Size problems when plotting xy/(x^2+2y^2)












8















I would like to plot the funtion xy/(x^2+2y^2) using PGFPlots. Here is what I want:



What I want



Please consider this MWE:



documentclass{article}
usepackage[english]{babel}
usepackage[utf8]{inputenc}
usepackage[T1]{fontenc}
usepackage[a4paper,margin=1in,footskip=0.25in]{geometry}
usepackage{amssymb}
usepackage{amsmath}

usepackage{pgfplots}
pgfplotsset{compat=1.15}
pgfplotsset{soldot/.style={color=black,only marks,mark=*}}
pgfplotsset{holdot/.style={color=red,fill=white,very thick,only marks,mark=*}}

begin{document}
begin{center}
begin{tikzpicture}[declare function={f(x,y)=(x*y)/(x*x+2*y*y);}]
begin{axis} [
axis on top,
axis equal image,
axis lines=center,
xlabel=$x$,
ylabel=$y$,
zlabel=$z$,
zmin=-1,
zmax=1,
ztick={-1,0,0.33,1},
zticklabels={$-1$,$0$,$1/3$,$1$},
ticklabel style={font=tiny},
legend pos=outer north east,
legend style={cells={align=left}},
legend cell align={left},
view={-135}{25},
]
addplot3[surf,mesh/ordering=y varies,shader=interp,domain=-1:1,domain y=-1:1,samples=61, samples y=61] {f(x,y)};;
end{axis}
end{tikzpicture}
end{center}

end{document}


What I have done



The MWE output has an incredible big zoom, so I would like to resize the plot but not using scale but another commands, like enlarge limits. However, all the results are in vain; I can not reproduce the visual appearance of what I want.



Thanks!!










share|improve this question




















  • 1





    Are you sure you want to plot the function in this way? If you use polar coordinates in the x-y plane, x=r cos(phi) and y=r sin(phi), you see that the function does not depend on r but only on the angle. This explains the behavior at 0, where the function is not well-defined. And otherwise the function depends only on one variable, so I am wondering if you would be better off if you plotted a function of one variable only, or at least use a different parametrization.

    – marmot
    Feb 24 at 7:02











  • @marmot please see the edit. The function has a "normal" behaviour when changing domain y=-1.2:1 to domain y=-1:1. If you want to use change of variables go ahead :). Do you know how to "enlarge" the axis without rescaling the entire function in order to make it a little more bigger?

    – manooooh
    Feb 24 at 7:05








  • 1





    Normally you can set width=15cm or something like this. Of course, with axis equal image, one needs to be a bit careful. What I meant to say is that the function is not well-defined at x=y=0 and otherwise only a function that depends on one variable, not on two. You see this actually rather well in the upper plot.

    – marmot
    Feb 24 at 7:11






  • 1





    I need to sleep so I will just post some 1d plot.

    – marmot
    Feb 24 at 7:28






  • 1





    Please do not alter the question that essentially by editing. It is much better to ask a new question.

    – TeXnician
    Feb 24 at 7:45
















8















I would like to plot the funtion xy/(x^2+2y^2) using PGFPlots. Here is what I want:



What I want



Please consider this MWE:



documentclass{article}
usepackage[english]{babel}
usepackage[utf8]{inputenc}
usepackage[T1]{fontenc}
usepackage[a4paper,margin=1in,footskip=0.25in]{geometry}
usepackage{amssymb}
usepackage{amsmath}

usepackage{pgfplots}
pgfplotsset{compat=1.15}
pgfplotsset{soldot/.style={color=black,only marks,mark=*}}
pgfplotsset{holdot/.style={color=red,fill=white,very thick,only marks,mark=*}}

begin{document}
begin{center}
begin{tikzpicture}[declare function={f(x,y)=(x*y)/(x*x+2*y*y);}]
begin{axis} [
axis on top,
axis equal image,
axis lines=center,
xlabel=$x$,
ylabel=$y$,
zlabel=$z$,
zmin=-1,
zmax=1,
ztick={-1,0,0.33,1},
zticklabels={$-1$,$0$,$1/3$,$1$},
ticklabel style={font=tiny},
legend pos=outer north east,
legend style={cells={align=left}},
legend cell align={left},
view={-135}{25},
]
addplot3[surf,mesh/ordering=y varies,shader=interp,domain=-1:1,domain y=-1:1,samples=61, samples y=61] {f(x,y)};;
end{axis}
end{tikzpicture}
end{center}

end{document}


What I have done



The MWE output has an incredible big zoom, so I would like to resize the plot but not using scale but another commands, like enlarge limits. However, all the results are in vain; I can not reproduce the visual appearance of what I want.



