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<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8"/>
<link type="text/css" rel="stylesheet" href="shared/css/style.css"/>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({ TeX: { equationNumbers: {autoNumber: "all"} } });
</script>
<script type="text/javascript" src="shared/js/MathJax/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
</head>
<body>
<div style="display:none">
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</div>
<h1 id="geometryprocessing–introduction">Geometry Processing – Introduction</h1>
<blockquote>
<p><strong>To get started:</strong> Fork this repository then issue</p>
<pre><code>git clone --recursive http://github.com/[username]/geometry-processing-introduction.git
</code></pre>
</blockquote>
<p>Welcome to Geometry Processing! The purpose of this assignment will be to get
you up and running with the two C++ libraries we will be using:
<a href="http://eigen.tuxfamily.org">Eigen</a> for dense and sparse linear algebra
routines and <a href="https://github.com/libigl/libigl">libigl</a> for geometry processing
routines. We will make use of the OpenGL-based viewer used by the <a href="http://libigl.github.io/libigl/tutorial/tutorial.html">libigl
tutorial</a>. This viewer
also depends on <a href="http://www.glfw.org/download.html">glfw</a>, a library for
managing windows on Linux, Mac OS X and windows.</p>
<h2 id="prerequisiteinstallation">Prerequisite installation</h2>
<p>On all platforms, we will assume you have installed cmake and a modern c++
compiler on Mac OS X<a href="#¹macusers">¹</a>, Linux<a href="#²linuxusers">²</a>, or
Windows<a href="#³windowsusers">³</a>.</p>
<p>We also assume that you have cloned this repository using the <code>--recursive</code>
flag (if not then issue <code>git submodule update --init --recursive</code>). </p>
<h2 id="layout">Layout</h2>
<p>All assignments will have a similar directory and file layout:</p>
<pre><code>README.md
CMakeLists.txt
main.cpp
include/
function1.h
function2.h
...
src/
function1.cpp
function2.cpp
...
shared/
libigl/
include/
igl/
...
...
</code></pre>
<p>The <code>README.md</code> file will describe the background, contents and tasks of the
assignment.</p>
<p>The <code>CMakeLists.txt</code> file setups up the cmake build routine for this
assignment.</p>
<p>The <code>main.cpp</code> file will include the headers in the <code>include/</code> directory and
link to the functions compiled in the <code>src/</code> directory. This file contains the
<code>main</code> function that is executed when the program is run from the command line.</p>
<p>The <code>include/</code> directory contains one file for each function that you will
implement as part of the assignment. <strong><em>Do not change</em></strong> these files.</p>
<p>The <code>src/</code> directory contains <em>empty implementations</em> of the functions
specified in the <code>include/</code> directory. This is where you will implement the
parts of the assignment.</p>
<p>The <code>shared/</code> directory will contain shared resources: cmake files, dependences
(e.g., libigl) and data. Feel free to poke around in here, but you shouldn’t
change any of these files.</p>
<h2 id="compilation">Compilation</h2>
<p>This and all following assignments will follow a typical cmake/make build
routine. Starting in this directory, issue:</p>
<pre><code>mkdir build
cd build
cmake ..
make
</code></pre>
<p>Why don’t you try this right now?</p>
<h2 id="execution">Execution</h2>
<p>Once built, you can execute the assignment from inside the <code>build/</code> using </p>
<pre><code>./introduction
</code></pre>
<p>After <a href="#compilation">compiling according to the instructions above</a>, if you try
executing right now, then you’ll see a bunny:</p>
<figure>
<img src="images/bunny-screenshot.jpg" alt="Screenshot of viewer displaying a bunny" />
<figcaption>Screenshot of viewer displaying a bunny</figcaption>
</figure>
<p>You can click and drag to change the view.</p>
<p>Optionally, this program can input a path to a triangle mesh file (other than
the bunny):</p>
<pre><code>./introduction [path to input file]
</code></pre>
<h2 id="background">Background</h2>
<blockquote>
<p>Every assignment, including this one, will start with a <strong>Background</strong>
section. This will review the math and algorithms behind the task in the
assignment. Students following the lectures should already be familiar with
this material and may opt to skip this section.</p>
</blockquote>
<p>Let’s get familiar with the <em>explicit</em> mesh representation of a discrete
surface immersed in <span class="math">\(\R^3\)</span>. Throughout the course, we will store the set of
mesh vertices <span class="math">\(V\)</span><a href="#⁴latex">⁴</a> and the set of triangles (a.k.a.
