title: libigl Tutorial
author: Alec Jacobson, Daniele Pannozo and others
date: 20 June 2014
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# Introduction
Libigl is an open source C++ library for geometry processing research and
development.
Dropping the heavy data structures of tradition geometry libraries, libigl is
a simple header-only library of encapsulated functions. 
%
This combines the rapid prototyping familiar to \textsc{Matlab} or
\textsc{Python} programmers
with the performance and versatility of C++.
%
The tutorial is a self-contained, hands-on introduction to \libigl. 
%
Via live coding and interactive examples, we demonstrate how to accomplish
various common geometry processing tasks such as computation of differential
quantities and operators, real-time deformation, global parametrization,
numerical optimization and mesh repair.
%
Accompanying lecture notes contain further details and cross-platform example
applications for each topic.
\end{abstract}

# Index

* **100_FileIO**: Example of reading/writing mesh files
* **101_Serialization**: Example of using the XML serialization framework
* **102_DrawMesh**: Example of plotting a mesh
* [201 Normals](#norm)
* [202 Gaussian Curvature](#gaus)
* [203 Curvature Directions](#curv)
* [204 Gradient](#grad)
* [204 Laplacian](#lapl)

# Compilation Instructions

All examples depends on glfw, glew and anttweakbar. A copy
of the sourcecode of each library is provided together with libigl
and they can be precompiled using:

**Alec: Is this just compiling the dependencies? Then perhaps rename `compile_dependencies_*`**

    sh compile_macosx.sh (MACOSX)
    sh compile_linux.sh (LINUX)
    compile_windows.bat (Visual Studio 2012)

Every example can be compiled by using the cmake file provided in its folder.
On Linux and MacOSX, you can use the provided bash script:

    sh ../compile_example.sh

## (Optional: compilation with libigl as static library)

By default, libigl is a _headers only_ library, thus it does not require
compilation. However, one can precompile libigl as a statically linked library.
See `../README.md` in the main directory for compilations instructions to
produce `libigl.a` and other libraries. Once compiled, these examples can be
compiled using the `CMAKE` flag `-DLIBIGL_USE_STATIC_LIBRARY=ON`:

    ../compile_example.sh -DLIBIGL_USE_STATIC_LIBRARY=ON

# Chapter 2: Discrete Geometric Quantities and Operators
This chapter illustrates a few discrete quantities that libigl can compute on a
mesh. This also provides an introduction to basic drawing and coloring routines
in our example viewer. Finally, we construct popular discrete differential
geometry operators.

## <a id=norm></a> Normals

### Per-face 
Well defined normal orthogonal to triangle's plane. Produces piecewise-flat
rending: not smooth.

### Per-vertex
Using per-vertex normals, Phong or Gouraud shading will produce smooth(er)
renderings. Most techniques for computing per-vertex normals take an average of
incident face normals. The techniques vary with respect to their different
weighting schemes. Uniform weighting is heavily biased by the discretization
choice, where as area-based or angle-based weighting is more forgiving.

The typical half-edge style computation of area-based weights might look
something like this:

```cpp
N.setZero(V.rows(),3);
for(int i : vertices)
{
  for(face : incident_faces(i))
  {
    N.row(i) += face.area * face.normal;
  }
}
N.rowwise().normalize();
```

Without a half-edge data-structure it may seem at first glance that looping
over incident faces---and thus constructing the per-vertex normals---would be
inefficient. However, per-vertex normals may be _throwing_ each face normal to
running sums on its corner vertices:

```cpp
N.setZero(V.rows(),3);
for(int f = 0; f < F.rows();f++)
{
  for(int c = 0; c < 3;c++)
  {
    N.row(F(f,c)) += area(f) * face_normal.row(f);
  }
}
N.rowwise().normalize();
```

### Per-corner

Storing normals per-corner is an efficient an convenient way of supporting both
smooth and sharp (e.g. creases and corners) rendering. This format is common to
OpenGL and the .obj mesh file format. Often such normals are tuned by the mesh
designer, but creases and corners can also be computed automatically. Libigl
implements a simple scheme which computes corner normals as averages of
normals of faces incident on the corresponding vertex which do not deviate by a
specified dihedral angle (e.g. 20°).

![The `Normals` example computes per-face (left), per-vertex (middle) and
per-corner (right) normals](images/fandisk-normals.jpg)

## <a id=gaus></a> Gaussian Curvature
Gaussian curvature on a continuous surface is defined as the product of the
principal curvatures:

 $k_G = k_1 k_2.$

As an _intrinsic_ measure, it depends on the metric and
not the surface's embedding.