Thanks!!










share|improve this question




















  • 1





    Are you sure you want to plot the function in this way? If you use polar coordinates in the x-y plane, x=r cos(phi) and y=r sin(phi), you see that the function does not depend on r but only on the angle. This explains the behavior at 0, where the function is not well-defined. And otherwise the function depends only on one variable, so I am wondering if you would be better off if you plotted a function of one variable only, or at least use a different parametrization.

    – marmot
    Feb 24 at 7:02











  • @marmot please see the edit. The function has a "normal" behaviour when changing domain y=-1.2:1 to domain y=-1:1. If you want to use change of variables go ahead :). Do you know how to "enlarge" the axis without rescaling the entire function in order to make it a little more bigger?

    – manooooh
    Feb 24 at 7:05








  • 1





    Normally you can set width=15cm or something like this. Of course, with axis equal image, one needs to be a bit careful. What I meant to say is that the function is not well-defined at x=y=0 and otherwise only a function that depends on one variable, not on two. You see this actually rather well in the upper plot.

    – marmot
    Feb 24 at 7:11






  • 1





    I need to sleep so I will just post some 1d plot.

    – marmot
    Feb 24 at 7:28






  • 1





    Please do not alter the question that essentially by editing. It is much better to ask a new question.

    – TeXnician
    Feb 24 at 7:45














8












8








8


0






I would like to plot the funtion xy/(x^2+2y^2) using PGFPlots. Here is what I want:



What I want



Please consider this MWE:



documentclass{article}
usepackage[english]{babel}
usepackage[utf8]{inputenc}
usepackage[T1]{fontenc}
usepackage[a4paper,margin=1in,footskip=0.25in]{geometry}
usepackage{amssymb}
usepackage{amsmath}

usepackage{pgfplots}
pgfplotsset{compat=1.15}
pgfplotsset{soldot/.style={color=black,only marks,mark=*}}
pgfplotsset{holdot/.style={color=red,fill=white,very thick,only marks,mark=*}}

begin{document}
begin{center}
begin{tikzpicture}[declare function={f(x,y)=(x*y)/(x*x+2*y*y);}]
begin{axis} [
axis on top,
axis equal image,
axis lines=center,
xlabel=$x$,
ylabel=$y$,
zlabel=$z$,
zmin=-1,
zmax=1,
ztick={-1,0,0.33,1},
zticklabels={$-1$,$0$,$1/3$,$1$},
ticklabel style={font=tiny},
legend pos=outer north east,
legend style={cells={align=left}},
legend cell align={left},
view={-135}{25},
]
addplot3[surf,mesh/ordering=y varies,shader=interp,domain=-1:1,domain y=-1:1,samples=61, samples y=61] {f(x,y)};;
end{axis}
end{tikzpicture}
end{center}

end{document}


What I have done



The MWE output has an incredible big zoom, so I would like to resize the plot but not using scale but another commands, like enlarge limits. However, all the results are in vain; I can not reproduce the visual appearance of what I want.



Thanks!!










share|improve this question
















I would like to plot the funtion xy/(x^2+2y^2) using PGFPlots. Here is what I want:



What I want



Please consider this MWE:



documentclass{article}
usepackage[english]{babel}
usepackage[utf8]{inputenc}
usepackage[T1]{fontenc}
usepackage[a4paper,margin=1in,footskip=0.25in]{geometry}
usepackage{amssymb}
usepackage{amsmath}

usepackage{pgfplots}
pgfplotsset{compat=1.15}
pgfplotsset{soldot/.style={color=black,only marks,mark=*}}
pgfplotsset{holdot/.style={color=red,fill=white,very thick,only marks,mark=*}}

begin{document}
begin{center}
begin{tikzpicture}[declare function={f(x,y)=(x*y)/(x*x+2*y*y);}]
begin{axis} [
axis on top,
axis equal image,
axis lines=center,
xlabel=$x$,
ylabel=$y$,
zlabel=$z$,
zmin=-1,
zmax=1,
ztick={-1,0,0.33,1},
zticklabels={$-1$,$0$,$1/3$,$1$},
ticklabel style={font=tiny},
legend pos=outer north east,
legend style={cells={align=left}},
legend cell align={left},
view={-135}{25},
]
addplot3[surf,mesh/ordering=y varies,shader=interp,domain=-1:1,domain y=-1:1,samples=61, samples y=61] {f(x,y)};;
end{axis}
end{tikzpicture}
end{center}

end{document}


What I have done



The MWE output has an incredible big zoom, so I would like to resize the plot but not using scale but another commands, like enlarge limits. However, all the results are in vain; I can not reproduce the visual appearance of what I want.



