:class:`Embedding`: Embedding an abstract complex into space
.. currentmodule:: simplicial
.. autoclass:: Embedding
Important
Embeddings are also used to support simplicial's
:ref:`drawing routines <drawing>`.
The embedding space can have any number of dimensions, defaulting to two. (At present simplicial only supports Euclidian embedding spaces.)
.. automethod:: Embedding.dimension
.. automethod:: Embedding.origin
.. automethod:: Embedding.complex
The locations give rise to a notion of distance between positions.
.. automethod:: Embedding.distance
An embedding works by specifying the positions of the 0-simplices of a complex. Since all higher simplices have a :term:`basis` constructed of 0-simplices, these positions then induce positions for all the higher-order simplices.
There are two ways to specify an embedding. The first is to provide explicit co-ordinates for every 0-simplex in a complex. This is very general but misses some frequent commonalities, and so the second way is to provide a sub-class of :class:`Embedding` that overrides the :meth:`Embedding.computePositionOf` method. The two methods play well together: one can provide a sub-class to capture the regularities, and then perturb the complex by moving individual points if required. (Client code should only use :meth:`Embedding.positionOf` to access simplex positions, and not access to computation method directly. This ensures that explicit positions are returned correctly and cached.)
.. automethod:: Embedding.positionSimplex
.. automethod:: Embedding.computePositionOf
.. automethod:: Embedding.positionOf
.. automethod:: Embedding.positionsOf
.. automethod:: Embedding.clearPositions
:class:`Embedding` also exports a dict-like interface.
.. automethod:: Embedding.__len__
.. automethod:: Embedding.__getitem__
.. automethod:: Embedding.__setitem__
.. automethod:: Embedding.__contains__
Embeddings apply a geometry to a topological structure. It can be useful to go the other way: to take some geometric or spatial information and create a topological structure from it, where the topological structure encodes useful information. Typically this depends on a notion of distance between the located 0-simplices. The distance metric can then provide a way of constructing higher-dimensional simplices.
.. automethod:: Embedding.vietorisRipsComplex