How to visualize a map $f:R^3 \to R^2$

Question

I want to visualize $\{x_1,x_2,x_3\}\to\{x_1 + 2 x_2, 3 x_3-x_1\}$,but we cannot use ParametricPlot like following,because this map have three parameters.

ParametricPlot[{x1+2x2,3x3-x1},{x1,-10,10},{x2,-10,10},{x3,-10,10}]

We will get some error informations.The current method is plot some discrete points to know it:

Graphics[Point[
  Level[Table[{x1 + 2 x2, 3 x3 - x1}, {x1, -10, 10, .5}, {x2, -10, 
     10, .5}, {x3, -10, 10, .5}], {3}]]]

Are there better solution can visualize it not just by some discrete points?

Answer 1

Let's show how a unit cube is projected, that is quite explanatory:

Table[
  ParametricPlot[{x1 + 2 x2, 3 x3 - x1}, {#, 0, 1}, {#2, 0, 1}], 
  {#3, {0, 1}}
] & @@@ {{x1, x2, x3}, {x1, x3, x2}, {x2, x3, x1}} // Flatten // Show[
   #,
   Graphics @  Table[
      Inset[{##}, {#1 + 2 #2, 3 #3 - #1}] & @@ p, 
      {p, Tuples[{0, 1}, {3}]}
   ],
   PlotRange -> All, Axes -> False, PlotRangePadding -> Scaled[.1]
] &

We can try to use ViewMatrix to show it too:

p = N @ {{1, 2, 0, 0}, {-1, 0, 3, 0}, {0, 0, 1, 0}, {0, 0, 0, 30}};

{x1, x2, x3} = IdentityMatrix[3];

 t = N @ TransformationMatrix @ TranslationTransform[3 {1, 1, 2}];

Graphics3D[{
  Thick,
  FaceForm@Opacity@.3, EdgeForm@Thick, Cuboid[{1, 1, 1}], Cuboid[],
  Sphere[{-1, -1, -1}]
  },
 Boxed -> True, ViewMatrix -> {t, p}, 
 PlotRange -> 3
 ]

Answer 2

I am not sure is this an answer you are looking for, but the following graph does visualize the mapping.

cubePoints3D = 
  Flatten[Table[{x1 , x2,  x3 }, {x1, -10, 10, 5}, {x2, -10, 
          10, 5}, {x3, -10, 10, 5}], 2];
cubePoints2D = 
  Function[{x1, x2, x3}, {x1 + 2 x2, 3 x3 - x1}] @@@ cubePoints3D;
offset = {0, 0, 50};
cubePoints2Dto3D = Map[Append[#, 0] + offset &, cubePoints2D];

Graphics3D[{GrayLevel[0.4],
  Line[cubePoints2Dto3D],
  PointSize[0.02],
  MapIndexed[{Blend[{Blue, Red, Yellow}, #2[[1]]/
       Length[cubePoints2Dto3D]], Point[#1]} &, cubePoints3D],
  Gray, FaceForm[None], Red, PointSize[0.02], 
  MapIndexed[{Blend[{Blue, Red, Yellow}, #2[[1]]/
       Length[cubePoints2Dto3D]], Point[#1]} &, cubePoints2Dto3D],
  Black, MapIndexed[Text[#2[[1]], #1, 2 {1, 1}] &, cubePoints2Dto3D],
  GrayLevel[0.9], 
  MapThread[Arrow[{#1, #2}] &, {cubePoints3D, cubePoints2Dto3D}],
  Black, MapIndexed[
   Text[Style[#2[[1]], Background -> White], #1, 0 {1, 1}] &, 
   cubePoints3D]}, Boxed -> False, ImageSize -> 1000]

And here is a variation with more points, no labels, and a sample of arrows:

Answer 3

How about a vector field approach?

VectorPlot3D[{x1 + 2 x2, 3 x3 - x1, 0}, {x1, -10, 10}, {x2, -10, 
  10}, {x3, -10, 10}]

Animate[
 VectorPlot[{x1 + 2 x2, 3 x3 - x1}, {x1, -10, 10}, {x2, -10, 10}, 
  PlotLabel -> StringTemplate["x3 = ``"][x3]],
 {x3, -10, 10}
 ]

Animate[
 StreamPlot[{x1 + 2 x2, 3 x3 - x1}, {x1, -10, 10}, {x2, -10, 10}, 
  PlotLabel -> StringTemplate["x3 = ``"][x3]],
 {x3, -10, 10}
 ]

Answer 4

There are three other ways to do this. Here we use another region instead of Cuboid[].

• Method 1

Clear[reg, sol];
reg = ImplicitRegion[
   x^6 - 5 x^4 y + 3 x^4 y^2 + 10 x^2 y^3 + 3 x^2 y^4 - y^5 + y^6 + 
     z^2 <= 1, {x, y, z}];
(* reg=Cuboid[] *) 
sol = Resolve[
   Exists[{x1, x2, x3}, {x1, x2, x3} ∈ 
     reg , {u == x1 + 2 x2 , v == 3 x3 - x1}], Reals];
Grid[{{RegionPlot3D[reg, PlotPoints -> 30], 
   RegionPlot[sol, {u, -5, 5}, {v, -5, 5}, PlotPoints -> 30]}}]

• Method 2

Clear[reg,plot];
reg = ImplicitRegion[
   x^6 - 5 x^4 y + 3 x^4 y^2 + 10 x^2 y^3 + 3 x^2 y^4 - y^5 + y^6 + 
     z^2 <= 1, {x, y, z}];
(*reg=Cuboid[];*)
plot = RegionPlot3D[reg, PlotPoints -> 30];
{plot, Show[
  plot /. {x1_Real, x2_Real, x3_Real} :> {x1 + 2 x2, 3 x3 - x1, 0}, 
  ViewPoint -> {0, 0, 1}, ViewProjection -> "Orthographic", 
  Boxed -> False]}

DynamicModule[{vp = {1.3`, -2.4`, 2.`}, 
  vv = {0, 0, 1}}, {plot = 
   RegionPlot3D[reg, PlotPoints -> 30, ViewPoint -> Dynamic[vp], 
    ViewVertical -> Dynamic@vv, SphericalRegion -> True], 
  Show[plot /. {x1_Real, x2_Real, x3_Real} :> {x1 + 2 x2, 3 x3 - x1, 
      0}, ViewPoint -> Dynamic[vp], ViewProjection -> "Orthographic", 
   Boxed -> False]}]

The DynamicModule visualize method come from Rotating few figures in the same manner by hand

• Method 3 (Only work for some simple solid such as Cuboid.

ParametricRegion[{{x1 + 2 x2, -x1 + 3 x3}, {x1, x2, x3} ∈ 
    Cuboid[]}, {x1, x2, x3}] // Region

Answer 5

In version 13, this requirement is very easy to implement:

VectorDisplacementPlot3D[({{1, 2, 0}, {-1, 0, 3}, {0, 0, 0}} - 
    0.999 IdentityMatrix[3]) . {x, y, z}, {x, -1, 1}, {y, -1, 
  1}, {z, -1, 1}, VectorSizes -> Full, VectorPoints -> "Boundary"]