faces) <span class="math">\(F\)</span> as two matrices: <code>V</code> and <code>F</code>.</p>
<p>The matrix <code>V</code> is <span class="math">\(|V|\)</span> by 3 in size, where the ith row of this matrix contains
the x-, y- and z-coordinates of the ith vertex of the mesh. </p>
<p>The matrix <code>F</code> is <span class="math">\(|F|\)</span> by 3 in size, where the jth row of this matrix contains
the indices into the rows of <code>V</code> of the first, second and third corners of the
jth triangle as a non-negative number (remember in C++ arrays and matrices
start with index <code>0</code>). </p>
<p>The information in <code>V</code> alone is purely positional and encodes the
<em>geometry</em> of the surface.</p>
<p>The information in <code>F</code> alone is purely combinatoric and encodes the <em>topology</em>
of the surface. </p>
<p>By convention, the indices in each row of <code>F</code> are <em>ordered</em> counter-clockwise
around the triangle. Using the right-hand rule, we can define the normal of
each triangle as the vector that points <em>most away</em> from the surface.</p>
<figure>
<img src="images/right-hand-rule.jpg" alt="The right-hand rule and the counterclockwise ordering convention defines the
normal of a triangle." />
<figcaption>The right-hand rule and the counterclockwise ordering convention defines the
normal of a triangle.</figcaption>
</figure>
<p>Each oriented triangle also defines three <em>directed edges</em> between its three
vertices. Other triangles in the mesh may contain edges with the same incident
vertices, possibly in the opposite direction. A manifold mesh will have at most
two triangles incident on the same (undirected) <em>edge</em>, therefor we’ll refer to
each triangle’s directed edge as a <em>half-edge</em>.</p>
<figure>
<img src="images/half-edges.jpg" alt="Two neighboring triangles may share the same (unoriented) edge (thick black).
In a consistently oriented mesh, these triangles' corresponding half-edges
(orange) will have opposite orientation." />
<figcaption>Two neighboring triangles may share the same (unoriented) edge (thick black).
In a consistently oriented mesh, these triangles’ corresponding half-edges
(orange) will have opposite orientation.</figcaption>
</figure>
<p>The number of vertices <span class="math">\(|V|\)</span> and number of faces <span class="math">\(|F|\)</span> and number of unique
(undirected) edges <span class="math">\(|E|\)</span> are <em>intimately</em> related. Adding a new triangle to a
mesh may mean increasing the number of vertices and edges, too. The algebraic
relationship between the number of vertices, edges and faces reveals the
topological <em>genus</em> of the surface via the <em>Euler Characteristic</em> </p>
<p>$$
χ = 2c - 2h - b,
$$</p>
<p>where <span class="math">\(c\)</span> is the number of connected components, <span class="math">\(h\)</span> is number of
holes (as in donut), and <span class="math">\(b\)</span> is the number of connected components of the
boundary of the surface.</p>
<p>For meshes representing polyhedral surfaces, the Euler Characteristic can be
computed very simply:</p>
<pre><code>Chi = |V| - |E| + |F|.
