Coverage for local/lib/python2.7/site-packages/sage/groups/affine_gps/euclidean_group.py : 92%
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r""" Euclidean Groups
AUTHORS:
- Volker Braun: initial version """
############################################################################## # Copyright (C) 2013 Volker Braun <vbraun.name@gmail.com> # # Distributed under the terms of the GNU General Public License (GPL) # # The full text of the GPL is available at: # # http://www.gnu.org/licenses/ ##############################################################################
r""" an Euclidean group.
The Euclidean group `E(A)` (or general affine group) of an affine space `A` is the group of all invertible affine transformations from the space into itself preserving the Euclidean metric.
If we let `A_V` be the affine space of a vector space `V` (essentially, forgetting what is the origin) then the Euclidean group `E(A_V)` is the group generated by the general linear group `SO(V)` together with the translations. Recall that the group of translations acting on `A_V` is just `V` itself. The general linear and translation subgroups do not quite commute, and in fact generate the semidirect product
.. MATH::
E(A_V) = SO(V) \ltimes V.
As such, the group elements can be represented by pairs `(A,b)` of a matrix and a vector. This pair then represents the transformation
.. MATH::
x \mapsto A x + b.
We can also represent this as a linear transformation in `\dim(V) + 1` dimensional space as
.. MATH::
\begin{pmatrix} A & b \\ 0 & 1 \end{pmatrix}
and lifting `x = (x_1, \ldots, x_n)` to `(x_1, \ldots, x_n, 1)`.
.. SEEALSO::
- :class:`AffineGroup`
INPUT:
Something that defines an affine space. For example
- An affine space itself:
* ``A`` -- affine space
- A vector space:
* ``V`` -- a vector space
- Degree and base ring:
* ``degree`` -- An integer. The degree of the affine group, that is, the dimension of the affine space the group is acting on.
* ``ring`` -- A ring or an integer. The base ring of the affine space. If an integer is given, it must be a prime power and the corresponding finite field is constructed.
* ``var`` -- (default: ``'a'``) Keyword argument to specify the finite field generator name in the case where ``ring`` is a prime power.
EXAMPLES::
sage: E3 = EuclideanGroup(3, QQ); E3 Euclidean Group of degree 3 over Rational Field sage: E3(matrix(QQ,[(6/7, -2/7, 3/7), (-2/7, 3/7, 6/7), (3/7, 6/7, -2/7)]), vector(QQ,[10,11,12])) [ 6/7 -2/7 3/7] [10] x |-> [-2/7 3/7 6/7] x + [11] [ 3/7 6/7 -2/7] [12] sage: E3([[6/7, -2/7, 3/7], [-2/7, 3/7, 6/7], [3/7, 6/7, -2/7]], [10,11,12]) [ 6/7 -2/7 3/7] [10] x |-> [-2/7 3/7 6/7] x + [11] [ 3/7 6/7 -2/7] [12] sage: E3([6/7, -2/7, 3/7, -2/7, 3/7, 6/7, 3/7, 6/7, -2/7], [10,11,12]) [ 6/7 -2/7 3/7] [10] x |-> [-2/7 3/7 6/7] x + [11] [ 3/7 6/7 -2/7] [12]
Instead of specifying the complete matrix/vector information, you can also create special group elements::
sage: E3.linear([6/7, -2/7, 3/7, -2/7, 3/7, 6/7, 3/7, 6/7, -2/7]) [ 6/7 -2/7 3/7] [0] x |-> [-2/7 3/7 6/7] x + [0] [ 3/7 6/7 -2/7] [0] sage: E3.reflection([4,5,6]) [ 45/77 -40/77 -48/77] [0] x |-> [-40/77 27/77 -60/77] x + [0] [-48/77 -60/77 5/77] [0] sage: E3.translation([1,2,3]) [1 0 0] [1] x |-> [0 1 0] x + [2] [0 0 1] [3]
Some additional ways to create Euclidean groups::
sage: A = AffineSpace(2, GF(4,'a')); A Affine Space of dimension 2 over Finite Field in a of size 2^2 sage: G = EuclideanGroup(A); G Euclidean Group of degree 2 over Finite Field in a of size 2^2 sage: G is EuclideanGroup(2,4) # shorthand True
sage: V = ZZ^3; V Ambient free module of rank 3 over the principal ideal domain Integer Ring sage: EuclideanGroup(V) Euclidean Group of degree 3 over Integer Ring
sage: EuclideanGroup(2, QQ) Euclidean Group of degree 2 over Rational Field
TESTS::
sage: E6 = EuclideanGroup(6, QQ) sage: E6 is E6 True sage: V = QQ^6 sage: E6 is EuclideanGroup(V) True sage: G = EuclideanGroup(2, GF(5)); G Euclidean Group of degree 2 over Finite Field of size 5 sage: TestSuite(G).run()
REFERENCES:
- :wikipedia:`Euclidean_group` """ """ Verify that ``A``, ``b`` define an affine group element.
This is called from the group element constructor and can be overridden for subgroups of the affine group. It is guaranteed that ``A``, ``b`` are in the correct matrix/vector space.
INPUT:
- ``A`` -- an element of :meth:`matrix_space`.
- ``b`` -- an element of :meth:`vector_space`.
OUTPUT:
The return value is ignored. You must raise a ``TypeError`` if the input does not define a valid group element.
TESTS::
sage: E3 = EuclideanGroup(3, QQ) sage: A = E3.matrix_space()([6/7,-2/7,3/7,-2/7,3/7,6/7,3/7,6/7,-2/7]) sage: det(A) -1 sage: b = E3.vector_space().an_element() sage: E3._element_constructor_check(A, b)
sage: A = E3.matrix_space()([1,2,3,4,5,6,7,8,0]) sage: det(A) 27 sage: E3._element_constructor_check(A, b) Traceback (most recent call last): ... TypeError: A must be orthogonal (unitary) """
r""" EXAMPLES::
sage: G = EuclideanGroup(6, GF(5)) sage: latex(G) \mathrm{E}_{6}(\Bold{F}_{5}) """
""" String representation of this group.
EXAMPLES::
sage: EuclideanGroup(6, GF(5)) Euclidean Group of degree 6 over Finite Field of size 5 """
""" Return a random element of this group.
EXAMPLES::
sage: G = EuclideanGroup(4, GF(3)) sage: G.random_element() # random [2 1 2 1] [1] [1 2 2 1] [0] x |-> [2 2 2 2] x + [1] [1 1 2 2] [2] sage: G.random_element() in G True
TESTS::
sage: G.random_element().A().is_unitary() True """ except ValueError: # v has norm zero pass
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