Mesh.laplacian_matrix

Mesh.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]

$$\begin{array}{r}{\mathbf{L}}_{ij}=\{\begin{array}{ll}-1& i=j\\ \frac{1}{deg(i)}& (i,j)\in \mathbf{E}\\ 0& \text{otherwise}\end{array}\end{array}$$

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)