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tutorial:coordinates [2011/01/17 09:12] – created joswig | user_guide:tutorials:coordinates [2019/01/25 13:40] – ↷ Page moved from user_guide:coordinates to user_guide:tutorials:coordinates oroehrig | ||
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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: | + | Note that a facet defining hyperplane is not uniquely determined if the polyhedron is not full-dimensional. {{ user_guide: |
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. | ||
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According to this model two points in //W// are identical to polymake if they differ by a positive multiple. In particular, for a polytope point in the input data it is not required that the first coordinate is //1//; it just has to be some positive number. | According to this model two points in //W// are identical to polymake if they differ by a positive multiple. In particular, for a polytope point in the input data it is not required that the first coordinate is //1//; it just has to be some positive number. | ||
- | polymake | + | Up to and including version 2.9.9 polymake |
+ | |||
+ | ===== An example ===== | ||
+ | |||
+ | The following defines the positive orthant in 3-space. | ||
+ | < | ||
+ | polytope | ||
+ | </ | ||
+ | |||
+ | This lists the facet coordinates. | ||
+ | < | ||
+ | polytope > print $p-> | ||
+ | 1 0 0 0 | ||
+ | 0 1 0 0 | ||
+ | 0 0 1 0 | ||
+ | 0 0 0 1 | ||
+ | </ | ||
+ | |||
+ | Each line describes one linear inequality. | ||
+ | |||
+ | Clearly, | ||
+ | < | ||
+ | polytope > print $p-> | ||
+ | 0 | ||
+ | </ | ||
+ | |||
+ | Yet, the combinatorial data describe | ||
+ | < | ||
+ | polytope | ||
+ | {1 2 3} | ||
+ | {0 2 3} | ||
+ | {0 1 3} | ||
+ | {0 1 2} | ||
+ | </ | ||
+ | |||
+ | The rays span the //face at infinity//. | ||
+ | < | ||
+ | polytope > print $p-> | ||
+ | {1 2 3} | ||
+ | </ | ||
+ | |||
+ | By the way, unbounded polyhedra can be visualized just like bounded ones. '' | ||
+ | < | ||
+ | polytope > $p-> | ||
+ | </ | ||
+ | ===== Internal treatment | ||
+ | As described above polyhedra in '' | ||
+ | |||
+ | Until version 2.9.9 input generators with a negative first coordinate are just multiplied by -1. |