| Both sides previous revision Previous revision Next revision | Previous revisionLast revisionBoth sides next revision |
| tutorial:coordinates [2011/05/18 07:40] – [An example] joswig | user_guide:tutorials:coordinates [2019/01/25 13:40] – ↷ Page moved from user_guide:coordinates to user_guide:tutorials:coordinates oroehrig |
|---|
| Points from both sections can now be identified with infinite rays through the origin in //W//. Facets are identified with the a hyperplane containing the image of the facet in //V// and the origin in //W//. This hyperplane is represented by a normal vector. | Points from both sections can now be identified with infinite rays through the origin in //W//. Facets are identified with the a hyperplane containing the image of the facet in //V// and the origin in //W//. This hyperplane is represented by a normal vector. |
| |
| Note that a facet defining hyperplane is not uniquely determined if the polyhedron is not full-dimensional. {{ tutorial:coord.gif?326|}} | Note that a facet defining hyperplane is not uniquely determined if the polyhedron is not full-dimensional. {{ user_guide:coord.gif?326|}} |
| |
| A vertex is incident with a facet if and only if the scalar product of their representatives in //W// is zero. | A vertex is incident with a facet if and only if the scalar product of their representatives in //W// is zero. |
| 0 0 0 1 | 0 0 0 1 |
| </code> | </code> |
| | |
| | Each line describes one linear inequality. The encoding is as follows: (a_0,a_1,...,a_d) is the inequality a_0 + a_1 x_1 + ... + a_d x_d >= 0. This way a point in (oriented) homogeneous coordinates satisfies an inequality if and only if the scalar product of the point with the inequality gives a non-negative value. (Use the command ''print_constraints($m)'' to display the inequalities of the matrix ''$m'' in a nice way.) |
| |
| Clearly, the polyhedron is unbounded. | Clearly, the polyhedron is unbounded. |
| polytope > $p->VISUAL; | polytope > $p->VISUAL; |
| </code> | </code> |
| ===== Internal treatment of polytope generators (svn version only) ===== | ===== Internal treatment of polytope generators ===== |
| As described above polyhedra in ''polymake'' are modelled as the intersection of a cone with the affine hyperplane defined by //x<sub>0</sub>=1//. Hence, infinitely many cones give rise to the same polytope. The algorithms in ''polymake'' usually work with the //homogenized cone// ''homog(P)'' of a polyhedron. Hence, ''polymake'' takes care about the correct canonicalization of user input of polytope generators in the following way:\\ In order to construct ''homog(P)'', the cone defining the polyhedron is intersected with the hyperplane //H<sub>0</sub>: x<sub>0</sub>=0//. The rays defining the bounded part (R<sub>b</sub>) and rays with //x<sub>0</sub>=0// (R<sub>0</sub>) are just inherited. To obtain the rest of the generators for the unbounded part, it is necessary to carry out a "dual Fourier-Motzkin procedure": Any two rays with different signs are linearly combined to a new ray that is contained in //H<sub>0</sub>//. All these rays together with the rays in R<sub>b</sub> and R<sub>0</sub> then define the //homogenized cone// ''homog(P)''. | As described above polyhedra in ''polymake'' are modelled as the intersection of a cone with the affine hyperplane defined by //x<sub>0</sub>=1//. Hence, infinitely many cones give rise to the same polytope. The algorithms in ''polymake'' usually work with the //homogenized cone// ''homog(P)'' of a polyhedron. Hence, ''polymake'' takes care about the correct canonicalization of user input of polytope generators in the following way:\\ In order to construct ''homog(P)'', the cone defining the polyhedron is intersected with the hyperplane //H<sub>0</sub>: x<sub>0</sub>=0//. The rays defining the bounded part (R<sub>b</sub>) and rays with //x<sub>0</sub>=0// (R<sub>0</sub>) are just inherited. To obtain the rest of the generators for the unbounded part, it is necessary to carry out a "dual Fourier-Motzkin procedure": Any two rays with different signs are linearly combined to a new ray that is contained in //H<sub>0</sub>//. All these rays together with the rays in R<sub>b</sub> and R<sub>0</sub> then define the //homogenized cone// ''homog(P)''. |
| |
| Until version 2.9.9 input generators with a negative first coordinate are just multiplied by -1. | Until version 2.9.9 input generators with a negative first coordinate are just multiplied by -1. |