Intuitively, Gaussian curvature tells how locally spherical or _elliptic_ the
surface is ( $k_G>0$ ), how locally saddle-shaped or _hyperbolic_ the surface
is ( $k_G<0$ ), or how locally cylindrical or _parabolic_ ( $k_G=0$ ) the
surface is.

In the discrete setting, one definition for a ``discrete Gaussian curvature''
on a triangle mesh is via a vertex's _angular deficit_:

 $k_G(v_i) = 2π - \sum\limits_{j\in N(i)}θ_{ij},$

where $N(i)$ are the triangles incident on vertex $i$ and $θ_{ij}$ is the angle
at vertex $i$ in triangle $j$ [][#meyer_2003].

Just like the continuous analog, our discrete Gaussian curvature reveals
elliptic, hyperbolic and parabolic vertices on the domain.

![The `GaussianCurvature` example computes discrete Gaussian curvature and visualizes it in
pseudocolor.](images/bumpy-gaussian-curvature.jpg)

## <a id=curv></a> Curvature Directions
The two principal curvatures $(k_1,k_2)$ at a point on a surface measure how much the
surface bends in different directions. The directions of maximum and minimum
(signed) bending are call principal directions and are always
orthogonal.

Mean curvature is defined simply as the average of principal curvatures:

 $H = \frac{1}{2}(k_1 + k_2).$ 

One way to extract mean curvature is by examining the Laplace-Beltrami operator
applied to the surface positions. The result is a so-called mean-curvature
normal:

  $-\Delta \mathbf{x} = H \mathbf{n}.$ 

It is easy to compute this on a discrete triangle mesh in libigl using the cotangent
Laplace-Beltrami operator [][#meyer_2003].

```cpp
#include <igl/cotmatrix.h>
#include <igl/massmatrix.h>
#include <igl/invert_diag.h>
...
MatrixXd HN;
SparseMatrix<double> L,M,Minv;
igl::cotmatrix(V,F,L);
igl::massmatrix(V,F,igl::MASSMATRIX_VORONOI,M);
igl::invert_diag(M,Minv);
HN = -Minv*(L*V);
H = (HN.rowwise().squaredNorm()).array().sqrt();
```

Combined with the angle defect definition of discrete Gaussian curvature, one
can define principal curvatures and use least squares fitting to find
directions [][#meyer_2003].

Alternatively, a robust method for determining principal curvatures is via
quadric fitting [][#pannozo_2010]. In the neighborhood
around every vertex, a best-fit quadric is found and principal curvature values
and directions are sampled from this quadric. With these in tow, one can
compute mean curvature and Gaussian curvature as sums and products
respectively.

![The `CurvatureDirections` example computes principal curvatures via quadric
fitting and visualizes mean curvature in pseudocolor and principal directions
with a cross field.](images/fertility-principal-curvature.jpg)

This is an example of syntax highlighted code:

```cpp
#include <foo.html>
int main(int argc, char * argv[])
{
  return 0;
}
```

## <a id=grad></a> Gradient
Scalar functions on a surface can be discretized as a piecewise linear function
with values defined at each mesh vertex:

 $f(\mathbf{x}) \approx \sum\limits_{i=0}^n \phi_i(\mathbf{x})\, f_i,$

where $\phi_i$ is a piecewise linear hat function defined by the mesh so that
for each triangle $\phi_i$ is _the_ linear function which is one only at
vertex $i$ and zero at the other corners.

![Hat function $\phi_i$ is one at vertex $i$, zero at all other vertices, and
linear on incident triangles.](images/hat-function.jpg)

Thus gradients of such piecewise linear functions are simply sums of gradients
of the hat functions:

 $\nabla f(\mathbf{x}) \approx 
 \nabla \sum\limits_{i=0}^n \nabla \phi_i(\mathbf{x})\, f_i = 
 \sum\limits_{i=0}^n \nabla \phi_i(\mathbf{x})\, f_i.$

This reveals that the gradient is a linear function of the vector of $f_i$
values. Because $\phi_i$ are linear in each triangle their gradient are
_constant_ in each triangle. Thus our discrete gradient operator can be written
as a matrix multiplication taking vertex values to triangle values:

 $\nabla f \approx \mathbf{G}\,\mathbf{f},$

where $\mathbf{f}$ is $n\times 1$ and $\mathbf{G}$ is an $m\times n$ sparse
matrix. This matrix $\mathbf{G}$ can be derived geometrically, e.g.
[ch. 2][#jacobson_thesis_2013].
Libigl's `gradMat`**Alec: check name** function computes $\mathbf{G}$ for
triangle and tetrahedral meshes:

![The `Gradient` example computes gradients of an input function on a mesh and
visualizes the vector field.](images/cheburashka-gradient.jpg)

## <a id=lapl></a> Laplacian

The discrete Laplacian is an essential geometry processing tool. Many
interpretations and flavors of the Laplace and Laplace-Beltrami operator exist. 

