BP_Arnaud_Fauconnet.pdf

@@ -1 +0,0 @@
-bachelorproject.pdf

bachelorproject.tex

@@ -5,45 +5,25 @@
 \usepackage[]{subcaption}
 \usepackage[]{float}
 \usepackage[]{multicol}
-\usepackage{mathtools}
-\usepackage[]{listings}
 
 \usepackage{tikz}
 \usepackage{tkz-euclide}
-\usetikzlibrary{external,shapes,through,arrows}
+\usetikzlibrary{external,shapes,through}
 
 \tikzexternalize[prefix=figures/]
 \tikzstyle{none}=[]
 \input{./tikzs/styles.tikzstyles}
 
-\tikzset{>=stealth}
-
 \pgfdeclarelayer{nodelayer}
 \pgfdeclarelayer{edgelayer}
 \pgfsetlayers{edgelayer,nodelayer,main}
 
-
-\lstset{language=C++,
-	basicstyle=\ttfamily,
-	keywordstyle=\color{blue}\ttfamily,
-	stringstyle=\color{red}\ttfamily,
-	commentstyle=\color{green}\ttfamily,
-	morecomment=[l][\color{magenta}]{\#},
-	emph={std, vector, vec2d, polygon, collision},
-	emphstyle=\color{orange}\ttfamily,
-	numbers=left,
-	morekeywords={uint},
-	numberstyle=\footnotesize,
-	numbersep=5pt,
-	frame=lines,
-	% breaklines=true,
-	% breaklines=true,
-	% postbreak=\raisebox{0ex}[0ex][0ex]{\space\ensuremath{\hookrightarrow}},
-}
-
 \graphicspath{{../figures/}{./figures/}}
 
+
 \newcommand*{\vv}[1]{\overrightarrow{#1}}
+
+
 \newcommand*{\figref}[1]{\figurename~\ref{#1}}
 
 \captionsetup{labelfont={bf}}
@@ -67,25 +47,25 @@
 	}
 }
 
-\abstract {Physics engines are a fun and interesting way to learn
+\abstract {
-  about a the laws of physics, as well as computer science.  They
+	Physics engines are a fun and interesting way to learn about a lot of
-  provide a real-time simulation of common physical phenomena, and
+	different subjects. First the theoretical concepts, such as the equations
-  therefore illustrate theoretical concepts such as the equations that
+	that dictate the motion of the objects, together with their components, need
-  dictate the motion of objects, The goal of this project was to
+	to be thoroughly understood. Then there is the necessity of finding a way to
-  extend an existing physics engine built for demonstration purposes.
+	represent all of those concepts in a given programming language and to
-  This engine was initially designed and developed to simulate
+	make them as efficient as possible so that the simulation runs fluidly.
-  circular objects (``balls'') in 2D. With this project, we intended
+	The task to be completed here was to extend an already existing
-  to extend this engine to also simulate arbitrary polygons, again in
+	physics engine that only made circles bounce off each other. The extension
-  a physically accurate way. The main technical challenges of the
+	was focused on having the ability to generate some arbitrary polygons and
-  project is therefore the correct simulation of the dynamics of
+	make them bounce off each other in a physically accurate way. The main
-  rigid, polygonal objects.  In particular, we developed a model of
+	issues that rose up during the development of the extension: determining the
-  polygonal rigid objects; we implemented a simulation of their
+	inertia of a arbitrary polygon, which is important for realistic
-  inertial motion, possibly in the presence of a constant force field
+	impacts; having an accurate collision detection system, which allows the
-  such as gravity; we detect collisions between objects; we compute
+	engine to know when to make two polygons bounce off each other. Once those
-  and then simulate the dynamic effects of collisions.  The
+	aspects were worked on and polished, the rest of the implementation went
-  simulations are animated and displayed in real-time.  It is also
+	smoothly.
-  therefore crucial that the simulation code be efficient to obtain
+
-  smooth animations.}
+}
 
 \counterwithin{figure}{section}
 \counterwithin{equation}{section}

sections/appendix.tex

@@ -3,7 +3,7 @@
 \section{Calculations}
 \label{appendix:calculations}
 
-\subsection{Moment of inertia of rectangle}
+\paragraph{Moment of inertia of rectangle}
 \begin{equation}
 	\label{eq:rect_moment_long}
 	\begin{split}
@@ -22,7 +22,7 @@
 \end{equation}
 
