Plotting Planes as Surfaces in Mathematica

Copyright © 1997, 2001 by James F. Hurley, University of Connecticut, Department of Mathematics, Unit 3009, Storrs CT 06269-3009.  All rights reserved.

1. Planes whose equations are solvable for z as a function of x and y. As the text mentions, if c ~= 0 it is easy to solve the equation ax + by + cz = d of a plane for z as a function of x and y: z = (d -- ax -- by)/d. Mathematica can easily plot such a plane, as the following routine illustrates for the plane with equation --12x + 9y + 11z = 9. Note that it passes through the point (2, 0, 3) and has normal vector n = (--12, 9, 11) = --12 i + 9 j + 11 k.To see the plot, hit the Enter key at the extreme lower right of the keyboard, or press the Shift and Return keys simultaneously.

            (* Mathematica Routine to plot graph of a plane   
       ax + by + cz = d, where c is not 0 *)
     a = -12;
     b = 9;
     c = 11;
     x0 = 2;
     y0 = 0;
     z0 = 3;
     F[x_, y_] := (a x0 + b y0 + c z0 - a x - b y)/c;
     Plot3D[ F[x, y], {x, -2, 5}, {y, -2, 10},
             AxesLabel -> {"x", "y", "z"} ]

      The following more elaborate routine adds coordinate axes (in red).

          (* Mathematica Routine to plot graph of a plane   
      ax + by + cz = d, where c is not 0, with red
      coordinate axes *)
          
   a = -12;
   b = 9;
   c = 11;
   x0 = 2;
   y0 = 0;
   z0 = 3;
   F[x_, y_] := (a x0 + b y0 + c z0 - a x - b y)/c;
   planeplot = Graphics3D[
                 Plot3D[ F[x, y], {x, -2, 5}, {y, -2, 10},
                         AxesLabel -> {"x", "y", "z"} ]
                        ];
   coordaxes = Graphics3D[{
                          {RGBColor[1, 0, 0],
                           Line[{{-2, 0, 0}, {5, 0, 0}}],
                           Text["x", {6, 0, 0}]},
                          {RGBColor[1, 0, 0],
                           Line[{{0, -2, 0}, {0, 10, 0}}],
                           Text["y", {0, 11, 0}]},
                          {RGBColor[1, 0, 0],
                           Line[{{0, 0, -2}, {0, 0, 9}}],
                           Text["z", {0, 0, 11}]}
                            }];              
   Show[planeplot, coordaxes]                          
   

2. Plotting a vertical plane.  The equation of a plane perpendicular to the xy-plane has no z-term. Thus, it is not possible to plot such a plane as the graph of an equation z = F(x, y). However, the following simple parametric approach produces a Mathematica plot, and provides a preview of the important idea of parametric representation of surfaces. The plane has normal vector n = a i + b j + 0 k = (a, b, 0), so the normal-form equation of the plane is
     

                  (a i + b j) · [(x -- x i + (y -- y j + (z -- z k] = 0 ,
                                    
                                           a(x -- x + b(y -- y = 0
                                                  
                                        ax + by  = a + b
                                                   
(1)                                     ax + by = d ,                                        where d = n ·

            
In (1), not both a and b are zero, because the normal vector n is nonzero. Thus, one of them is expressible in terms of the other. Suppose that b is nonzero. Then y = d/b -- ax/b = (d -- ax)/b. Since the equation of the plane does not involve z, it can vary over all real numbers. The following triple of parametric equations thus describes the plane:

        x = u, y = (d -- au)/b, z = v, where  u ∈ (-- ∞, +∞)  and  v ∈ (-- ∞, +∞).
                   
Note that there are two parameters, reflecting the fact that a plane has two dimensions, rather than the single dimension of lines and curves. The following Mathematica routine plots the plane through P(1, 3, 2) with normal vector n = 3 i -- j = (3, -- 1, 0).  Since d = n · for this plane d = (3, -- 1, 0) · (1, 3, 2) = 3 -- 3 = 0.  Execute the routine to see the graph of the plane!

(* Mathematica Routine to plot graph of a plane   
   ax + by = d, where b is not 0 *)
   
 a := 3;
 b := -1;
 c := 0;
 x0 := 1;
 y0 := 3;
 z0 := 2;
 d := Dot[{a, b, c}, {x0, y0, z0}]
 ParametricPlot3D[{u, (d - a u)/b, v}, {u, -4, 4},
                  {v, -2, 8},AxesLabel -> {x, y, z} ]