In open Euclidean space, the _Laplace_ operator is the usual divergence of gradient
(or equivalently the Laplacian of a function is the trace of its Hessian):

 $\Delta f = 
 \frac{\partial^2 f}{\partial x^2} +
 \frac{\partial^2 f}{\partial y^2} +
 \frac{\partial^2 f}{\partial z^2}.$

The _Laplace-Beltrami_ operator generalizes this to surfaces.

When considering piecewise-linear functions on a triangle mesh, a discrete Laplacian may
be derived in a variety of ways. The most popular in geometry processing is the
so-called ``cotangent Laplacian'' $\mathbf{L}$, arising simultaneously from FEM, DEC and
applying divergence theorem to vertex one-rings. As a linear operator taking
vertex values to vertex values, the Laplacian $\mathbf{L}$ is a $n\times n$
matrix with elements:

$L_{ij} = \begin{cases}j \in N(i) &\cot \alpha_{ij} + \cot \beta_{ij},\\
j \notin N(i) & 0,\\
i = j & -\sum\limits_{k\neq i} L_{ik},
\end{cases}$

where $N(i)$ are the vertices adjacent to (neighboring) vertex $i$, and
$\alpha_{ij},\beta_{ij}$ are the angles opposite edge ${ij}$.
This oft
produced formula leads to a typical half-edge style implementation for
constructing $\mathbf{L}$:

```cpp
for(int i : vertices)
{
  for(int j : one_ring(i))
  {
    for(int k : triangle_on_edge(i,j))
    {
      L(i,j) = cot(angle(i,j,k));
      L(i,i) -= cot(angle(i,j,k));
    }
  }
}
```

Without a half-edge data-structure it may seem at first glance that looping
over one-rings, and thus constructing the Laplacian would be inefficient.
However, the Laplacian may be built by summing together contributions for each
triangle, much in spirit with its FEM discretization of the Dirichlet energy
(sum of squared gradients):

```cpp
for(triangle t : triangles)
{
  for(edge i,j : t)
  {
    L(i,j) += cot(angle(i,j,k));
    L(j,i) += cot(angle(i,j,k));
    L(i,i) -= cot(angle(i,j,k));
    L(j,j) -= cot(angle(i,j,k));
  }
}
```

Libigl implements discrete "cotangent" Laplacians for triangles meshes and
tetrahedral meshes, building both with fast geometric rules rather than "by the
book" FEM construction which involves many (small) matrix inversions, cf.
**Alec: cite Ariel reconstruction paper**.

The operator applied to mesh vertex positions amounts to smoothing by _flowing_
the surface along the mean curvature normal direction. This is equivalent to
minimizing surface area.

![The `Laplacian` example computes conformalized mean curvature flow using the
cotangent Laplacian [#kazhdan_2012][].](images/cow-curvature-flow.jpg)

### Mass matrix
The mass matrix $\mathbf{M}$ is another $n \times n$ matrix which takes vertex
values to vertex values. From an FEM point of view, it is a discretization of
the inner-product: it accounts for the area around each vertex. Consequently,
$\mathbf{M}$ is often a diagonal matrix, such that $M_{ii}$ is the barycentric
or voronoi area around vertex $i$ in the mesh [#meyer_2003][]. The inverse of
this matrix is also very useful as it transforms integrated quantities into
point-wise quantities, e.g.:

 $\nabla f \approx \mathbf{M}^{-1} \mathbf{L} \mathbf{f}.$

In general, when encountering squared quantities integrated over the surface,
the mass matrix will be used as the discretization of the inner product when
sampling function values at vertices:

 $\int_S x\, y\ dA \approx \mathbf{x}^T\mathbf{M}\,\mathbf{y}.$

[#meyer_2003]: Mark Meyer and Mathieu Desbrun and Peter Schröder and Alan H.  Barr,
"Discrete Differential-Geometry Operators for Triangulated
2-Manifolds," 2003.
[#pannozo_2010]: Daniele Pannozo, Enrico Puppo, Luigi Rocca,
"Efficient Multi-scale Curvature and Crease Estimation," 2010.
[#jacobson_thesis_2013]: Alec Jacobson,
_Algorithms and Interfaces for Real-Time Deformation of 2D and 3D Shapes_,
2013.
[#kazhdan_2012]: Michael Kazhdan, Jake Solomon, Mirela Ben-Chen,
"Can Mean-Curvature Flow Be Made Non-Singular," 2012.
