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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?

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How do you want the plane to be drawn? – Caramdir Jun 5 '11 at 17:42
Perhaps one need to distinguish different cases: If the plane intersects all three coordinate axis, I want to connect those intersection points and draw the resulting triangle. If this is not the case I want to have something like this – student Jun 5 '11 at 18:06

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:



% Set some defaults 
    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
    \expandafter\gdef\csname tsx@plane@#1\endcsname{

% Draw a plane.
% The optional first argument is passed as options to TikZ.
% The mandatory second argument is the name of the plane.
    \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
                    (0,0,0) --
                    (\tsx@plane@maxx,0,0) --
                    (\tsx@plane@maxx,\tsx@plane@maxy,0) --
                    (0,\tsx@plane@maxy,0) --
        \else % x=0, y != 0
            \ifdim\tsx@plane@zcoeff pt=0pt % x=0, z=0
                    (0,0,0) --
                    (\tsx@plane@maxx,0,0) --
                    (\tsx@plane@maxx,0,\tsx@plane@maxz) --
                    (0,0,\tsx@plane@maxz) --
            \else % x=0
                    (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) --
    \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
                    (0,0,0) --
                    (0,0,\tsx@plane@maxz) --
                    (0,\tsx@plane@maxy,\tsx@plane@maxz) --
                    (0,\tsx@plane@maxy,0) --
            \else % x!=0,y=0,z!=0
                    (\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) --
        \else % x!=0,y!=0
            \ifdim\tsx@plane@zcoeff pt=0pt % x!=0,y!=0,z=0
                    (\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) --
            \else % x!=0,y!=0,z!=0
                    (\tsx@plane@scalar/\tsx@plane@xcoeff,0,0) --
                    (0,\tsx@plane@scalar/\tsx@plane@ycoeff,0) --
                    (0,0,\tsx@plane@scalar/\tsx@plane@zcoeff) --

% 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.
    \coordinate (#1) at (#2,#3,#4);
    \expandafter\gdef\csname tsx@point@#1\endcsname{

% Project a point to a plane.
% #1 = name of the new point
% #2 = name of old point
% #3 = name of plane
    \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}

    % 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}

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


\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);


    \draw (mypoint) circle [radius=1pt];

    \fill (proj) circle [radius=1pt];

    \draw[->, shorten <=1pt,shorten >=1pt] (mypoint) -- (proj);

project a point to a plane

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

    \draw ($(proj)!-2cm!(mypoint)$) -- (proj);
    \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.
share|improve this answer
Fine! It's a good starting point. I started a package about 3D geometry and this answer is interesting. – Alain Matthes Jun 6 '11 at 8:40
@Astramundus, do you know the tikz-3dplot package? I use it all the time, might also be a good source for your 3d package? Check it out at: archive.cs.uu.nl/mirror/CTAN/graphics/pgf/contrib/tikz-3dplot/… – romeovs Jun 6 '11 at 8:48
@Altermundus: I was wondering whether I should invest more time in this and add more things for 3D geometry (e.g. intersect various things) or if you are already working on a 3D version of TkZ. – Caramdir Jun 6 '11 at 15:56
@romeovs: tikz-3dplot is nice, but as far as I am aware it doesn't have features to define planes and do intersections and such. – Caramdir Jun 6 '11 at 15:58
@romeovs Yes :) but the author of the tikz-3dplot package uses some of my ideas. The author found interesting my "annotated-3d-box" example on texample.net: texample.net/tikz/examples/annotated-3d-box – Alain Matthes Jun 6 '11 at 16:14

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