Thanks!!







tikz-pgf






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited Feb 24 at 7:57







manooooh

















asked Feb 24 at 6:34









manoooohmanooooh

1,1941517




1,1941517








  • 1





    Are you sure you want to plot the function in this way? If you use polar coordinates in the x-y plane, x=r cos(phi) and y=r sin(phi), you see that the function does not depend on r but only on the angle. This explains the behavior at 0, where the function is not well-defined. And otherwise the function depends only on one variable, so I am wondering if you would be better off if you plotted a function of one variable only, or at least use a different parametrization.

    – marmot
    Feb 24 at 7:02











  • @marmot please see the edit. The function has a "normal" behaviour when changing domain y=-1.2:1 to domain y=-1:1. If you want to use change of variables go ahead :). Do you know how to "enlarge" the axis without rescaling the entire function in order to make it a little more bigger?

    – manooooh
    Feb 24 at 7:05








  • 1





    Normally you can set width=15cm or something like this. Of course, with axis equal image, one needs to be a bit careful. What I meant to say is that the function is not well-defined at x=y=0 and otherwise only a function that depends on one variable, not on two. You see this actually rather well in the upper plot.

    – marmot
    Feb 24 at 7:11






  • 1





    I need to sleep so I will just post some 1d plot.

    – marmot
    Feb 24 at 7:28






  • 1





    Please do not alter the question that essentially by editing. It is much better to ask a new question.

    – TeXnician
    Feb 24 at 7:45














  • 1





    Are you sure you want to plot the function in this way? If you use polar coordinates in the x-y plane, x=r cos(phi) and y=r sin(phi), you see that the function does not depend on r but only on the angle. This explains the behavior at 0, where the function is not well-defined. And otherwise the function depends only on one variable, so I am wondering if you would be better off if you plotted a function of one variable only, or at least use a different parametrization.

    – marmot
    Feb 24 at 7:02











  • @marmot please see the edit. The function has a "normal" behaviour when changing domain y=-1.2:1 to domain y=-1:1. If you want to use change of variables go ahead :). Do you know how to "enlarge" the axis without rescaling the entire function in order to make it a little more bigger?

    – manooooh
    Feb 24 at 7:05








  • 1





    Normally you can set width=15cm or something like this. Of course, with axis equal image, one needs to be a bit careful. What I meant to say is that the function is not well-defined at x=y=0 and otherwise only a function that depends on one variable, not on two. You see this actually rather well in the upper plot.

    – marmot
    Feb 24 at 7:11






  • 1





    I need to sleep so I will just post some 1d plot.

    – marmot
    Feb 24 at 7:28






  • 1





    Please do not alter the question that essentially by editing. It is much better to ask a new question.

    – TeXnician
    Feb 24 at 7:45








1




1





Are you sure you want to plot the function in this way? If you use polar coordinates in the x-y plane, x=r cos(phi) and y=r sin(phi), you see that the function does not depend on r but only on the angle. This explains the behavior at 0, where the function is not well-defined. And otherwise the function depends only on one variable, so I am wondering if you would be better off if you plotted a function of one variable only, or at least use a different parametrization.

– marmot
Feb 24 at 7:02





Are you sure you want to plot the function in this way? If you use polar coordinates in the x-y plane, x=r cos(phi) and y=r sin(phi), you see that the function does not depend on r but only on the angle. This explains the behavior at 0, where the function is not well-defined. And otherwise the function depends only on one variable, so I am wondering if you would be better off if you plotted a function of one variable only, or at least use a different parametrization.

– marmot
Feb 24 at 7:02













@marmot please see the edit. The function has a "normal" behaviour when changing domain y=-1.2:1 to domain y=-1:1. If you want to use change of variables go ahead :). Do you know how to "enlarge" the axis without rescaling the entire function in order to make it a little more bigger?

