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tutorial:polytope_semantics [2011/11/23 17:56] – moved case distinction for VERTICES to extra paragraph joswig | user_guide:tutorials:polytope_semantics [2019/02/04 22:55] (current) – external edit 127.0.0.1 | ||
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- | ===== Semantics of Cones and Polytopes ===== | + | {{page>.:latest:@FILEID@}} |
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- | ==== General Remarks ==== | + | |
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- | The general semantics of a big object in polymake is as follows: | + | |
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- | All big objects are immutable as mathematical objects. | + | |
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- | As an example an object of class Polytope defined by '' | + | |
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- | If a user asks for a property which cannot be derived this property is set to undef. | + | |
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- | ===== Objects of type '' | + | |
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- | Polytope theory is nice because this is where combinatorics meets metric geometry. | + | |
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- | ==== With coordinates: | + | |
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- | Being non-empty is recorded in the property '' | + | |
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- | A non-empty polytope in R^n is encoded as its homogenization in R^{n+1}. | + | |
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- | FIXME: cdd does not return the facet [1,0,0,...] in redundancy checks. It does so in the dual description. As our model should be completely equivalent in the primal and dual description we will try to use the dual description of cdd for redundancy both in the primal and dual case. | + | |
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- | ==== Without coordinates: Combinatorics ==== | + | |
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- | '' | + | |
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- | Each property must clearly specify if it depends on the geometry or only on the combinatorics. | + | |
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- | ==== Special Cases ==== | + | |
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- | Most of what comes below is a consequence of the design decisions explained above. | + | |
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- | === Empty polytopes === | + | |
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- | With the introduction of the '' | + | |
- | This is a bit subtle as the cone over an empty polytope does not have a canonical definition. | + | |
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- | However, this was changed for the reason that often people generate systems of inequalities and then look at the feasible region. | + | |
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- | An empty polytope is recognized by '' | + | |
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- | === Zero-dimensional polytopes === | + | |
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- | A zero-dimensional polytope is a single point. | + | |
- | '' | + | |
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- | Such a polytope is both simple and simplicial, i.e. it is a simplex. | + | |
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- | ==== Summing Up ==== | + | |
- | For instance we have four possibilities which can occur for '' | + | |
- | * does not exist (it is not listed in '' | + | |
- | * exists and is set to '' | + | |
- | * exists and is empty: So the polytope is empty. | + | |
- | * exists and is neither set to '' | + | |