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29 | 29 |
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30 | 30 | where |
31 | 31 |
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32 | | - * **vertices** `v_1`, `\dots`, `v_k` are a finite number of |
| 32 | + * **vertices** `v_1,\dots,v_k` are a finite number of |
33 | 33 | points. Each vertex is specified by an arbitrary vector, and two |
34 | 34 | points are equal if and only if the vector is the same. |
35 | 35 |
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36 | | - * **rays** `r_1`, `\dots`, `r_m` are a finite number of directions |
| 36 | + * **rays** `r_1,\dots,r_m` are a finite number of directions |
37 | 37 | (directions of infinity). Each ray is specified by a non-zero |
38 | 38 | vector, and two rays are equal if and only if the vectors are |
39 | 39 | the same up to rescaling with a positive constant. |
40 | 40 |
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41 | | - * **lines** `\ell_1`, `\dots`, `\ell_n` are a finite number of |
| 41 | + * **lines** `\ell_1,\dots,\ell_n` are a finite number of |
42 | 42 | unoriented directions. In other words, a line is equivalent to |
43 | 43 | the set `\{r, -r\}` for a ray `r`. Each line is specified by a |
44 | 44 | non-zero vector, and two lines are equivalent if and only if the |
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159 | 159 |
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160 | 160 | The base ring of the polyhedron can be specified by the ``base_ring`` |
161 | 161 | optional keyword argument. If not specified, a suitable common base |
162 | | -ring for all coordinates/coefficients will be chosen |
| 162 | +ring for all coordinates and coefficients will be chosen |
163 | 163 | automatically. Important cases are: |
164 | 164 |
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165 | | -* ``base_ring=QQ`` uses a fast implementation for exact rational |
166 | | - numbers. |
| 165 | +* ``base_ring=QQ`` uses a fast implementation for exact rational numbers. |
167 | 166 |
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168 | 167 | * ``base_ring=ZZ`` is similar to ``QQ``, but the resulting polyhedron |
169 | 168 | object will have extra methods for lattice polyhedra. |
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174 | 173 | Polyhedra with symmetries often are defined over some algebraic field |
175 | 174 | extension of the rationals. As a simple example, consider the |
176 | 175 | equilateral triangle whose vertex coordinates involve `\sqrt{3}`. An |
177 | | -exact way to work with roots in Sage is the :mod:`Algebraic Real Field |
178 | | -<sage.rings.qqbar>` :: |
| 176 | +exact way to work with roots in Sage is the |
| 177 | +:mod:`Algebraic Real Field <sage.rings.qqbar>` :: |
179 | 178 |
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180 | 179 | sage: triangle = Polyhedron([(0,0), (1,0), (1/2, sqrt(3)/2)], base_ring=AA) # optional - sage.rings.number_field # optional - sage.symbolic |
181 | 180 | sage: triangle.Hrepresentation() # optional - sage.rings.number_field # optional - sage.symbolic |
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