– manooooh
Feb 24 at 7:05







@marmot please see the edit. The function has a "normal" behaviour when changing domain y=-1.2:1 to domain y=-1:1. If you want to use change of variables go ahead :). Do you know how to "enlarge" the axis without rescaling the entire function in order to make it a little more bigger?

– manooooh
Feb 24 at 7:05






1




1





Normally you can set width=15cm or something like this. Of course, with axis equal image, one needs to be a bit careful. What I meant to say is that the function is not well-defined at x=y=0 and otherwise only a function that depends on one variable, not on two. You see this actually rather well in the upper plot.

– marmot
Feb 24 at 7:11





Normally you can set width=15cm or something like this. Of course, with axis equal image, one needs to be a bit careful. What I meant to say is that the function is not well-defined at x=y=0 and otherwise only a function that depends on one variable, not on two. You see this actually rather well in the upper plot.

– marmot
Feb 24 at 7:11




1




1





I need to sleep so I will just post some 1d plot.

– marmot
Feb 24 at 7:28





I need to sleep so I will just post some 1d plot.

– marmot
Feb 24 at 7:28




1




1





Please do not alter the question that essentially by editing. It is much better to ask a new question.

– TeXnician
Feb 24 at 7:45





Please do not alter the question that essentially by editing. It is much better to ask a new question.

– TeXnician
Feb 24 at 7:45










1 Answer
1






active

oldest

votes


















7





+50









Not an answer to the (LaTeX part of the) question. However, if you use polar coordinates in the x-y plane, x = r cos(ϕ) and y = r sin(ϕ), you see that the function does not depend on r but only on the angle. So away from the origin x = y = 0 all the information is already in a one-dimensional plot.



documentclass[tikz,border=3.14mm]{standalone}
usepackage{pgfplots}
pgfplotsset{compat=1.15}
begin{document}
begin{tikzpicture}[declare function={fan(t)=-(sin(2*t)/(-3 + cos(2*t)));}]
begin{axis}
addplot[domain=0:360,smooth,samples=101] {fan(x)};
end{axis}
end{tikzpicture}
end{document}


enter image description here



And this yields a 3d smooth plot.



documentclass[tikz,border=3.14mm]{standalone}

usepackage{pgfplots}
pgfplotsset{compat=1.15}
pgfplotsset{soldot/.style={color=black,only marks,mark=*}}
pgfplotsset{holdot/.style={color=red,fill=white,very thick,only marks,mark=*}}

begin{document}
begin{tikzpicture}[declare function={f(x,y)=(x*y)/(x*x+2*y*y);
fan(t)=-(sin(2*t)/(-3 + cos(2*t)));}]
begin{axis} [width=18cm,
axis on top,
axis equal image,
axis lines=center,
xlabel=$x$,
ylabel=$y$,
zlabel=$z$,
zmin=-1,
zmax=1,
ztick={-1,0,0.33,1},
zticklabels={$-1$,$0$,$1/3$,$1$},
ticklabel style={font=tiny},
legend pos=outer north east,
legend style={cells={align=left}},
legend cell align={left},
view={-135}{25},
data cs=polar,
]

addplot3[surf,mesh/ordering=y varies,shader=interp,domain=0:360,
domain y=0:1,samples=61, samples y=21,
z buffer=sort] { fan(x)};
addlegendentry{{$f(x,y)$}}
end{axis}
end{tikzpicture}
end{document}


enter image description here






share|improve this answer


























  • I edited your answer a bit (you can go to the revisions to see what I changed). If it is not good, be free to roll back to the previous version!

    – JouleV
    Feb 27 at 9:32












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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









7





+50









Not an answer to the (LaTeX part of the) question. However, if you use polar coordinates in the x-y plane, x = r cos(ϕ) and y = r sin(ϕ), you see that the function does not depend on r but only on the angle. So away from the origin x = y = 0 all the information is already in a one-dimensional plot.



documentclass[tikz,border=3.14mm]{standalone}
usepackage{pgfplots}
pgfplotsset{compat=1.15}
begin{document}
begin{tikzpicture}[declare function={fan(t)=-(sin(2*t)/(-3 + cos(2*t)));}]
begin{axis}
addplot[domain=0:360,smooth,samples=101] {fan(x)};
end{axis}
end{tikzpicture}
end{document}


enter image description here



And this yields a 3d smooth plot.