 \newpage
-\subsection{Moment of inertia of sub-triangle of regular polygon}
+\paragraph{Moment of inertia of sub-triangle of regular polygon}
 Before starting the calculations, it is to be noted that according to Figure
 \ref{fig:subtriangle}, we have that
 $$ \tan\left(\frac{\theta}{2}\right) = \frac{\frac{l}{2}}{h} = \frac{l}{2h} $$
@@ -46,7 +46,7 @@ it will be useful to simplify the result of the integral.
 \end{equation}
 
 \newpage
-\subsection{Moment of inertia of sub-triangle of arbitrary polygon} Recall
+\paragraph{Moment of inertia of sub-triangle of arbitrary polygon} Recall
 equation \ref{eq:r} defines
 $$ \vec r = \alpha \vv{CA} + \beta \alpha \vv{AB} $$
 \begin{equation}
@@ -62,37 +62,3 @@ $$ \vec r = \alpha \vv{CA} + \beta \alpha \vv{AB} $$
 		&= \frac{\rho hb}{4} \left(\frac{1}{3}\vv{AB}^2 + \vv{AB} \cdot \vv{CA} + \vv{CA}^2\right) \\
 	\end{split}
 \end{equation}
-
-\newpage
-\subsection{Solving for impulse parameter}
-\label{app:impulse_long}
-We start with equation \ref{eq:vp2n}:
-\begin{equation*}
-	\begin{split}
-		\vec v_{p2} \cdot \vec n &= - e \vec v_{p1} \cdot \vec n\\
-		\left(\vec v_{ap2} - \vec v_{bp2}\right)\cdot \vec n &= - e \vec v_{p1} \cdot \vec n\\
-		\left( \vec v_{a2} + \omega_{a2} \times \vec r_{ap} - \vec v_{b2} - \omega_{b2} \times \vec r_{bp}
-		\right)\cdot \vec n &= - e \vec v_{p1} \cdot \vec n\\
-	\end{split}
-\end{equation*}
-
-We now expand the bracket on the left-hand side using the equations \ref{eq:va2}
-- \ref{eq:omega_b2} and then simplify with equation \ref{eq:vp1}.
-
-\[
-	\begin{split}
-		\left( \vec v_{a1} +  \frac{j\vec n}{m_a} + \omega_{a1} + \frac{\vec r_{ap} \times j\vec n}{I_a} \times \vec r_{ap} - \vec v_{b1} +  \frac{j\vec n}{m_b} - \omega_{b1} + \frac{\vec r_{bp} \times j\vec n}{I_b} \times \vec r_{bp} \right)\cdot \vec n &= - e \vec v_{p1} \cdot \vec n\\
-		\left( \frac{j\vec n}{m_a} + \frac{\vec r_{ap} \times j\vec n}{I_a} \times \vec r_{ap} +  \frac{j\vec n}{m_b} + \frac{\vec r_{bp} \times j\vec n}{I_b} \times \vec r_{bp} \right)\cdot \vec n &= - (1+e) \vec v_{p1} \cdot \vec n\\
-	\end{split}
-\]
-Using the triple scalar product rule, we can derive that
-\[
-	j \left( \frac{1}{m_a} + \frac{1}{m_b} + \frac{\left(\vec r_{ap} \times \vec n\right)^2}{I_a} + \frac{\left( \vec r_{bp} \times \vec n \right)^2}{I_b}
-	\right) = -(1+e) \vec v_{p1} \cdot \vec n
-\]
-and therefore
-\begin{equation}
-	j = \frac{ - (1+e) \cdot \vec v_{ap1} \cdot \vec n }{\frac{1}{m_a} + \frac{1}{m_b} +
-		\frac{\left( \vec r_{ap} \times \vec n \right)^2}{I_a} + \frac{\left( \vec
-			r_{bp} \times \vec n \right)^2}{I_b}}
-\end{equation}

