simplicial
Embedding
Important
Embeddings are also used to support simplicial's drawing routines <drawing>.
The embedding space can have any number of dimensions, defaulting to two. (At present simplicial only supports Euclidian embedding spaces.)
Embedding.dimension
Embedding.origin
Embedding.complex
The locations give rise to a notion of distance between positions.
Embedding.distance
An embedding works by specifying the positions of the 0-simplices of a complex. Since all higher simplices have a 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 Embedding that overrides the 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 Embedding.positionOf to access simplex positions, and not access to computation method directly. This ensures that explicit positions are returned correctly and cached.)
Embedding.positionSimplex
Embedding.computePositionOf
Embedding.positionOf
Embedding.positionsOf
Embedding.clearPositions
Embedding also exports a dict-like interface.
Embedding.__len__
Embedding.__getitem__
Embedding.__setitem__
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.
Embedding.vietorisRipsComplex