deleted trailing whitespace

src/doc/en/thematic_tutorials/geometry/lectures.rst

@@ -33,7 +33,7 @@ Lecture 0: Basic definitions and constructions
 ==============================================
 
 A real :math:`(k\times d)`-matrix :math:`A` and a real vector :math:`b`
-in :math:`\mathbb{R}^d` define a (convex) **polyhedron** :math:`P` as the set of solutions 
+in :math:`\mathbb{R}^d` define a (convex) **polyhedron** :math:`P` as the set of solutions
 of the system of linear inequalities:
 
 .. MATH::
@@ -43,22 +43,22 @@ of the system of linear inequalities:
 Each row of :math:`A` defines a closed half-space of :math:`\mathbb{R}^d`.
 Hence a polyhedron is the intersection of finitely many closed half-spaces in
 :math:`\mathbb{R}^d`. The matrix :math:`A` may contain equal rows, which may lead to a
-set of *equalities* satisfied by the polyhedron. If there are no redundant rows 
-in the above definition, this definition is refered to as the 
+set of *equalities* satisfied by the polyhedron. If there are no redundant rows
+in the above definition, this definition is refered to as the
 :math:`\mathbf{H}` **-representation** of a polyhedron.
 
 The maximal affine subspace :math:`L` contained in a polyhedron is the
-**lineality** space. Fixing a point :math:`o` of the lineality space to act 
+**lineality** space. Fixing a point :math:`o` of the lineality space to act
 as the *origin*, one can write every point :math:`p` inside a polyhedron as a combination
 
 .. MATH::
 
     p = \ell +\sum_{i=1}^{n}\lambda_iv_i+\sum_{i=1}^{m}\mu_ir_i,
 
-where :math:`\ell\in L` (using :math:`o` as the origin), :math:`\sum_{i=1}^n\lambda_i=1`, 
-:math:`\mu_i\geq0`, and :math:`r_i\neq0` for all :math:`0\leq i\leq m` and the 
-set of :math:`r_i` 's are positively independant (the origin is not in their positive span). 
-There are many equivalent ways write the above, so one asks :math:`n` and :math:`m` 
+where :math:`\ell\in L` (using :math:`o` as the origin), :math:`\sum_{i=1}^n\lambda_i=1`,
+:math:`\mu_i\geq0`, and :math:`r_i\neq0` for all :math:`0\leq i\leq m` and the
+set of :math:`r_i` 's are positively independant (the origin is not in their positive span).
+There are many equivalent ways write the above, so one asks :math:`n` and :math:`m`
 to be minimal with that property.
 
 The points :math:`v_i` 's are called the *vertices* of :math:`P` and the points
@@ -71,7 +71,7 @@ polyhedral cone* generated by finitely many rays.
 When the lineality space and the rays are reduced to a point (i.e. no rays and
 no lines) the object is often refered to as a **polytope**.
 
-.. note :: As mentioned in the documentation of the constructor when typing :code:`Polyhedron?`, 
+.. note :: As mentioned in the documentation of the constructor when typing :code:`Polyhedron?`,
 
     *You may either define it with vertex/ray/line or
     inequalities/equations data, but not both. Redundant data will
@@ -231,7 +231,7 @@ polyhedron over it:
 
 .. end of output
 
-Similarly, it is not possible to create polyhedron objects over :code:`RR` 
+Similarly, it is not possible to create polyhedron objects over :code:`RR`
 (no matter how many bits of precision).
 
 ::
@@ -281,8 +281,8 @@ without having specified the base ring :code:`RDF` by the user.
 `H`-representation
 ------------------
 
-If a polyhedron object was constructed via a :math:`V`-representation, Sage can provide 
-the :math:`H`-representation of the object. 
+If a polyhedron object was constructed via a :math:`V`-representation, Sage can provide
+the :math:`H`-representation of the object.
 
 ::
 
@@ -309,7 +309,7 @@ defined. The :math:`H`-representation may contain equations:
 
 The construction of a polyhedron object via its :math:`H`-representation,
 requires a precise format. Each inequality :math:`(a_{i1}, \dots, a_{id})\cdot
-x + b_i \geq 0` must be written as :code:`[b_i,a_i1, ..., a_id]`. 
+x + b_i \geq 0` must be written as :code:`[b_i,a_i1, ..., a_id]`.
 