</code></pre>
<p>Assuming no <em>unreferenced</em> vertices in <code>V</code>, each of the qunatitites in the
right-hand side can be determined from <code>F</code> alone. This indicates that its a
purely topological property. Changing the geometric positions (i.e., changing
the vertex positions in <code>V</code>) will not affect the Euler Characteristic. Due to
this, we say that the Euler Characteristic is a <em>topological invariant</em>.</p>
<h2 id="tasks">Tasks</h2>
<p>Every assignment, including this one, will contain a <strong>Tasks</strong> section. This
will enumerate all of the tasks a student will need to complete for this
assignment. These tasks will match the header/implementation pairs in the
<code>include/</code>/<code>src/</code> directories.</p>
<h3 id="groundrules">Groundrules</h3>
<p>Libigl has implemented many of the tasks you’ll find in this course. As a
result, for some assignments, including this one, you’ll see a <strong>Groundrules</strong>
section that lists which functions you can and should use from libigl and/or
functions you may not use (and should avoid copying your answers from).</p>
<h4 id="blacklistlibiglfunctions">Blacklist libigl functions</h4>
<p>For this assignment you may not use</p>
<ul>
<li><code>igl::all_edges</code></li>
<li><code>igl::edge_flaps</code></li>
<li><code>igl::edge_topology</code></li>
<li><code>igl::edges</code></li>
<li><code>igl::euler_characteristic</code></li>
<li><code>igl::exterior_edges</code></li>
<li><code>igl::is_boundary_edge</code></li>
<li><code>igl::unique_edge_map</code></li>
<li>or any other libigl function that returns a list of edges.</li>
</ul>
<h3 id="srcedges.cpp"><code>src/edges.cpp</code></h3>
<p>From a list of triangles <code>F</code>, construct a <span class="math">\(|E|\)</span> by 2 matrix <code>E</code>, where the kth
row of this matrix contains the indices into the rows of <code>V</code> of the start and
end point of the kth edge in the mesh. <code>E</code> should contain every <em>undirected
edge</em> exactly once. </p>
<h3 id="srceuler_characteristic.cpp"><code>src/euler_characteristic.cpp</code></h3>
<p>From the list of triangles <code>F</code>, return the Euler Characteristic <code>X</code> of the
triangle mesh. You may and should use your <code>edges</code> function from the previous
taks.</p>
<h3 id="submission">Submission</h3>
<p>Submit your completed homework as a <a href="https://github.com/alecjacobson/geometry-processing-introduction/pulls">New Pull Request to this repository</a>.</p>
<h3 id="questions">Questions?</h3>
<p>Direct your questions to the <a href="https://github.com/alecjacobson/geometry-processing-introduction/issues">Issues page of this
repository</a>.</p>
<h3 id="answers">Answers?</h3>
<p>Help your fellow students by answering questions or positions helpful tips on
the <a href="https://github.com/alecjacobson/geometry-processing-introduction/issues">Issues page of this
repository</a>.</p>
<hr />
<blockquote>
<h4 id="¹macusers">¹ Mac Users</h4>
<p>You will need to install Xcode if you haven’t already. </p>
<h4 id="²linuxusers">² Linux Users</h4>
<p>Many linux distributions do not include gcc and the basic development tools
in their default installation. On Ubuntu, you need to install the following
packages:</p>
<pre><code>sudo apt-get install git
sudo apt-get install build-essential
sudo apt-get install cmake
sudo apt-get install libx11-dev
sudo apt-get install mesa-common-dev libgl1-mesa-dev libglu1-mesa-dev
sudo apt-get install libxrandr-dev
sudo apt-get install libxi-dev
sudo apt-get install libxmu-dev
sudo apt-get install libblas-dev
</code></pre>
<h4 id="³windowsusers">³ Windows Users</h4>
<p>libigl only supports the Microsoft Visual Studio
2015 compiler in 64bit mode. It will not work with a 32bit build and it will
not work with older versions of visual studio.</p>
<h4 id="⁴latex">⁴ LaTeX</h4>
<p>This markdown document, and those for all other assignments, contains
<span class="math">\(\LaTeX\)</span> math. GitHub just shows the un-evaluated LaTeX code, but other
markdown browsers will show the typeset math. You can also generate
<code>README.html</code> using
<a href="http://fletcherpenney.net/multimarkdown/">multimarkdown</a>:</p>
<pre><code>cat shared/markdown/header.md README.md | multimarkdown --process-html -o README.html
</code></pre>
</blockquote>
</body>
</html>