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@@ -36,6 +36,7 @@ applications for each topic.
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* [202 Gaussian Curvature](#gaus)
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* [203 Curvature Directions](#curv)
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* [204 Gradient](#grad)
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+* [204 Laplacian](#lapl)
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# Compilation Instructions
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@@ -192,6 +193,101 @@ triangle and tetrahedral meshes:
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+## <a id=lapl></a> Laplacian
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+
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+The discrete Laplacian is an essential geometry processing tool. Many
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+interpretations and flavors of the Laplace and Laplace-Beltrami operator exist.
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+
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+In open Euclidean space, the _Laplace_ operator is the usual divergence of gradient
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+(or equivalently the Laplacian of a function is the trace of its Hessian):
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+
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+ $\Delta f =
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+ \frac{\partial^2 f}{\partial x^2} +
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+ \frac{\partial^2 f}{\partial y^2} +
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+ \frac{\partial^2 f}{\partial z^2}.$
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+
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+The _Laplace-Beltrami_ operator generalizes this to surfaces.
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+
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+When considering piecewise-linear functions on a triangle mesh, a discrete Laplacian may
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+be derived in a variety of ways. The most popular in geometry processing is the
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+so-called ``cotangent Laplacian'' $\mathbf{L}$, arising simultaneously from FEM, DEC and
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+applying divergence theorem to vertex one-rings. As a linear operator taking
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+vertex values to vertex values, the Laplacian $\mathbf{L}$ is a $n\times n$
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+matrix with elements:
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+
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+$L_{ij} = \begin{cases}j \in N(i) &\cot \alpha_{ij} + \cot \beta_{ij},\\
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+j \notin N(i) & 0,\\
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+i = j & -\sum\limits_{k\neq i} L_{ik},
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+\end{cases}$
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+
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+where $N(i)$ are the vertices adjacent to (neighboring) vertex $i$, and
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+$\alpha_{ij},\beta_{ij}$ are the angles opposite edge ${ij}$.
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+This oft
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+produced formula leads to a typical half-edge style implementation for
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+constructing $\mathbf{L}$:
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+
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+```cpp
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+for(int i : vertices)
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+{
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+ for(int j : one_ring(i))
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+ {
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+ for(int k : triangle_on_edge(i,j))
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+ {
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+ L(i,j) = cot(angle(i,j,k));
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+ L(i,i) -= cot(angle(i,j,k));
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+ }
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+ }
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+}
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+```
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+
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+Without a half-edge data-structure it may seem at first glance that looping
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+over one-rings, and thus constructing the Laplacian would be inefficient.
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+However, the Laplacian may be built by summing together contributions for each
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+triangle, much in spirit with its FEM discretization of the Dirichlet energy
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+(sum of squared gradients):
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+
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+```cpp
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+for(triangle t : triangles)
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+{
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+ for(edge i,j : t)
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+ {
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+ L(i,j) += cot(angle(i,j,k));
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+ L(j,i) += cot(angle(i,j,k));
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+ L(i,i) -= cot(angle(i,j,k));
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+ L(j,j) -= cot(angle(i,j,k));
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+ }
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+}
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+```
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+
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+Libigl implements discrete "cotangent" Laplacians for triangles meshes and
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+tetrahedral meshes, building both with fast geometric rules rather than "by the
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+book" FEM construction which involves many (small) matrix inversions, cf.
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+**Alec: cite Ariel reconstruction paper**.
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+
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+The operator applied to mesh vertex positions amounts to smoothing by _flowing_
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+the surface along the mean curvature normal direction. This is equivalent to
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+minimizing surface area.
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+
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+![The `Laplacian` example computes conformalized mean curvature flow using the
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+cotangent Laplacian [#kazhdan_2012][].](images/cow-curvature-flow.jpg)
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+
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+### Mass matrix
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+The mass matrix $\mathbf{M}$ is another $n \times n$ matrix which takes vertex
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+values to vertex values. From an FEM point of view, it is a discretization of
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+the inner-product: it accounts for the area around each vertex. Consequently,
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+$\mathbf{M}$ is often a diagonal matrix, such that $M_{ii}$ is the barycentric
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+or voronoi area around vertex $i$ in the mesh [#meyer_2003][]. The inverse of
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+this matrix is also very useful as it transforms integrated quantities into
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+point-wise quantities, e.g.:
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+
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+ $\nabla f \approx \mathbf{M}^{-1} \mathbf{L} \mathbf{f}.$
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+
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+In general, when encountering squared quantities integrated over the surface,
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+the mass matrix will be used as the discretization of the inner product when
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+sampling function values at vertices:
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+
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+ $\int_S x\, y\ dA \approx \mathbf{x}^T\mathbf{M}\,\mathbf{y}.$
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+
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[#meyer_2003]: Mark Meyer and Mathieu Desbrun and Peter Schröder and Alan H. Barr,
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"Discrete Differential-Geometry Operators for Triangulated
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2-Manifolds," 2003.
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@@ -200,3 +296,5 @@ visualizes the vector field.](images/cheburashka-gradient.jpg)
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[#jacobson_thesis_2013]: Alec Jacobson,
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_Algorithms and Interfaces for Real-Time Deformation of 2D and 3D Shapes_,
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2013.
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+[#kazhdan_2012]: Michael Kazhdan, Jake Solomon, Mirela Ben-Chen,
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+"Can Mean-Curvature Flow Be Made Non-Singular," 2012.
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Alec Jacobson
