Block.laplacian_matrix

Block.laplacian_matrix(rtype='array')

Compute the Laplacian matrix of the mesh.

Parameters:

rtypeLiteral[‘array’, ‘csc’, ‘csr’, ‘coo’, ‘list’], optional

Format of the result.

Returns:

array-like

The Laplacian matrix.

Notes

The \(n \times n\) uniform Laplacian matrix \(\mathbf{L}\) of a mesh with vertices \(\mathbf{V}\) and edges \(\mathbf{E}\) is defined as follows [1]

\[\begin{split}\mathbf{L}_{ij} = \begin{cases} -1 & i = j \\ \frac{1}{deg(i)} & (i, j) \in \mathbf{E} \\ 0 & \text{otherwise} \end{cases}\end{split}\]

with \(deg(i)\) the degree of vertex \(i\).

Therefore, the uniform Laplacian of a vertex \(\mathbf{v}_{i}\) points to the centroid of its neighboring vertices.

References

[1]

Nealen A., Igarashi T., Sorkine O. and Alexa M. Laplacian Mesh Optimization.

Examples

>>> from compas.datastructures import Mesh
>>> mesh = Mesh.from_polyhedron(6)
>>> L = mesh.laplacian_matrix(rtype='array')
>>> type(L)
<class 'numpy.ndarray'>

>>> from numpy import asarray
>>> xyz = asarray(mesh.vertices_attributes('xyz'))
>>> L = mesh.laplacian_matrix(mesh)
>>> d = L.dot(xyz)