Editing Discrete Laplace operator (section)

===Mesh Laplacians===

In addition to considering the connectivity of nodes and edges in a graph, mesh Laplace operators take into account the geometry of a surface (e.g. the angles at the nodes). For a two-dimensional [[manifold]] triangle mesh, the [[Laplace–Beltrami operator]] of a scalar function <math>u</math> at a vertex <math>i</math> can be approximated as

:<math>
(\Delta u)_{i} \equiv \frac{1}{2A_i} \sum_{j} (\cot \alpha_{ij} + \cot \beta_{ij}) (u_j - u_i),
</math>

where the sum is over all adjacent vertices <math>j</math> of <math>i</math>, <math>\alpha_{ij}</math> and <math>\beta_{ij}</math> are the two angles opposite of the edge <math>ij</math>, and <math>A_i</math> is the ''vertex area'' of <math>i</math>; that is, e.g. one third of the summed areas of triangles incident to <math>i</math>.
It is important to note that the sign of the discrete [[Laplace–Beltrami operator]] is conventionally opposite the sign of the ordinary [[Laplace operator]].
The above cotangent formula can be derived using many different methods among which are [[finite element method|piecewise linear finite elements]], [[finite volume method|finite volumes]], and [[discrete exterior calculus]].<ref name="crane13">
{{cite conference
 | last1= Crane
 |first1=K.
 |last2=de Goes
 |first2=F.
 |last3=Desbrun
 |first3=M.
 |last4=Schröder
 |first4=P.
 | year = 2013
 | title = Digital geometry processing with discrete exterior calculus
 | conference = SIGGRAPH '13
 | book-title = ACM SIGGRAPH 2013 Courses
 | volume = 7
 | pages = 1–126
 | url = http://doi.acm.org/10.1145/2504435.2504442
 | doi = 10.1145/2504435.2504442
 }}
</ref>

To facilitate computation, the Laplacian is encoded in a matrix <math>L\in\mathbb{R}^{|V|\times|V|}</math> such that <math> Lu = (\Delta u)_i </math>. Let <math>C</math> be the (sparse) ''cotangent matrix'' with entries

<math>
C_{ij} =  
\begin{cases} 
      \frac{1}{2}(\cot \alpha_{ij} + \cot \beta_{ij}) & ij \text{ is an edge, that is } j \in N(i), \\
      -\sum\limits_{k \in N(i)}C_{ik} & i = j,  \\
      0 & \text{otherwise}
     
\end{cases}
</math>

where <math>N(i) </math> denotes the neighborhood of <math> i</math>, and let <math> M </math> be the diagonal ''mass matrix'' <math> M </math> whose <math>i</math>-th entry along the diagonal is the vertex area <math> A_i </math>. Then <math> L=M^{-1}C </math> is the sought discretization of the Laplacian.

A more general overview of mesh operators is given in.<ref name="reuter06">{{cite journal
 |last1= Reuter |first1=M. |last2=Biasotti |first2=S. |last3=Giorgi |first3=D. |last4=Patane |first4=G. |last5=Spagnuolo |first5=M.
 | year = 2009
 | title = Discrete Laplace–Beltrami operators for shape analysis and segmentation
 | journal = Computers & Graphics
 | volume = 33
 | issue = 3
 | pages = 381–390df
 | doi=10.1016/j.cag.2009.03.005
| citeseerx = 10.1.1.157.757
 }}</ref>