# CUBE+PLANES = POLYHEDRONS

I have an cube and an collention of planes ( normal and one point, Ax = b ). how can I get the polyhedrons formed by the intersection of the cube and the planes?

CUBE+PLANES = POLYHEDRONS

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Thanks for the reponse: in the above solution, the plane intersects or cut to the cube generating two polyhedra, so how could get these polihedros to manipulate?

I don't understand your question. The intersection of two convex sets is always a single convex set (perhaps empty).

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Define the cube as a polyhedron (its a canned example):

```
sage: cube = polytopes.n_cube(3)
sage: cube.Hrepresentation()
(An inequality (0, 0, -1) x + 1 >= 0, An inequality (0, -1, 0) x + 1 >= 0, An inequality (-1, 0, 0) x + 1 >= 0, An inequality (1, 0, 0) x + 1 >= 0, An inequality (0, 0, 1) x + 1 >= 0, An inequality (0, 1, 0) x + 1 >= 0)
```

Define the plane as a polyhedron:

```
sage: plane = Polyhedron(eqns=[(0,1,0,0)])
sage: plane.Hrepresentation()
(An equation (1, 0, 0) x + 0 == 0,)
```

Compute the intersection:

```
sage: cube.intersection(plane)
A 2-dimensional polyhedron in QQ^3 defined as the convex hull of 4 vertices
```

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Asked: ** 2012-07-31 16:21:52 +0200 **

Seen: **293 times**

Last updated: **Jul 31 '12**

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