Taking in a $3\times3$ matrix and then graphing its vectors

Question

I am trying to graph various vectors of a $3\times3$ matrix to make a box by writing a function that takes in a $3\times3$ matrix and then plots the polygons created from it. Let the 3 column vectors be $\vec{A}$, $\vec{B}$, and $\vec{C}$, then for example one face of the box is given by the polygon

P1 = Polygon[O, A, A + B, B]

where O is the origin. There are 5 additional polygons that make up the surface of the box; however, I can't figure out how to set up function correctly. Here is my code so far:

plotBox[{ {a_, b_, c_}, {d_, e_, f_ }, {g_, h_, i_} }] := 
P1 = Polygon[{{0, 0, 0}, {a, d, g}, {a + b, d + e, g + h}, {b, e, 
h}}]    
Graphics3D[{P1}]

Now if I call plotBox and pass it a $3\times3$ matrix, it will simply output the Polygon that was created. However, if I go to where I actually wrote the function and hit Shift+Enter, then it will Graphics3D the plotBox I entered in. What exactly am I doing wrong here?

Answer 1

The main problem with your code was that you got P1 in there with your function definition.

plotBox[{{a_, b_, c_}, {d_, e_, f_}, {g_, h_, i_}}] := 
  Polygon[{{0, 0, 0}, {a, d, g}, {a + b, d + e, g + h}, {b, e, h}}]

will work to get you one of the polygons.

For a more general function to get all six polygons, you can do

plotbox[matrix_] := With[{trmat = Transpose @ matrix}, 
  Polygon /@ 
   Transpose[
    {#, Total @ trmat - #} & /@ {{0, 0, 0}, #1, #1 + #2, #2}
   ] & @@@ Subsets[trmat, {2}]
 ]

Then

m = RandomReal[1, {3, 3}];
p1 = plotbox[m];
Graphics3D[{PointSize[Large], Point[{{0, 0, 0}, Total @ Transpose @ m}], p1}]

A brief explanation

plotbox might not be particularly intuitive to read, but it actually breaks down quite simply. First, suppose that our matrix is

matrix = {{a, b, c}, {d, e, f}, {g, h, i}};

Then trmat = Transpose @ matrix just gives a list of the column vectors (if you want to plot the row vector box, just leave out the Transpose), and

Subsets[trmat, {2}]

(* { {{a, d, g}, {b, e, h}}, 
     {{a, d, g}, {c, f, i}}, 
     {{b, e, h}, {c, f, i}} } *)

gives all the vector pairs. Each pair is used to construct two polygons. The vertices for one polygon from the first pair are

{{0, 0, 0}, {a, d, g}, {a + b, d + e, g + h}, {b, e, h}}

(as in the OP's original function). The second set of vertices are these same points subtracted from the total of all the column vectors of matrix:

Total @ trmat (* or Total /@ matrix *)

(* {a + b + c, d + e + f, g + h + i} *)

Then plotbox sets up the pure function

Transpose[{#, Total @ trmat - #} & /@ {{0, 0, 0}, #1, #1 + #2, #2}] &

which, when applied to a pair of points in Subsets[trmat, {2}] (corresponding to the slots #1 and #2), gives the vertices of the two polygons derived from that pair. The @@@ simply Applys the function to each pair, giving a list of polygon vertices. Polygon /@ then turns these vertices into actual polygons.

Answer 2

If a visualization is all that's wanted, use Parallelepiped[]:

BlockRandom[SeedRandom[128]; (* for reproducibility *)
            Graphics3D[Parallelepiped[{0, 0, 0}, Transpose[RandomReal[1, {3, 3}]]]]]

(Compare with the result of Graphics3D[plotbox[RandomReal[1, {3, 3}]]] as in aardvark's answer.)

If you really need the polygons, you can start with PolyhedronData["Cube"] instead:

BlockRandom[SeedRandom[128]; 
            MapAt[(# + 1/2).Transpose[RandomReal[1, {3, 3}]] &, 
                  PolyhedronData["Cube", "GraphicsComplex"], {1}] // Normal]
   {{Polygon[{{1.65941, 1.11438, 1.2699}, {0.872272, 0.730417, 1.00725},
              {0.0192757, 0.71586, 0.166596}, {0.80641, 1.09982, 0.429248}}], 
     Polygon[{{1.65941, 1.11438, 1.2699}, {0.80641, 1.09982, 0.429248},
              {0.787135, 0.383961, 0.262652}, {1.64013, 0.398517, 1.1033}}], 
     Polygon[{{1.65941, 1.11438, 1.2699}, {1.64013, 0.398517, 1.1033},
              {0.852996, 0.0145563, 0.840653}, {0.872272, 0.730417, 1.00725}}], 
     Polygon[{{0.872272, 0.730417, 1.00725}, {0.852996, 0.0145563, 0.840653},
              {0., 0., 0.}, {0.0192757, 0.71586, 0.166596}}], 
     Polygon[{{0., 0., 0.}, {0.852996, 0.0145563, 0.840653},
              {1.64013, 0.398517, 1.1033}, {0.787135, 0.383961, 0.262652}}], 
     Polygon[{{0.0192757, 0.71586, 0.166596}, {0., 0., 0.},
              {0.787135, 0.383961, 0.262652}, {0.80641, 1.09982, 0.429248}}]}}

and then evaluating Graphics3D[%] should yield the expected picture.