 ::
 
@@ -354,20 +354,20 @@ second group of four rows.
 
 .. end of output
 
-Of course, this is a toy example, but it is generally worth to preprocess 
+Of course, this is a toy example, but it is generally worth to preprocess
 the data before defining the polyhedron if possible.
 
 Lecture 1: Representation objects
 ===================================
 
-Many objects are related to the :math:`H`- and :math:`V`-representations. Sage 
+Many objects are related to the :math:`H`- and :math:`V`-representations. Sage
 has classes implemented for them.
 
 `H`-representation
 ------------------
 
 You can store the :math:`H`-representation in a variable and use the
-inequalities and equalities as objects. 
+inequalities and equalities as objects.
 
 ::
 
@@ -409,7 +409,7 @@ as follows.
 
 .. NOTE ::
 
-    It is recommended to use :code:`equations` or :code:`equation_generator` 
+    It is recommended to use :code:`equations` or :code:`equation_generator`
     (and similarly for inequalities) if one wants to iterate over them instead
     of :code:`equations_list`.
 
@@ -465,7 +465,7 @@ as follows.
 
 .. NOTE ::
 
-    It is recommended to use :code:`vertices` or :code:`vertex_generator` 
+    It is recommended to use :code:`vertices` or :code:`vertex_generator`
     (and similarly for rays and lines) if one wants to iterate over them instead
     of :code:`vertices_list`.
 
@@ -491,13 +491,13 @@ These backends offer various functionalities and have their own specific strengt
 
    - This is a :code:`python` backend that provides an implementation of
      polyhedron over irrational coordinates.
- 
+
  - :ref:`sage.geometry.polyhedron.backend_normaliz`, (requires the optional package :code:`pynormaliz`)
 
    - `Normaliz Homepage <https://www.normaliz.uni-osnabrueck.de/>`_
 
 
-The default backend is :code:`ppl`. Whenever one needs **speed** it is good to try out 
+The default backend is :code:`ppl`. Whenever one needs **speed** it is good to try out
 the different backends. The backend :code:`field` is **not** specifically designed
 for dealing with extremal computations but can do computations in exact
 coordinates.
@@ -617,7 +617,7 @@ The default backend for polyhedron objects i :code:`ppl`.
 
 .. end of output
 
-As you see, it does not accepts values in :code:`RDF` and the polyhedron constructor 
+As you see, it does not accepts values in :code:`RDF` and the polyhedron constructor
 used the :code:`cdd` backend.
 
 The :code:`polymake` backend
@@ -647,7 +647,7 @@ An example with quadratic field:
 The :code:`field` backend
 -------------------------
 
-As it turns out, the rational numbers do not suffice to represent all combinatorial 
+As it turns out, the rational numbers do not suffice to represent all combinatorial
 types of polytopes. For example, Perles constructed a `8`-dimensional polytope with
 `12` vertices which does not have a realization with rational coordinates, see
 Example 6.21 p. 172 of [Zie2007]_.
@@ -710,7 +710,7 @@ This backend does not work with :code:`RDF`, or algebraic numbers or the :code:`
 
     sage: P4_normaliz = Polyhedron(vertices = [[sqrt_2, 0], [0, cbrt_2]], backend='normaliz')       # optional - pynormaliz
     Traceback (most recent call last):
-    ... 
+    ...
     ValueError: No such backend (=normaliz) implemented for given basering (=Algebraic Real Field).
 
     sage: P5_normaliz = Polyhedron(vertices = [[sqrt_2s, 0], [0, cbrt_2s]], backend='normaliz')     # optional - pynormaliz
@@ -720,7 +720,7 @@ This backend does not work with :code:`RDF`, or algebraic numbers or the :code:`
 
 .. end of output
 
-The backend :code:`normaliz` provides other methods such as 
+The backend :code:`normaliz` provides other methods such as
 :code:`integral_hull`, which also works on unbounded polyhedron.
 
