Documentation for zonotopes, parallelepipeds and OctaSym

doc/manual/manual/contractors/index.rst

@@ -1,3 +1,4 @@
+.. _sec-ctc:
 Contractors, separators
 =======================
 

doc/manual/manual/geometry/zonotope.rst

@@ -12,15 +12,69 @@ The following classes represent zonotopes.
 Zonotope
 --------
 
-.. doxygenclass:: codac2::Zonotope
-  :project: codac
-  :members:
+A zonotope is a convex and symmetric polytope. 
+It can be represented as the Minkowski sum of a finite number of line segments, which are called its generators.
+
+In Codac, zonotopes are represented by the class :class:`Zonotope`. A zonotope is defined by its center and its generators.
+The center is a :class:`Vector`, noted :math:`z`, and the generators are stored in a :class:`Matrix`, noted :math:`A`. 
+Each column of the matrix corresponds to a generator.
+
+The resulting zonotope is:
+
+.. math::
+  Z = z + A \cdot [-1,1]^m
+
+Where :math:`m` is the number of generators (i.e., the number of columns of the matrix :math:`A`).
+
+For example, let us consider the following zonotope in :math:`\mathbb{R}^2`:
+
+.. math::
+  Z = \left(\begin{array}{c} 1\\ 2\\ \end{array}\right) + \left(\begin{array}{ccc} 1 & 0 & 2 \\ 0 & 1 & 1 \end{array}\right) \cdot [-1,1]^3
+
+It can be constructed in Codac as follows:
+
+.. tabs::
+
+  .. code-tab:: py
+
+    z = Vector([1,2])
+    A = Matrix([[1,0,2],[0,1,1]])
+
+    Z = Zonotope(z, A)
+
+  .. code-tab:: c++
+
+    Vector z({1,2});
+    Matrix A({{1,0,2},{0,1,1}});
+
+    Zonotope Z(z, A);
+
+  .. code-tab:: matlab
+
+    z = Vector({1,2});
+    A = Matrix({{1,0,2},{0,1,1}});
+
+    Z = Zonotope(z,A);
+
+The resulting zonotope is represented in the figure below:
+
+.. figure:: zonotope.png
+    :width: 400px
+
+Additional methods are provided to handle zonotopes.
+
+- `box()`: gives the bounding box of the zonotope. Returns an :class:`IntervalVector`.
+- `proj(v)`: Projects the zonotope on a given subspace, defined by the vector of indices `v` (:class:`std::vector\<Index\>`). Returns a :class:`Zonotope`.
 
 .. _subsec-zonotope-parallelepiped:
 
 Parallelepiped
 --------------
 
-.. doxygenclass:: codac2::Parallelepiped
-  :project: codac
-  :members:
+A parallelepiped is a special case of zonotope, where the number of generators is equal or less than the dimension of the space.
+
+It inherits from the `Zonotope` class, and thus has the same properties and methods. In addition, it defines the following methods:
+
+- `vertices()`: gives the vertices of the parallelepiped. Returns a :class:`std::vector\<Vector\>`.
+- `contains(v)`: checks if the point `v` is contained in the parallelepiped. Returns a :class:`BoolInterval`. **The matrix A must be invertible**
+- `is_superset(x)`: checks if the parallelepiped is a superset of the :class:`IntervalVector` `x`. Returns a :class:`BoolInterval`. **The matrix A must be invertible**