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@@ -1,3 +1,6 @@
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+title: libigl Tutorial
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+author: Alec Jacobson, Daniele Pannozo and others
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+date: 20 June 2014
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css: style.css
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html header: <script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
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<link rel="stylesheet" href="http://yandex.st/highlightjs/7.3/styles/default.min.css">
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@@ -5,8 +8,25 @@ html header: <script type="text/javascript" src="http://cdn.mathjax.org/mathja
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<script>hljs.initHighlightingOnLoad();</script>
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# Introduction
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-
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-TODO
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+Libigl is an open source C++ library for geometry processing research and
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+development.
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+Dropping the heavy data structures of tradition geometry libraries, libigl is
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+a simple header-only library of encapsulated functions.
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+%
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+This combines the rapid prototyping familiar to \textsc{Matlab} or
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+\textsc{Python} programmers
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+with the performance and versatility of C++.
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+%
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+The tutorial is a self-contained, hands-on introduction to \libigl.
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+%
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+Via live coding and interactive examples, we demonstrate how to accomplish
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+various common geometry processing tasks such as computation of differential
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+quantities and operators, real-time deformation, global parametrization,
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+numerical optimization and mesh repair.
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+%
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+Accompanying lecture notes contain further details and cross-platform example
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+applications for each topic.
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+\end{abstract}
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# Index
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@@ -14,6 +34,8 @@ TODO
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* **101_Serialization**: Example of using the XML serialization framework
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* **102_DrawMesh**: Example of plotting a mesh
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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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# Compilation Instructions
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@@ -68,7 +90,7 @@ on a triangle mesh is via a vertex's _angular deficit_:
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$k_G(v_i) = 2π - \sum\limits_{j\in N(i)}θ_{ij},$
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where $N(i)$ are the triangles incident on vertex $i$ and $θ_{ij}$ is the angle
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-at vertex $i$ in triangle $j$. (**Alec: cite Meyer or something**)
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+at vertex $i$ in triangle $j$ [][#meyer_2003].
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Just like the continuous analog, our discrete Gaussian curvature reveals
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elliptic, hyperbolic and parabolic vertices on the domain.
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@@ -76,7 +98,7 @@ elliptic, hyperbolic and parabolic vertices on the domain.
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-## <a id=principal></a> Curvature Directions
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+## <a id=curv></a> Curvature Directions
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The two principal curvatures $(k_1,k_2)$ at a point on a surface measure how much the
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surface bends in different directions. The directions of maximum and minimum
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(signed) bending are call principal directions and are always
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@@ -93,7 +115,7 @@ normal:
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$-\Delta \mathbf{x} = H \mathbf{n}.$
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It is easy to compute this on a discrete triangle mesh in libigl using the cotangent
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-Laplace-Beltrami operator (**Alec: cite Meyer**):
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+Laplace-Beltrami operator [][#meyer_2003].
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```cpp
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#include <igl/cotmatrix.h>
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@@ -111,10 +133,10 @@ H = (HN.rowwise().squaredNorm()).array().sqrt();
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Combined with the angle defect definition of discrete Gaussian curvature, one
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can define principal curvatures and use least squares fitting to find
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-directions (**Alec: cite meyer**).
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+directions [][#meyer_2003].
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Alternatively, a robust method for determining principal curvatures is via
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-quadric fitting (**Alec: cite whatever we're using**). In the neighborhood
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+quadric fitting [][#pannozo_2010]. In the neighborhood
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around every vertex, a best-fit quadric is found and principal curvature values
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and directions are sampled from this quadric. With these in tow, one can
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compute mean curvature and Gaussian curvature as sums and products
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@@ -133,3 +155,48 @@ int main(int argc, char * argv[])
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return 0;
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}
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```
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+
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+## <a id=grad></a> Gradient
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+Scalar functions on a surface can be discretized as a piecewise linear function
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+with values defined at each mesh vertex:
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+
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+ $f(\mathbf{x}) \approx \sum\limits_{i=0}^n \phi_i(\mathbf{x})\, f_i,$
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+
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+where $\phi_i$ is a piecewise linear hat function defined by the mesh so that
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+for each triangle $\phi_i$ is _the_ linear function which is one only at
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+vertex $i$ and zero at the other corners.
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+
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+
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+
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+Thus gradients of such piecewise linear functions are simply sums of gradients
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+of the hat functions:
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+
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+ $\nabla f(\mathbf{x}) \approx
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+ \nabla \sum\limits_{i=0}^n \nabla \phi_i(\mathbf{x})\, f_i =
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+ \sum\limits_{i=0}^n \nabla \phi_i(\mathbf{x})\, f_i.$
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+
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+This reveals that the gradient is a linear function of the vector of $f_i$
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+values. Because $\phi_i$ are linear in each triangle their gradient are
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+_constant_ in each triangle. Thus our discrete gradient operator can be written
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+as a matrix multiplication taking vertex values to triangle values:
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+
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+ $\nabla f \approx \mathbf{G}\,\mathbf{f},$
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+
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+where $\mathbf{f}$ is $n\times 1$ and $\mathbf{G}$ is an $m\times n$ sparse
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+matrix. This matrix $\mathbf{G}$ can be derived geometrically, e.g.
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+[ch. 2][#jacobson_thesis_2013].
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+Libigl's `gradMat`**Alec: check name** function computes $\mathbf{G}$ for
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+triangle and tetrahedral meshes:
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+
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+
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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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+[#pannozo_2010]: Daniele Pannozo, Enrico Puppo, Luigi Rocca,
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+"Efficient Multi-scale Curvature and Crease Estimation," 2010.
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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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Alec Jacobson
