Laplacian smoothing is an algorithm to smooth a polygonal mesh. 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:
x ¯ i = 1 N ∑ j = 1 N x ¯ j {\displaystyle {\bar {x}}_{i}={\frac {1}{N}}\sum _{j=1}^{N}{\bar {x}}_{j}}
Where N {\displaystyle N} is the number of adjacent vertices to node i {\displaystyle i} , x ¯ j {\displaystyle {\bar {x}}_{j}} is the position of the j {\displaystyle j} -th adjacent vertex and x ¯ i {\displaystyle {\bar {x}}_{i}} is the new position for node i {\displaystyle i} .
See also Tutte embedding, an embedding of a planar mesh in which each vertex is already at the average of its neighbours' positions
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