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Fill points into a pre-rotated convex Dodecahedron

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2

$begingroup$

I am trying to fill a rotated convex polyhedron with points. This functions well in the unrotated case:

pts = Flatten[Table[x, y, z, x, -2, 2, 0.1, y, -2, 2, 0.1, z, -2, 2, 0.1], 2];
inside = Select[pts,Apply@PolyhedronData["Dodecahedron","RegionFunction"]];
Graphics3D[Sphere[inside, 0.1], Axes -> True, AxesLabel -> "x", "y", "z", Ticks -> -2, 2, -2, 2, -2, 2]

Now, I would like to do the same but with a previously rotated polyhedron as it can be displayed with:

Graphics3D[
 GeometricTransformation[PolyhedronData["Dodecahedron", 
 "GraphicsComplex"], RotationMatrix[-36 Degree, 0, 0, 1]], 
 Axes -> True, AxesLabel -> "x", "y", "z", 
 Ticks -> -2, 2, -2, 2, -2, 2]

So far, I have tried using GeometricTransformation and Rotate on PolyhedronData["Cuboctahedron"], which didn't work out.

graphics3d regions computational-geometry polyhedra

share|improve this question

edited May 1 at 16:58

Carl Woll

77.1k3101202

asked May 1 at 15:29

Jeff71Jeff71

253

$endgroup$

add a comment | 

2

$begingroup$

I am trying to fill a rotated convex polyhedron with points. This functions well in the unrotated case:

pts = Flatten[Table[x, y, z, x, -2, 2, 0.1, y, -2, 2, 0.1, z, -2, 2, 0.1], 2];
inside = Select[pts,Apply@PolyhedronData["Dodecahedron","RegionFunction"]];
Graphics3D[Sphere[inside, 0.1], Axes -> True, AxesLabel -> "x", "y", "z", Ticks -> -2, 2, -2, 2, -2, 2]

Now, I would like to do the same but with a previously rotated polyhedron as it can be displayed with:

Graphics3D[
 GeometricTransformation[PolyhedronData["Dodecahedron", 
 "GraphicsComplex"], RotationMatrix[-36 Degree, 0, 0, 1]], 
 Axes -> True, AxesLabel -> "x", "y", "z", 
 Ticks -> -2, 2, -2, 2, -2, 2]

So far, I have tried using GeometricTransformation and Rotate on PolyhedronData["Cuboctahedron"], which didn't work out.

graphics3d regions computational-geometry polyhedra

share|improve this question

edited May 1 at 16:58

Carl Woll

77.1k3101202

asked May 1 at 15:29

Jeff71Jeff71

253

$endgroup$

add a comment | 

$begingroup$

I am trying to fill a rotated convex polyhedron with points. This functions well in the unrotated case:

pts = Flatten[Table[x, y, z, x, -2, 2, 0.1, y, -2, 2, 0.1, z, -2, 2, 0.1], 2];
inside = Select[pts,Apply@PolyhedronData["Dodecahedron","RegionFunction"]];
Graphics3D[Sphere[inside, 0.1], Axes -> True, AxesLabel -> "x", "y", "z", Ticks -> -2, 2, -2, 2, -2, 2]

Now, I would like to do the same but with a previously rotated polyhedron as it can be displayed with:

Graphics3D[
 GeometricTransformation[PolyhedronData["Dodecahedron", 
 "GraphicsComplex"], RotationMatrix[-36 Degree, 0, 0, 1]], 
 Axes -> True, AxesLabel -> "x", "y", "z", 
 Ticks -> -2, 2, -2, 2, -2, 2]

So far, I have tried using GeometricTransformation and Rotate on PolyhedronData["Cuboctahedron"], which didn't work out.

