Laplacian smoothing explained

Laplacian smoothing is an algorithm to smooth a polygonal mesh.^{[1] [2]} For each vertex in a mesh, a new position is chosen based on local information (such as the position of neighbours) and the vertex is moved there. In the case that a mesh is topologically a rectangular grid (that is, each internal vertex is connected to four neighbours) then this operation produces the Laplacian of the mesh.

More formally, the smoothing operation may be described per-vertex as:

\bar{x} _{i} = (1)/(N) \sum ^{N} _{j =1} \bar{x} _{j}

Where

N is the number of adjacent vertices to node i , \bar{x} _{j} is the position of the j -th adjacent vertex and \bar{x} _{i} is the new position for node i .^[3]

See also

• Tutte embedding, an embedding of a planar mesh in which each vertex is already at the average of its neighbours' positions

References

1. .
2. Book: Sorkine, O., Cohen-Or, D., Lipman, Y., Alexa, M., Rössl, C., Seidel, H.-P.. Laplacian Surface Editing. Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing. 2004. 175–184. ACM. Nice, France. SGP '04. 10.1145/1057432.1057456. 3-905673-13-4. 1980978. 1 December 2013.
3. Book: Mesh enhancement . limited . Hansen . Glen A. . Douglass . R. W . Andrew . Zardecki . 2005 . Imperial College Press . 404 . Glen05.