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Plot a polyhedron/region trapped between 4 planes?

Amir Baghban

Posted 7 years ago

I need to plot the region trapped between 4 planes x = y = z = x + y + z - 1 = 0. Here, is the code that I used: RegionPlot3D[ContourPlot3D[x == 0, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}], ContourPlot3D[y == 0, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}], ContourPlot3D[z == 0, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}], ContourPlot3D[x + y + z - 1 == 0, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}]] But, there are other extra bits that I do not know how to delete them.

I need to plot the region trapped between 4 planes x = y = z = x + y + z - 1 = 0. Here, is the code that I used:

RegionPlot3D[ContourPlot3D[x == 0, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}], 
 ContourPlot3D[y == 0, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}], 
 ContourPlot3D[z == 0, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}],
 ContourPlot3D[x + y + z - 1 == 0, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}]]

But, there are other extra bits that I do not know how to delete them.

POSTED BY: Amir Baghban

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Valeriu Ungureanu

Valeriu Ungureanu, Moldova State University

Posted 7 years ago

Do you want to plot this one? RegionPlot3D[x + y + z <= 1, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}] Or, may be this one? Graphics3D[{EdgeForm[{Thick, Blue}], FaceForm[{Pink, Opacity[0.7]}], Tetrahedron[{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {0, 0, 0}}]}, Boxed -> False]

Do you want to plot this one?

RegionPlot3D[x + y + z <= 1, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}]

enter image description here

Or, may be this one?

Graphics3D[{EdgeForm[{Thick, Blue}], FaceForm[{Pink, Opacity[0.7]}], 
  Tetrahedron[{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {0, 0, 0}}]}, 
 Boxed -> False]

enter image description here

POSTED BY: Valeriu Ungureanu

Amir Baghban

Posted 7 years ago

Dear Valeriu, I would like to plot a polyhedron trapped between four arbitrary surfaces not only a polyhedron between the coordinate surfaces and other one.

POSTED BY: Amir Baghban

Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 7 years ago

As suggested by Valeriu, you can use `Tetrahedron` if you give the vertices: vertices = RandomReal[{0, 1}, {4, 3}]; Graphics3D[Tetrahedron[vertices]] Graphics3D[Map[InfinitePlane, Subsets[vertices, {3}]], PlotRange -> CoordinateBounds[vertices], PlotRangePadding -> Scaled[.2]] If you start from the equations of the 4 faces, you can extract the vertices by solving the equations first: equations = {x == 0, y == 0, z == 0, x + y + z - 1 == 0}; vertices = Map[{x, y, z} /. First@Solve[#] &, Subsets[equations, {3}]]; Graphics3D[Tetrahedron[vertices]]

As suggested by Valeriu, you can use Tetrahedron if you give the vertices:

vertices = RandomReal[{0, 1}, {4, 3}];
Graphics3D[Tetrahedron[vertices]]
Graphics3D[Map[InfinitePlane, Subsets[vertices, {3}]], 
 PlotRange -> CoordinateBounds[vertices], 
 PlotRangePadding -> Scaled[.2]]

If you start from the equations of the 4 faces, you can extract the vertices by solving the equations first:

equations = {x == 0, y == 0, z == 0, x + y + z - 1 == 0};
vertices = Map[{x, y, z} /. First@Solve[#] &, Subsets[equations, {3}]];
Graphics3D[Tetrahedron[vertices]]

POSTED BY: Gianluca Gorni

Valeriu Ungureanu

Valeriu Ungureanu, Moldova State University

Posted 7 years ago

Dear Amir, You may use the following approach: RegionPlot3D[ Subscript[x, 1] + Subscript[x, 2] + Subscript[x, 3] <= 3 && Subscript[x, 1] - Subscript[x, 2] >= -1 && Subscript[x, 2] - Subscript[x, 3] >= -1 && Subscript[x, 3] >= 0, {Subscript[x, 1], -2, 4}, {Subscript[x, 2], -2, 4}, {Subscript[x, 3], -2, 4}, PlotPoints -> 75, PlotStyle -> Directive[Orange, Opacity[0.5]], Mesh -> None]

Dear Amir,

You may use the following approach:

RegionPlot3D[
 Subscript[x, 1] + Subscript[x, 2] + Subscript[x, 3] <= 3 && 
  Subscript[x, 1] - Subscript[x, 2] >= -1 && 
  Subscript[x, 2] - Subscript[x, 3] >= -1 && 
  Subscript[x, 3] >= 0, {Subscript[x, 1], -2, 4}, {Subscript[x, 
  2], -2, 4}, {Subscript[x, 3], -2, 4}, PlotPoints -> 75, 
 PlotStyle -> Directive[Orange, Opacity[0.5]], Mesh -> None]

enter image description here

POSTED BY: Valeriu Ungureanu

Amir Baghban

Posted 7 years ago

Dear friends, sorry for my special question of my general one. I think my specialization was not suitable. My main problem is that: I have a set of some planes in 3D. I want to plot the region trapped between them (if there exists) and the quantity "4" was an example for my general question.

POSTED BY: Amir Baghban

Valeriu Ungureanu

Valeriu Ungureanu, Moldova State University

Posted 7 years ago

You may use the same approach, e.g.: RegionPlot3D[ x^2 + y^2 + z^2 <= 1 && -x + y + Sin[z]^2 <= 1 && x - y + z <= 1 && x + y - z <= 1 && -x - y + z <= 1 && -x + y - z <= 1 && x - y - z <= 1 && -x^2 - y - z^2 <= 1, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, PlotPoints -> 75, PlotStyle -> Directive[Orange, Opacity[0.777]], Mesh -> None] Nevertheless, there are some difficulties. If you have 4 surfaces, then the number of possible "polyhedra" is 16. Some of them are not polyhedra but polyhedral sets, i.e.unbounded sets.

You may use the same approach, e.g.:

RegionPlot3D[
 x^2 + y^2 + z^2 <= 1 && -x + y + Sin[z]^2 <= 1 && x - y + z <= 1 && 
  x + y - z <= 1 && -x - y + z <= 1 && -x + y - z <= 1 && 
  x - y - z <= 1 && -x^2 - y - z^2 <= 1, {x, -1, 1}, {y, -1, 
  1}, {z, -1, 1}, PlotPoints -> 75, 
 PlotStyle -> Directive[Orange, Opacity[0.777]], Mesh -> None]

enter image description here

Nevertheless, there are some difficulties. If you have 4 surfaces, then the number of possible "polyhedra" is 16. Some of them are not polyhedra but polyhedral sets, i.e.unbounded sets.

POSTED BY: Valeriu Ungureanu

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