documentclass[tikz,border=3.14mm]{standalone}

usepackage{pgfplots}
pgfplotsset{compat=1.15}
pgfplotsset{soldot/.style={color=black,only marks,mark=*}}
pgfplotsset{holdot/.style={color=red,fill=white,very thick,only marks,mark=*}}

begin{document}
begin{tikzpicture}[declare function={f(x,y)=(x*y)/(x*x+2*y*y);
fan(t)=-(sin(2*t)/(-3 + cos(2*t)));}]
begin{axis} [width=18cm,
axis on top,
axis equal image,
axis lines=center,
xlabel=$x$,
ylabel=$y$,
zlabel=$z$,
zmin=-1,
zmax=1,
ztick={-1,0,0.33,1},
zticklabels={$-1$,$0$,$1/3$,$1$},
ticklabel style={font=tiny},
legend pos=outer north east,
legend style={cells={align=left}},
legend cell align={left},
view={-135}{25},
data cs=polar,
]

addplot3[surf,mesh/ordering=y varies,shader=interp,domain=0:360,
domain y=0:1,samples=61, samples y=21,
z buffer=sort] { fan(x)};
addlegendentry{{$f(x,y)$}}
end{axis}
end{tikzpicture}
end{document}


enter image description here






share|improve this answer


























  • I edited your answer a bit (you can go to the revisions to see what I changed). If it is not good, be free to roll back to the previous version!

    – JouleV
    Feb 27 at 9:32
















7





+50









Not an answer to the (LaTeX part of the) question. However, if you use polar coordinates in the x-y plane, x = r cos(ϕ) and y = r sin(ϕ), you see that the function does not depend on r but only on the angle. So away from the origin x = y = 0 all the information is already in a one-dimensional plot.



documentclass[tikz,border=3.14mm]{standalone}
usepackage{pgfplots}
pgfplotsset{compat=1.15}
begin{document}
begin{tikzpicture}[declare function={fan(t)=-(sin(2*t)/(-3 + cos(2*t)));}]
begin{axis}
addplot[domain=0:360,smooth,samples=101] {fan(x)};
end{axis}
end{tikzpicture}
end{document}


enter image description here



And this yields a 3d smooth plot.



documentclass[tikz,border=3.14mm]{standalone}

usepackage{pgfplots}
pgfplotsset{compat=1.15}
pgfplotsset{soldot/.style={color=black,only marks,mark=*}}
pgfplotsset{holdot/.style={color=red,fill=white,very thick,only marks,mark=*}}

begin{document}
begin{tikzpicture}[declare function={f(x,y)=(x*y)/(x*x+2*y*y);
fan(t)=-(sin(2*t)/(-3 + cos(2*t)));}]
begin{axis} [width=18cm,
axis on top,
axis equal image,
axis lines=center,
xlabel=$x$,
ylabel=$y$,
zlabel=$z$,
zmin=-1,
zmax=1,
ztick={-1,0,0.33,1},
zticklabels={$-1$,$0$,$1/3$,$1$},
ticklabel style={font=tiny},
legend pos=outer north east,
legend style={cells={align=left}},
legend cell align={left},
view={-135}{25},
data cs=polar,
]

addplot3[surf,mesh/ordering=y varies,shader=interp,domain=0:360,
domain y=0:1,samples=61, samples y=21,
z buffer=sort] { fan(x)};
addlegendentry{{$f(x,y)$}}
end{axis}
end{tikzpicture}
end{document}


enter image description here






share|improve this answer


























  • I edited your answer a bit (you can go to the revisions to see what I changed). If it is not good, be free to roll back to the previous version!

