Draw a plane in space and a coordinate system using tikz

Question

Suppose I have a plane $E: a*x_1 + b*x_2 + c*x_3 = d$ (a,..,d should be concrete numbers) in space and a point P. Is there a way to draw a 3-d coordinate system, E, P and perpendicular line to E through P?

Answer 1

TikZ has a built-in 3d coordinate system. You can set the directions of the unit vectors with the x, y and z options. Then you could just do the math by hand and draw the points and lines. Or, you can define some macros to do the job:

\documentclass{article}
\usepackage{tikz}

\makeatletter

% Set some defaults 
\tikzset{
    plane max x/.initial=2,
    plane max y/.initial=2,
    plane max z/.initial=2
}

\tikzset{plane/.style={fill opacity=0.5}}

% Define a plane.
% #1 = name of the plane
% #2*x + #3*y + #4*z = #5 is the equation of the plane
\newcommand*\definePlaneByEquation[5]{
    \expandafter\gdef\csname tsx@plane@#1\endcsname{
        \def\tsx@plane@xcoeff{#2}
        \def\tsx@plane@ycoeff{#3}
        \def\tsx@plane@zcoeff{#4}
        \def\tsx@plane@scalar{#5}
    }
}

% Draw a plane.
% The optional first argument is passed as options to TikZ.
% The mandatory second argument is the name of the plane.
\newcommand\drawPlane[2][]{
    \tikzset{plane max x/.get=\tsx@plane@maxx}
    \tikzset{plane max y/.get=\tsx@plane@maxy}
    \tikzset{plane max z/.get=\tsx@plane@maxz}
    \csname tsx@plane@#2\endcsname

    \ifdim\tsx@plane@xcoeff pt=0pt
        \ifdim\tsx@plane@ycoeff pt=0pt
            \ifdim\tsx@plane@zcoeff pt=0pt
                %invalid plane
            \else % x=0, y=0
                \filldraw[plane,#1,shift={(0,0,\tsx@plane@scalar/\tsx@plane@zcoeff)}]
                    (0,0,0) --
                    (\tsx@plane@maxx,0,0) --
                    (\tsx@plane@maxx,\tsx@plane@maxy,0) --
                    (0,\tsx@plane@maxy,0) --
                    cycle;
            \fi
        \else % x=0, y != 0
            \ifdim\tsx@plane@zcoeff pt=0pt % x=0, z=0
                \filldraw[plane,#1,shift={(0,\tsx@plane@scalar/\tsx@plane@ycoeff,0)}]
                    (0,0,0) --
                    (\tsx@plane@maxx,0,0) --
                    (\tsx@plane@maxx,0,\tsx@plane@maxz) --
                    (0,0,\tsx@plane@maxz) --
                    cycle;
            \else % x=0
                \filldraw[plane,#1]
                    (0,\tsx@plane@scalar/\tsx@plane@ycoeff,0) --
                    (0,0,\tsx@plane@scalar/\tsx@plane@zcoeff) --
                    (\tsx@plane@maxx,0,\tsx@plane@scalar/\tsx@plane@zcoeff) --
                    (\tsx@plane@maxx,\tsx@plane@scalar/\tsx@plane@ycoeff,0) --
                    cycle;
            \fi
        \fi
    \else % x!=0
        \ifdim\tsx@plane@ycoeff pt=0pt % x!=0,y=0
            \ifdim\tsx@plane@zcoeff pt=0pt % x!=0,y=0,z=0
                \filldraw[plane,#1,shift={(\tsx@plane@scalar/\tsx@plane@xcoeff,0,0)}]
                    (0,0,0) --
                    (0,0,\tsx@plane@maxz) --
                    (0,\tsx@plane@maxy,\tsx@plane@maxz) --
                    (0,\tsx@plane@maxy,0) --
                    cycle;
            \else % x!=0,y=0,z!=0
                \filldraw[plane,#1]
                    (\tsx@plane@scalar/\tsx@plane@xcoeff,0) --
                    (0,0,\tsx@plane@scalar/\tsx@plane@zcoeff) --
                    (0,\tsx@plane@maxy,\tsx@plane@scalar/\tsx@plane@zcoeff) --
                    (\tsx@plane@scalar/\tsx@plane@xcoeff,\tsx@plane@maxy,0) --
                    cycle;
            \fi
        \else % x!=0,y!=0
            \ifdim\tsx@plane@zcoeff pt=0pt % x!=0,y!=0,z=0
                \filldraw[plane,#1]
                    (\tsx@plane@scalar/\tsx@plane@xcoeff,0) --
                    (0,\tsx@plane@scalar/\tsx@plane@ycoeff,0) --
                    (0,\tsx@plane@scalar/\tsx@plane@ycoeff,\tsx@plane@maxz) --
                    (\tsx@plane@scalar/\tsx@plane@xcoeff,0,\tsx@plane@maxz) --
                    cycle;
            \else % x!=0,y!=0,z!=0
                \filldraw[plane,#1]
                    (\tsx@plane@scalar/\tsx@plane@xcoeff,0,0) --
                    (0,\tsx@plane@scalar/\tsx@plane@ycoeff,0) --
                    (0,0,\tsx@plane@scalar/\tsx@plane@zcoeff) --
                    cycle;
            \fi
        \fi
    \fi
}