graphics3d regions computational-geometry polyhedra

share|improve this question

edited May 1 at 16:58

Carl Woll

77.1k3101202

asked May 1 at 15:29

Jeff71Jeff71

253

$endgroup$

pts = Flatten[Table[x, y, z, x, -2, 2, 0.1, y, -2, 2, 0.1, z, -2, 2, 0.1], 2];
inside = Select[pts,Apply@PolyhedronData["Dodecahedron","RegionFunction"]];
Graphics3D[Sphere[inside, 0.1], Axes -> True, AxesLabel -> "x", "y", "z", Ticks -> -2, 2, -2, 2, -2, 2]

Graphics3D[
 GeometricTransformation[PolyhedronData["Dodecahedron", 
 "GraphicsComplex"], RotationMatrix[-36 Degree, 0, 0, 1]], 
 Axes -> True, AxesLabel -> "x", "y", "z", 
 Ticks -> -2, 2, -2, 2, -2, 2]

share|improve this question

edited May 1 at 16:58

Carl Woll

77.1k3101202

asked May 1 at 15:29

Jeff71Jeff71

253

share|improve this question

edited May 1 at 16:58

Carl Woll

77.1k3101202

asked May 1 at 15:29

Jeff71Jeff71

253

edited May 1 at 16:58

Carl Woll

77.1k3101202

edited May 1 at 16:58

Carl Woll

77.1k3101202

asked May 1 at 15:29

Jeff71Jeff71

253

asked May 1 at 15:29

Jeff71Jeff71

253

2 Answers
2

active

oldest

votes

5

$begingroup$

If you use a BoundaryMeshRegion, you can transform the region, and then create a RegionMemberFunction from it.

mesh = PolyhedronData["Dodecahedron", "BoundaryMeshRegion"];
transform = TransformedRegion[mesh, RotationTransform[-36 Degree, 0, 0, 1]];
rmf = RegionMember[transform];

Then, use rmf in your Select:

pts = Flatten[Table[x, y, z, x, -2, 2, 0.1, y, -2, 2, 0.1, z, -2, 2, 0.1], 2];
inside = Select[pts, rmf];
Graphics3D[Sphere[inside, 0.1], Axes -> True, AxesLabel -> "x", "y", "z", Ticks -> -2, 2, -2, 2, -2, 2]

It is also possible to use RandomPoint to get random points in the dodecahedron:

Graphics3D[Sphere[RandomPoint[transform, 10000],.1]]

share|improve this answer

answered May 1 at 15:54

Carl WollCarl Woll

77.1k3101202

$endgroup$

add a comment | 

1

$begingroup$

Here's another approach:

reg = Dodecahedron[-36 Degree, 0];
RegionImage[reg, Quiet @ RegionBounds[reg]]

share|improve this answer

answered May 1 at 17:25

Chip HurstChip Hurst

24k15996

$endgroup$

add a comment | 

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5

$begingroup$

If you use a BoundaryMeshRegion, you can transform the region, and then create a RegionMemberFunction from it.

mesh = PolyhedronData["Dodecahedron", "BoundaryMeshRegion"];
transform = TransformedRegion[mesh, RotationTransform[-36 Degree, 0, 0, 1]];
rmf = RegionMember[transform];

Then, use rmf in your Select:

pts = Flatten[Table[x, y, z, x, -2, 2, 0.1, y, -2, 2, 0.1, z, -2, 2, 0.1], 2];
inside = Select[pts, rmf];
Graphics3D[Sphere[inside, 0.1], Axes -> True, AxesLabel -> "x", "y", "z", Ticks -> -2, 2, -2, 2, -2, 2]

It is also possible to use RandomPoint to get random points in the dodecahedron:

Graphics3D[Sphere[RandomPoint[transform, 10000],.1]]

share|improve this answer

answered May 1 at 15:54

Carl WollCarl Woll

77.1k3101202

$endgroup$

add a comment | 

5

$begingroup$

If you use a BoundaryMeshRegion, you can transform the region, and then create a RegionMemberFunction from it.

mesh = PolyhedronData["Dodecahedron", "BoundaryMeshRegion"];
transform = TransformedRegion[mesh, RotationTransform[-36 Degree, 0, 0, 1]];
rmf = RegionMember[transform];