    – JouleV
    Feb 27 at 9:32














7





+50







7





+50



7




+50





Not an answer to the (LaTeX part of the) question. However, if you use polar coordinates in the x-y plane, x = r cos(ϕ) and y = r sin(ϕ), you see that the function does not depend on r but only on the angle. So away from the origin x = y = 0 all the information is already in a one-dimensional plot.



documentclass[tikz,border=3.14mm]{standalone}
usepackage{pgfplots}
pgfplotsset{compat=1.15}
begin{document}
begin{tikzpicture}[declare function={fan(t)=-(sin(2*t)/(-3 + cos(2*t)));}]
begin{axis}
addplot[domain=0:360,smooth,samples=101] {fan(x)};
end{axis}
end{tikzpicture}
end{document}


enter image description here



And this yields a 3d smooth plot.



documentclass[tikz,border=3.14mm]{standalone}

usepackage{pgfplots}
pgfplotsset{compat=1.15}
pgfplotsset{soldot/.style={color=black,only marks,mark=*}}
pgfplotsset{holdot/.style={color=red,fill=white,very thick,only marks,mark=*}}

begin{document}
begin{tikzpicture}[declare function={f(x,y)=(x*y)/(x*x+2*y*y);
fan(t)=-(sin(2*t)/(-3 + cos(2*t)));}]
begin{axis} [width=18cm,
axis on top,
axis equal image,
axis lines=center,
xlabel=$x$,
ylabel=$y$,
zlabel=$z$,
zmin=-1,
zmax=1,
ztick={-1,0,0.33,1},
zticklabels={$-1$,$0$,$1/3$,$1$},
ticklabel style={font=tiny},
legend pos=outer north east,
legend style={cells={align=left}},
legend cell align={left},
view={-135}{25},
data cs=polar,
]

addplot3[surf,mesh/ordering=y varies,shader=interp,domain=0:360,
domain y=0:1,samples=61, samples y=21,
z buffer=sort] { fan(x)};
addlegendentry{{$f(x,y)$}}
end{axis}
end{tikzpicture}
end{document}


enter image description here






share|improve this answer















Not an answer to the (LaTeX part of the) question. However, if you use polar coordinates in the x-y plane, x = r cos(ϕ) and y = r sin(ϕ), you see that the function does not depend on r but only on the angle. So away from the origin x = y = 0 all the information is already in a one-dimensional plot.



documentclass[tikz,border=3.14mm]{standalone}
usepackage{pgfplots}
pgfplotsset{compat=1.15}
begin{document}
begin{tikzpicture}[declare function={fan(t)=-(sin(2*t)/(-3 + cos(2*t)));}]
begin{axis}
addplot[domain=0:360,smooth,samples=101] {fan(x)};
end{axis}
end{tikzpicture}
end{document}


enter image description here



And this yields a 3d smooth plot.



documentclass[tikz,border=3.14mm]{standalone}

usepackage{pgfplots}
pgfplotsset{compat=1.15}
pgfplotsset{soldot/.style={color=black,only marks,mark=*}}
pgfplotsset{holdot/.style={color=red,fill=white,very thick,only marks,mark=*}}

begin{document}
begin{tikzpicture}[declare function={f(x,y)=(x*y)/(x*x+2*y*y);
fan(t)=-(sin(2*t)/(-3 + cos(2*t)));}]
begin{axis} [width=18cm,
axis on top,
axis equal image,
axis lines=center,
xlabel=$x$,
ylabel=$y$,
zlabel=$z$,
zmin=-1,
zmax=1,
ztick={-1,0,0.33,1},
zticklabels={$-1$,$0$,$1/3$,$1$},
ticklabel style={font=tiny},
legend pos=outer north east,
legend style={cells={align=left}},
legend cell align={left},
view={-135}{25},
data cs=polar,
]

addplot3[surf,mesh/ordering=y varies,shader=interp,domain=0:360,
domain y=0:1,samples=61, samples y=21,
z buffer=sort] { fan(x)};
addlegendentry{{$f(x,y)$}}
end{axis}
end{tikzpicture}
end{document}


enter image description here







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edited Feb 27 at 9:32









JouleV

8,64222154




8,64222154










answered Feb 24 at 7:29









marmotmarmot

112k5144270




112k5144270













  • I edited your answer a bit (you can go to the revisions to see what I changed). If it is not good, be free to roll back to the previous version!

    – JouleV
    Feb 27 at 9:32



















  • I edited your answer a bit (you can go to the revisions to see what I changed). If it is not good, be free to roll back to the previous version!

    – JouleV
    Feb 27 at 9:32

















I edited your answer a bit (you can go to the revisions to see what I changed). If it is not good, be free to roll back to the previous version!

– JouleV
Feb 27 at 9:32





I edited your answer a bit (you can go to the revisions to see what I changed). If it is not good, be free to roll back to the previous version!

– JouleV
Feb 27 at 9:32


















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