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
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
of the mesh.
More formally, the smoothing operation may be described per-vertex as:
:
Where
is the number of adjacent vertices to node
,
is the position of the
-th adjacent vertex and
is the new position for node
.
See also
*
Tutte embedding
In graph drawing and geometric graph theory, a Tutte embedding or barycentric embedding of a simple, 3-vertex-connected, planar graph is a crossing-free straight-line embedding with the properties that the outer face is a convex polygon and tha ...
, an embedding of a planar mesh in which each vertex is already at the average of its neighbours' positions
References
Mesh generation
Geometry processing
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