipout(bbox(5mm,invisible));

Answer

新的：相图有 3 种类型：斜率场、方向场、矢量场。在这种情况下，我选择矢量长度标准化的方向场（符合我的口味 ^^，如下所示.1*unit(...)）。

path vector(pair z) {return .1*unit((f(z.x,z.y),g(z.x,z.y)));}

更多：所选框中的剪辑允许根据 OP 的要求填充不变域。

// http://asymptote.ualberta.ca/
size(8cm);
import contour;
import graph;
real f(real x, real y) {return y+x*x-x*(y-1+2*x*x)/4;}
real g(real x, real y) {return -2x*(1+y);}
real p(real x) {return 1-2x^2;}
real a=3.5;
pair A=(-a,-a), B=(a,a);
guide[][] pf=contour(f,A,B, new real[] {0});
guide[][] pg=contour(g,A,B, new real[] {0});
path pp=graph(p,-2,2);

path invdomain=pp--(a,-a)--(a,-1)--(-a,-1)--(-a,-a)--cycle;
fill(invdomain,yellow);

// set up the vector of the vector field
path vector(pair z) {return .1*unit((f(z.x,z.y),g(z.x,z.y)));}
// set up the condition for the vector will be drawn
bool cond(pair z) {
if (g(z.x,z.y)==0) {return false;}   
else {return true;}  
}
add(vectorfield(vector,A,B,28,cond,paleblue,Arrow()));

draw(pf[0],blue);
draw(pg[0],magenta);
draw(pp,magenta);

clip(box(A,B));
xaxis("$x$",BottomTop,LeftTicks);
yaxis("$y$",LeftRight,RightTicks);

shipout(bbox(5mm,invisible));

老的：渐近线方法：隐函数的图形/轮廓/水平曲线，矢量场中的矢量（带条件）。

// http://asymptote.ualberta.ca/
size(8cm);
import contour;
import graph;
real f(real x, real y) {return y+x*x-x*(y-1+2*x*x)/4;}
real g(real x, real y) {return -2x*(1+y);}
real a=3.5;
pair A=(-a,-a), B=(a,a);

guide[][] pf=contour(f,A,B, new real[] {0});
guide[][] pg=contour(g,A,B, new real[] {0});

// set up the vector of the vector field
path vector(pair z) {return (f(z.x,z.y),g(z.x,z.y));}

// set up the condition for the vector will be drawn
bool cond(pair z) {
if (g(z.x,z.y)==0) {return false;}   
else {return true;}  
}

add(vectorfield(vector,A,B,20,cond,blue,Arrow()));

draw(pf[0],orange);
draw(pg[0],magenta);

//xaxis("$x$",BottomTop,LeftTicks);
//yaxis("$y$",LeftRight,RightTicks);
shipout(bbox(5mm,invisible));

Question 2

我不确定这是否是您想要的，但您可以使用 Metapost 执行以下操作：

s := 1.5cm ;

vardef u(expr x,y) = (y+x*x-0.25*x*(y-1+2*x*x)) enddef ;
vardef v(expr x,y) = (-2*x*(1+y)) enddef ;

path tmparr ;
for x = -3.5 step 0.3 until 3.5 :
    for y = -3.5 step 0.3 until 3.5 :
        if ((x,y) = (0,0)):
        else:
            tmparr := (origin -- unitvector (u(x,y), v(x,y))) scaled 0.28s ;
            tmparr := tmparr shifted - center tmparr ;
            drawarrow tmparr shifted ((x,y) scaled s) withcolor 0.37white;
        fi;
    endfor ;
endfor ;


% parabola

vardef f(expr x) = (1-2*x*x) enddef ;

path tmppara ; 
tmppara = (-1.5, f(-1.5)) for x = -1.5 step 0.01 until 1.5: .. (x, f(x)) endfor ;
draw tmppara scaled s withcolor red ;


% line

draw (-3.5,-1) scaled s -- (3.5,-1) scaled s withcolor red ;

使用 PGFplots 的方法是：

\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}

\begin{document}

\begin{tikzpicture}
\begin{axis}[domain=-3.5:3.5, ymin=-3.5, view={0}{90}]
\addplot3 [quiver={
        u={(y+x*x-0.25*x*(y-1+2*x*x))/
            veclen(-2*x*(1+y), y+x*x-0.25*x*(y-1+2*x*x))}, 
        v={(-2*x*(1+y))/
            veclen(-2*x*(1+y), y+x*x-0.25*x*(y-1+2*x*x))}, 
        scale arrows=0.28
    }, -stealth, samples=20] {0};
\addplot [smooth, red] {1-2*x*x};
\addplot [smooth, red] {-1};
\end{axis}
\end{tikzpicture}

\end{document}

我不确定抛物线和直线是否正确，并且箭筒可能需要一些调整......

在这种情况下可能没有必要，但通常要绘制隐函数，可以使用gnuplotPGF 机制（也许可以浏览此站点以获取更多示例）：

\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}

\begin{document}

\begin{tikzpicture}
\begin{axis}[domain=-3.5:3.5, ymin=-3.5, ymax=3.5, view={0}{90}]
\addplot3 [quiver={
        u={(y+x*x-0.25*x*(y-1+2*x*x))/
            veclen(-2*x*(1+y), y+x*x-0.25*x*(y-1+2*x*x))}, 
        v={(-2*x*(1+y))/
            veclen(-2*x*(1+y), y+x*x-0.25*x*(y-1+2*x*x))}, 
        scale arrows=0.28
    }, -stealth, samples=20] {0};
\addplot [smooth, red] {1-2*x*x};
\addplot [smooth, red] {-1};

\addplot3 [raw gnuplot, smooth, blue] gnuplot {
      set contour base;
      set cntrparam levels discrete 0.003;
      unset surface;
      splot y+x*x-0.25*x*(y-1+2*x*x);
    };
\end{axis}
\end{tikzpicture}

\end{document}

至于 Metapost，我不得不承认，我不确定在没有参数化的情况下是否可以轻松绘制隐函数。

Answer