libigl_Martin/tutorial/readme.md

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Introduction

TODO

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
• 202 Gaussian Curvature

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:

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

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.

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:

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

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

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