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:

${\displaystyle {\bar {x))_{i}={\frac {1}{N))\sum _{j=1}^{N}{\bar {x))_{j))$

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