trimesh_cotangent_laplacian_matrix

compas.datastructures.trimesh_cotangent_laplacian_matrix(mesh, rtype='csr')[source]

Construct the Laplacian of a triangular mesh with cotangent weights.

Parameters

mesh (compas.datastructures.Mesh) – Instance of mesh.

Returns

array – The Laplacian matrix with cotangent weights.

Notes

The cotangent laplacian of a vertex $\mathbf{v}_{i}$ points from the vertex to the projection of the vertex into the 1-ring plane. The cotangent laplacian vectors of a mesh thus provide an approximation of the per-vertex normals.

The $n \times n$ cotangent Laplacian matrix $\mathbf{L}$ of a mesh with vertices $\mathbf{V}$ and edges $\mathbf{E}$ is defined as follows 1

$$\begin{split}\mathbf{L}_{ij} = \begin{cases} -1 & \text{if i = j} \\ w_{ij} & \text{if (i, j) \in \mathbf{E}} \\ 0 & \text{otherwise} \end{cases}\end{split}$$

with

$$w_{ij} = \frac{\omega_{ij}}{\sum_{(i, k) \in \mathbf{E}_{i}} \omega_{ik}}$$

Examples

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References

1

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