igl/docs/tutorial/chapter-2.md

Chapter 2: Discrete Geometric Quantities and Operators

This chapter illustrates a few discrete quantities that libigl can compute on a mesh and the libigl functions that construct popular discrete differential geometry operators. It also provides an introduction to basic drawing and coloring routines of our viewer.

Normals

Surface normals are a basic quantity necessary for rendering a surface. There are a variety of ways to compute and store normals on a triangle mesh. [Example 201]({{ repo_url }}/tutorial/201_Normals/main.cpp) demonstrates how to compute and visualize normals with libigl.

Per-face

Normals are well defined on each triangle of a mesh as the vector orthogonal to triangle's plane. These piecewise-constant normals produce piecewise-flat renderings: the surface appears non-smooth and reveals its underlying discretization.

Per-vertex

Normals can be computed and stored on vertices, and interpolated in the interior of the triangles to produce smooth renderings (Phong shading). Most techniques for computing per-vertex normals take an average of incident face normals. The main difference between these techniques is their weighting scheme: Uniform weighting is heavily biased by the discretization choice, whereas area-based or angle-based weighting is more forgiving.

The typical half-edge style computation of area-based weights has this structure:

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

At first glance, it might seem inefficient to loop over incident faces---and thus constructing the per-vertex normals--- without using an half-edge data structure. However, per-vertex normals may be throwing each face normal to running sums on its corner vertices:

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 and 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 more than a specified dihedral angle (e.g. 20°).

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 ¹ .

Just like the continuous analog, our discrete Gaussian curvature reveals elliptic, hyperbolic and parabolic vertices on the domain, as demonstrated in [Example 202]({{ repo_url }}/tutorial/202_GaussianCurvature/main.cpp).

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 called principal directions and are always orthogonal.

Mean curvature is defined 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 ¹ .

#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_TYPE_VORONOI,M);
igl::invert_diag(M,Minv);
HN = -Minv*(L*V);
H = HN.rowwise().norm(); //up to sign

Combined with the angle defect definition of discrete Gaussian curvature, one can define principal curvatures and use least squares fitting to find directions ¹ .

Alternatively, a robust method for determining principal curvatures is via quadric fitting ² . In the neighborhood around every vertex, a best-fit quadric is found and principal curvature values and directions are analytically computed on this quadric ([Example 203]({{ repo_url }}/tutorial/203_curvatureDirections/main.cpp)).

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=1}^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.

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

$\nabla f(\mathbf{x}) \approx \nabla \sum\limits_{i=1}^n \phi_i(\mathbf{x}), f_i = \sum\limits_{i=1}^n \nabla \phi_i(\mathbf{x}), f_i.$

This reveals that the gradient is a linear function of the vector of f_i values. Because the \phi_i are linear in each triangle, their gradients 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 md\times n sparse matrix. This matrix \mathbf{G} can be derived geometrically, e.g. ch. 2³ . Libigl's grad function computes \mathbf{G} for triangle and tetrahedral meshes ([Example 204]({{ repo_url }}/tutorial/204_Gradient/main.cpp)):

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 to edge {ij}. This formula leads to a typical half-edge style implementation for constructing \mathbf{L}:

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));
    }
  }
}

Similarly as before, it may seem to loop over one-rings without having an half-edge data structure. However, this is not the case, since 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):

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. ⁴ .

The operator applied to mesh vertex positions amounts to smoothing by flowing the surface along the mean curvature normal direction ([Example 205]({{ repo_url }}/tutorial/205_Laplacian/main.cpp)). Note that this is equivalent to minimizing surface area.

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 ¹ . The inverse of this matrix is also very useful as it transforms integrated quantities into point-wise quantities, e.g.:

\Delta 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}.

An alternative mass matrix \mathbf{T} is a md \times md matrix which takes triangle vector values to triangle vector values. This matrix represents an inner-product accounting for the area associated with each triangle (i.e. the triangles true area).

Alternative construction of Laplacian

An alternative construction of the discrete cotangent Laplacian is by "squaring" the discrete gradient operator. This may be derived by applying Green's identity (ignoring boundary conditions for the moment):

\int_S \|\nabla f\|^2 dA = \int_S f \Delta f dA

Or in matrix form which is immediately translatable to code:

$\mathbf{f}^T \mathbf{G}^T \mathbf{T} \mathbf{G} \mathbf{f} = \mathbf{f}^T \mathbf{M} \mathbf{M}^{-1} \mathbf{L} \mathbf{f} = \mathbf{f}^T \mathbf{L} \mathbf{f}.$

So we have that \mathbf{L} = \mathbf{G}^T \mathbf{T} \mathbf{G}. This also hints that we may consider \mathbf{G}^T as a discrete divergence operator, since the Laplacian is the divergence of the gradient. Naturally, \mathbf{G}^T is a n \times md sparse matrix which takes vector values stored at triangle faces to scalar divergence values at vertices.

Geodesic

The discrete geodesic distance between two points is the length of the shortest path between then restricted to the surface. For triangle meshes, such a path is made of a set of segments which can be either edges of the mesh or crossing a triangle.

Libigl includes a wrapper for the exact geodesic algorithm ⁶ developed by Danil Kirsanov (https://code.google.com/archive/p/geodesic/), exposing it through an Eigen-based API. The function

igl::exact_geodesic(V,F,VS,FS,VT,FT,d);

computes the closest geodesic distances of each vertex in VT or face in FT, from the source vertices VS or faces FS of the input mesh V,F. The output is writted in the vector d, which lists first the distances for the vertices in VT, and then for the faces in FT. For example, if you want to compute the distance from the vertex with id vid, to all vertices of F you can use:

Eigen::VectorXi VS,FS,VT,FT;
// The selected vertex is the source
VS.resize(1);
VS << vid;
// All vertices are the targets
VT.setLinSpaced(V.rows(),0,V.rows()-1);
Eigen::VectorXd d;
igl::exact_geodesic(V,F,VS,FS,VT,FT,d);

References

---

1. Mark Meyer, Mathieu Desbrun, Peter Schröder and Alan H. Barr, Discrete Differential-Geometry Operators for Triangulated 2-Manifolds, 2003. ↩︎

2. Daniele Panozzo, Enrico Puppo, Luigi Rocca, Efficient Multi-scale Curvature and Crease Estimation, 2010. ↩︎

3. Alec Jacobson, Algorithms and Interfaces for Real-Time Deformation of 2D and 3D Shapes, 2013. ↩︎

4. Andrei Sharf, Thomas Lewiner, Gil Shklarski, Sivan Toledo, and Daniel Cohen-Or. Interactive topology-aware surface reconstruction, 2007. ↩︎

5. Michael Kazhdan, Jake Solomon, Mirela Ben-Chen, Can Mean-Curvature Flow Be Made Non-Singular, 2012. ↩︎

6. Joseph S. B. Mitchell, David M. Mount, Christos H. Papadimitriou. The Discrete Geodesic Problem, 1987 ↩︎