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Projection along affine hull

asked 2016-11-08 13:09:57 -0500

done_with_fish gravatar image

Let n_1,...,n_r be integral points in a polyhedron P. The paper I am reading refers to "the projection of P along aff(n_1,...,n_r)". Here aff(n_1,...,n_r) is the affine hull of n_1,...,n_r. Can I accomplish this projection in sage?

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Could we have a precise definition of what is meant by "projection of P along the affine hull"? This would make the question more precise.

jipilab gravatar imagejipilab ( 2017-10-10 12:55:27 -0500 )edit

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answered 2017-10-10 13:51:09 -0500

jipilab gravatar image

Here is an example. It requires to load the Projection function:

sage: from sage.geometry.polyhedron.plot import Projection

Let's say we have the following polytope:

sage: P = Polyhedron(vertices=[[-2,0,3],[2,0,3],[0,2,1],[0,-2,1]])

and we are interested in the affine space spanned by the points

sage: affine_basis = [vector([1,1,2]),vector([1,-1,2]),vector([-1,-1,2])]

we first get a linear subspace and create the projection matrix:

sage: linear_subspace = [ap - affine_points[0] for ap in affine_basis[1:]]
sage: VS = VectorSpace(QQ,3)
sage: proj_matrix = matrix([sum([v.inner_product(lv)*lv/(lv.inner_product(lv)) for lv in linear_subspace]) for v in VS.basis()]).transpose()

Then, we create a function that will emulate the affine projection (notice the addition of the first element of the affine basis).

sage: def my_proj(x): return proj_matrix*x + affine_basis[0]

Then, we project the polytope and use the transformed coordinates of the vertices to create a new polytope.

sage: proj_p = Projection(P,proj=my_proj)
sage: projected_P = Polyhedron(vertices=proj_p.transformed_coords)

Now, we can see the difference between

sage: P
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 4 vertices
sage: P.vertices()
(A vertex at (-2, 0, 3),
 A vertex at (0, -2, 1),
 A vertex at (0, 2, 1),
 A vertex at (2, 0, 3))

and the resulting polytope:

sage: projected_P
A 2-dimensional polyhedron in QQ^3 defined as the convex hull of 4 vertices
sage: projected_P.vertices()
(A vertex at (0, -2, 2),
 A vertex at (0, 0, 2),
 A vertex at (2, 2, 2),
 A vertex at (2, 4, 2))
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Asked: 2016-11-08 13:09:57 -0500

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Last updated: Oct 10 '17