Then, use rmf in your Select:

pts = Flatten[Table[x, y, z, x, -2, 2, 0.1, y, -2, 2, 0.1, z, -2, 2, 0.1], 2];
inside = Select[pts, rmf];
Graphics3D[Sphere[inside, 0.1], Axes -> True, AxesLabel -> "x", "y", "z", Ticks -> -2, 2, -2, 2, -2, 2]

It is also possible to use RandomPoint to get random points in the dodecahedron:

Graphics3D[Sphere[RandomPoint[transform, 10000],.1]]

share|improve this answer

answered May 1 at 15:54

Carl WollCarl Woll

77.1k3101202

$endgroup$

add a comment | 

$begingroup$

If you use a BoundaryMeshRegion, you can transform the region, and then create a RegionMemberFunction from it.

mesh = PolyhedronData["Dodecahedron", "BoundaryMeshRegion"];
transform = TransformedRegion[mesh, RotationTransform[-36 Degree, 0, 0, 1]];
rmf = RegionMember[transform];

Then, use rmf in your Select:

pts = Flatten[Table[x, y, z, x, -2, 2, 0.1, y, -2, 2, 0.1, z, -2, 2, 0.1], 2];
inside = Select[pts, rmf];
Graphics3D[Sphere[inside, 0.1], Axes -> True, AxesLabel -> "x", "y", "z", Ticks -> -2, 2, -2, 2, -2, 2]

It is also possible to use RandomPoint to get random points in the dodecahedron:

Graphics3D[Sphere[RandomPoint[transform, 10000],.1]]

share|improve this answer

answered May 1 at 15:54

Carl WollCarl Woll

77.1k3101202

$endgroup$

mesh = PolyhedronData["Dodecahedron", "BoundaryMeshRegion"];
transform = TransformedRegion[mesh, RotationTransform[-36 Degree, 0, 0, 1]];
rmf = RegionMember[transform];

pts = Flatten[Table[x, y, z, x, -2, 2, 0.1, y, -2, 2, 0.1, z, -2, 2, 0.1], 2];
inside = Select[pts, rmf];
Graphics3D[Sphere[inside, 0.1], Axes -> True, AxesLabel -> "x", "y", "z", Ticks -> -2, 2, -2, 2, -2, 2]

Graphics3D[Sphere[RandomPoint[transform, 10000],.1]]

share|improve this answer

answered May 1 at 15:54

Carl WollCarl Woll

77.1k3101202

answered May 1 at 15:54

Carl WollCarl Woll

77.1k3101202

answered May 1 at 15:54

Carl WollCarl Woll

77.1k3101202

1

$begingroup$

Here's another approach:

reg = Dodecahedron[-36 Degree, 0];
RegionImage[reg, Quiet @ RegionBounds[reg]]

share|improve this answer

answered May 1 at 17:25

Chip HurstChip Hurst

24k15996

$endgroup$

add a comment | 

1

$begingroup$

Here's another approach:

reg = Dodecahedron[-36 Degree, 0];
RegionImage[reg, Quiet @ RegionBounds[reg]]

share|improve this answer

answered May 1 at 17:25

Chip HurstChip Hurst

24k15996

$endgroup$

add a comment | 

$begingroup$

Here's another approach:

reg = Dodecahedron[-36 Degree, 0];
RegionImage[reg, Quiet @ RegionBounds[reg]]

share|improve this answer

answered May 1 at 17:25

Chip HurstChip Hurst

24k15996

$endgroup$

reg = Dodecahedron[-36 Degree, 0];
RegionImage[reg, Quiet @ RegionBounds[reg]]

share|improve this answer

answered May 1 at 17:25

Chip HurstChip Hurst

24k15996

answered May 1 at 17:25

Chip HurstChip Hurst

24k15996

answered May 1 at 17:25

Chip HurstChip Hurst

24k15996