 ::
@@ -746,7 +746,7 @@ In order to **know all the methods that a polyhedron object has** one has to loo
  - :ref:`sage.geometry.polyhedron.base_RDF`
 
 Don't be surprised if the classes look empty! The classes mainly contain private
-methods that implement some comparison methods: to verify equality and inequality 
+methods that implement some comparison methods: to verify equality and inequality
 of numbers in the base ring and other internal functionalities.
 
 To get a full overview of methods offered to you, :ref:`sage.geometry.polyhedron.base` is the first place you want to go.
@@ -788,5 +788,5 @@ Bibliography
              978-1-4613-0019-9
 
 .. [Zie2007] Ziegler, G. M., Lectures on polytopes, volume 152 of Graduate
-             Texts in Mathematics. Springer-Verlag, New York, 2007. 
+             Texts in Mathematics. Springer-Verlag, New York, 2007.
              ISBN 978-0-387-94365-7

src/doc/en/thematic_tutorials/geometry/polytope_tikz.rst

@@ -45,7 +45,7 @@ Then, you can either copy-paste it to your article by typing ``Img`` in Sage or
 
 .. end of output
 
-Then in the pwd (present working directory of sage, the one of your article) 
+Then in the pwd (present working directory of sage, the one of your article)
 you will have a file named ``Img_poly.tex`` containing the tikzpicture of your polytope.
 
 Customization
@@ -77,7 +77,7 @@ In Sage, you type:
 
 Then, you visualize the polytope by typing ``P.show(aspect_ratio=1)``
 
-When you found a good angle, follow the above procedure to obtain the values 
+When you found a good angle, follow the above procedure to obtain the values
 [674,108,-731] and angle=112, for example.
 
 ::
@@ -117,7 +117,7 @@ It is also possible to replace it by the following 4 lines (and adding ``\usetik
 
 .. end of output
 
-Finally, you may want to tweak your picture my adding labels, elements on 
+Finally, you may want to tweak your picture my adding labels, elements on
 vertices, edges, facets, etc.
 
 Automatize using SageTex
@@ -129,7 +129,7 @@ For this you need to put
 
 in the preamble of your article
 
-There are different ways to use sagetex and you may create your own. Here are 
+There are different ways to use sagetex and you may create your own. Here are
 some possibilities.
 
 1) You can directly type in a sagestr in the article:

src/doc/en/thematic_tutorials/geometry/polytutorial.rst

@@ -11,7 +11,7 @@ A Brief Introduction to Polytopes in Sage
 
 If you already know some convex geometry  *a la*  Grünbaum or
 Brøndsted, then you may have itched to get your hands dirty with some
-polytope calculations.  Here is a mini\-guide to doing just that. 
+polytope calculations.  Here is a mini\-guide to doing just that.
 
 Basics
 """"""
@@ -114,7 +114,7 @@ Surely you want to compute the polar dual:
 Check it out\-\-\-we started with an integer\-lattice polytope and dualized
 to a rational\-lattice polytope.  Let's look at that.
 
- 
+
 
 
 ::
@@ -171,7 +171,7 @@ at this...
 
 .. end of output
 
-Here is what's going on. 
+Here is what's going on.
 
 If a polytope ``P`` is in `\ZZ`, then...
 
@@ -190,7 +190,7 @@ However, if a polytope is  *not*  in `\ZZ`, for example if it's in `\QQ` or
 
 Keep all of this in mind as you take polar duals.
 
- 
+
 
 Polytope Constructions
 """"""""""""""""""""""
@@ -350,7 +350,7 @@ Once you've constructed some polytope, you can ask Sage questions about it.
 
 .. end of output
 
-Face information can be useful.  
+Face information can be useful.
 
 
 ::

src/doc/en/thematic_tutorials/geometry/visualization.rst

@@ -39,7 +39,7 @@ This plots the graph (with unbounded edges) of the polyhedron
 
 .. end of output
 
-:code:`plot` 
+:code:`plot`
 ==================================================
 
 The :code:`plot` method draws the graph, the polygons and vertices of the