simplicial-complex

simplicial-complex

This CommonJS module

Topological operations and indexing for simplicial complexes (ie graphs, triangular and tetrahedral meshes, etc.) in node.js.

Usage

First, you need to install the library using npm:

npm install simplical-complex

And then in your scripts, you can just require it like usual:

var top = require("simplicial-complex");

simplicial-complex represents cell complexes as arrays of arrays of vertex indices. For example, here is a triangular mesh:

var tris = [
    [0,1,2],
    [1,2,3],
    [2,3,4]
  ];
  

And here is how you would compute its edges using mesh-topology:

var edges = top.skeleton(tris, 1);

//Result:
//  edges = [ [0,1],
//            [0,2],
//            [1,2],
//            [1,3],
//            [2,3],
//            [2,4],
//            [3,4] ]

The functionality in this library can be broadly grouped into the following categories:

Generic

dimension(cells)

Returns: The dimension of the cell complex.

Time complexity: O(cells.length)

countVertices(cells)

An optimized way to get the number of 0-cells in a cell complex with dense, sequentially indexed vertices. If cells has these properties, then:

top.countVertices(cells)

Is equivalent to:

top.skeleton(cells, 0).length

• cells is a cell complex

Returns: The number of vertices in the cell complex

Time complexity: O(cells.length * d), where d = dimension(cells)

cloneCells(cells)

Makes a copy of a cell complex

• cells is an array of cells

Returns: A deep copy of the cell complex

Time complexity: O(cells.length * d)

Indexing and Incidence

compareCells(a, b)

Ranks a pair of cells relative to one another up to permutation.

• a is a cell
• b is a cell

Returns a signed integer representing the relative rank:

• < 0 : a comes before b
• = 0 : a = b up to permutation

• 0 : b comes before a

Time complexity: O( a.length * log(a.length) )

normalize(cells[, attr])

Canonicalizes a cell complex so that it is possible to compute findCell queries. Note that this function is done in place. cells will be mutated. If this is not acceptable, you should make a copy first using cloneCells.

• cells is a complex.
• attr is an optional array of per-cell properties which is permuted alongside cells

Returns cells

Time complexity: O(d * log(cells.length) * cells.length )

findCell(cells, c)

Finds a lower bound on the first index of cell c in a normalized array of cells.

• cells is a normalize'd array of cells
• c is a cell represented by an array of vertex indices

Returns: The index of c in the array if it exists, otherwise -1

Time complexity: O(d * log(d) * log(cells.length)), where d is the max of the dimension of cells and c

buildIndex(from_cells, to_cells)

Builds an index for neighborhood queries. This allows you to quickly find cells the in to_cells which are incident to cells in from_cells.

• from_cells a normalized array of cells
• to_cells a list of cells which we are going to query against

Returns: An array with the same length as from_cells, the ith entry of which is an array of indices into to_cells which are incident to from_cells[i].

Time complexity: O(from_cells.length + d * 2^d * log(from_cells.length) * to_cells.length), where d = max(dimension(from_cells), dimension(to_cells)).

stars(cells[, vertex_count])

A more optimized way to build an index for vertices for cell complexes with sequentially enumerated vertices. If cells is a complex with each occuring exactly once, then:

top.stars(cells)

Is equivalent to doing:

top.buildIndex(cells, top.skeleton(cells, 0), 0)

• cells is a cell complex
• vertex_count is an optional parameter giving the number of vertices in the cell complex. If not specified, then countVertices() is used internally to get the size of the cell complex.

Returns: An array of elements with the same length as vertex_count giving the vertex stars of the mesh as indexed arrays of cells.

Time complexity: O(d * cells.length)

Basic Topology

subcells(cells, n)

Enumerates all n cells in the complex.

• cells is an array of cells
• n is the dimension of the cycles to compute

Returns: A list of all cycles

Time complexity: O(d^n * cells.length)

skeleton(cells, n)

Computes the n-skeleton of an unoriented simplicial complex. This is the set of all unique n-cells up to permutation.

• cells is a cell complex
• n is the dimension of the skeleton to compute.

Returns: A normalize array of all n-cells which are unique up to permutation.

Example:

• skeleton(tris, 1) returns all the edge in a triangular mesh
• skeleton(tets, 2) returns all the faces of a tetrahedral mesh
• skeleton(cells, 0) returns all the vertices of a mesh

Time complexity: O( n^d * cells.length ), where d is the dimension of the cell complex

boundary(cells, n)

Computes the d-dimensional boundary of a cell complex. For example, in a triangular mesh boundary(tris, 1) gives an array of all the boundary edges of the mesh; or boundary(tets, 2) gives an array of all boundary faces. Algebraically, this is the same as evaluating the boundary operator in the Z/2 homology.

• cells is a cell complex.
• n is the dimension of the boundary we are computing.

Returns: A normalized array of n-dimensional cells representing the boundary of the cell complex.

Time complexity: O((d^n + log(cells.length)) * cells.length)

connectedComponents(cells[, vertex_count])

Splits a simplicial complex into its connected components. If vertex_count is specified, we assume that the cell complex is dense -- or in other words the vertices of the cell complex is the set of integers [0, vertex_count). This allows for a slightly more efficient implementation. If unspecified, a more general but less efficient sparse algorithm is used.

• cells is an array of cells
• vertex_count (optional) is the result of calling countVertices(cells) or in other words is the total number of vertices.

Returns: An array of cell complexes, one per each connected component. Note that these complexes are not normalized.

Time complexity:

• If vertex_count is specified: O(vertex_count + d^2 * cells.length)
• If vertex_count is not specified: O(d^3 * log(cells.length) * cells.length)

Credits

(c) 2013 Mikola Lysenko. MIT License