% Define a point.
% #1 = name of the point
% (#2,#3,#4) is the location.
% Also creates a coordinate node of name #1 at the location.
\newcommand\definePointByXYZ[4]{
    \coordinate (#1) at (#2,#3,#4);
    \expandafter\gdef\csname tsx@point@#1\endcsname{
        \def\tsx@point@x{#2}
        \def\tsx@point@y{#3}
        \def\tsx@point@z{#4}
    }
}

% Project a point to a plane.
% #1 = name of the new point
% #2 = name of old point
% #3 = name of plane
\newcommand\projectPointToPlane[3]{{
    \csname tsx@point@#2\endcsname
    \csname tsx@plane@#3\endcsname

    % square of norm of the normal vector
    \pgfmathparse{\tsx@plane@xcoeff*\tsx@plane@xcoeff + \tsx@plane@ycoeff*\tsx@plane@ycoeff + \tsx@plane@zcoeff*\tsx@plane@zcoeff}
    \let\nnormsq\pgfmathresult

    % Calculate distance in terms of the (non-normalized) normal vector
    \pgfmathparse{(\tsx@point@x*\tsx@plane@xcoeff + \tsx@point@y*\tsx@plane@ycoeff + \tsx@point@z*\tsx@plane@zcoeff - \tsx@plane@scalar) / \nnormsq}
    \let\distance\pgfmathresult

    % Calculate point
    \pgfmathparse{\tsx@point@x - \distance*\tsx@plane@xcoeff}
    \let\x\pgfmathresult
    \pgfmathparse{\tsx@point@y - \distance*\tsx@plane@ycoeff}
    \let\y\pgfmathresult
    \pgfmathparse{\tsx@point@z - \distance*\tsx@plane@zcoeff}
    \let\z\pgfmathresult

    \definePointByXYZ{#1}{\x}{\y}{\z}
}}
\makeatother

\begin{document}
\begin{tikzpicture}[x={(240:0.8cm)}, y={(-10:1cm)}, z={(0,1cm)},
        plane max z=3]
    \draw[->] (0,0,0) -- (3,0,0);
    \draw[->] (0,0,0) -- (0,3,0);
    \draw[->] (0,0,0) -- (0,0,3);

    \definePlaneByEquation{myplane}{1}{1.5}{0}{2}
    \drawPlane[thick,fill=blue]{myplane}

    \definePointByXYZ{mypoint}{1}{1.5}{2};
    \draw (mypoint) circle [radius=1pt];

    \projectPointToPlane{proj}{mypoint}{myplane}
    \fill (proj) circle [radius=1pt];

    \draw[->, shorten <=1pt,shorten >=1pt] (mypoint) -- (proj);
\end{tikzpicture}
\end{document}

Some comments:

• I hope I didn't mess up some case.
• With the plane max x/y/z options, you can specify how far a plane with zero parameters should extend.

• If you want to draw a line through the point, normal to the plane, you can use

  \definePlaneByEquation{myplane}{1}{1.5}{0}{2}
  \definePointByXYZ{mypoint}{1}{1.5}{2};
  \projectPointToPlane{proj}{mypoint}{myplane}
  
  \draw ($(proj)!-2cm!(mypoint)$) -- (proj);
  \drawPlane[thick,fill=blue]{myplane}
  \draw (proj) -- ($(mypoint)!-2cm!(proj)$);
  

  The order is important to get correct overpainting in case the plane is opaque or the line is non-black.

• The tests for 0 should really be tests for being smaller than epsilon.
• Maybe it would be useful to apply \pgfmathparse to the arguments of the definition macros.