sections/conclusion.tex

@@ -1,26 +1 @@
 \section{Conclusion}
-
-The overall development of this extension was very instructive. Getting to
-discover some theoretical concepts, having to understand them and then being
-able to apply them in the code is a very rewarding experience. The objective of
-project is considered accomplished. In figure \ref{fig:before_after}, we can see
-a comparison of the state of the application before and after the extension.
-
-\begin{figure}[H]
-	\centering
-	\hfill
-	\begin{subfigure}[]{.4\textwidth}
-		\centering
-		\includegraphics[width=.8\textwidth]{before}
-		\caption{Original state of the simulation}
-	\end{subfigure}
-	\hfill
-	\begin{subfigure}[]{.4\textwidth}
-		\centering
-		\includegraphics[width=.8\textwidth]{after}
-		\caption{State after the extension}
-	\end{subfigure}
-	\hfill\null
-	\caption{Before v. After}
-	\label{fig:before_after}
-\end{figure}

sections/implementation.tex

@@ -1,178 +1 @@
 \section{Implementation}
-
-This section will dive into the actual implementing of the project, trying to
-describe how the theoretical concepts in section \ref{sec:theory}.
-
-\subsection{Structure}
-
-The added code to the project can be divided into three parts
-\begin{itemize}
-	\item polygon generator;
-	\item collision detection;
-	\item collision resolution.
-\end{itemize}
-
-Each part will be explained in the following sub-sections.
-
-\subsubsection{Polygon generator}
-Before talking about how we generate polygons, let's first talk about how the
-polygons are defined in this project.
-
-In the file \texttt{polygons.h} we define a polygon as
-
-\begin{lstlisting}[caption={Polygon class (simplified)},label={lst:polygon}]
-class polygon {
-    std::vector<vec2d> points;
-
-    vec2d center;
-    double angle;
-
-    double inertia;
-    double mass;
-
-    std::vector<vec2d> global_points();
-
-    vec2d speed;
-    double angular_speed;
-}
-\end{lstlisting}
-
-Polygons possesses multiple fields. Firstly we have
-\lstinline{std::vector<vec2d> points}, a collection of
-\lstinline{vec2d} objects, which represent the set of ordered points that
-compose the polygon. Those points are expressed in local coordinates, and the
-center of mass of the polygon is placed the origin.
-
-Now that we know how the polygon is composed, we want to move it around the
-space it lives in, that is the purpose of the \lstinline{vec2d center} field.
-It represents where the center of mass of the polygon is located in simulation.
-Since the shapes also rotate, we need to keep track of the rotation of the
-polygon, hence the use of \lstinline{double angle}. All of these fields are used
-when computing the result of the method \lstinline{std::vector<vec2d> global_points()}.
-
-Finally, we have \lstinline{double inertia} and \lstinline{double mass} which,
-as their name suggest, store the value for the polygon's inertia and mass, that
-are used to calculate the polygons final speed and angular speed, who are
-represented by \lstinline{vec2d speed} and \lstinline{double angular_speed}.
-
-\paragraph{Polygon generator} Now that we know how polygons are represented in
-our simulation, we can generate them. In \texttt{polygon\_generator.h} (and its
-implementation file \texttt{polygon\_generator.cc}), there are some functions
-for that. In Listing \ref{lst:polygon_gen} we can see the signature of the
-functions that generate
-\begin{itemize}
-	\item a rectangle of a certain width and height;
-	\item a square of a certain side length;
-	\item a triangle given two side lengths and an angle between them;
-	\item a regular polygon;
-	\item an arbitrary polygon.
-\end{itemize}
-
-\begin{lstlisting}[caption={Polygon Generator header
-file},label={lst:polygon_gen}]
-namespace poly_generate {
-    polygon rectangle(double width, double height);
-
-    inline polygon square(double width) {
-        assert(width > 0);
-        return rectangle(width, width, label);
-    };
-
-    polygon triangle(double side1, double side2, double angle);
-
-    polygon regular(double radius, uint n_sides);
-
-    polygon general(std::vector<vec2d> points);
-};
-\end{lstlisting}
-
-The implementation of those functions are fairly straight forward, they generate
-the appropriate number of points in the correct place. After what they calculate
-the mass and subsequent inertia of the polygon as shown in section
-\ref{sub:moment}.
-
-\subsubsection{Collision}
-\label{sub:implementation-collision-detection}
-
-The algorithm described in section~\ref{sub:vertex-collision} is implemented in
-\texttt{collision.cc} and exposed to other modules with \texttt{collision.h}.
-
-The module exposes a collision structure and a collides function.
-
-\begin{lstlisting}[caption={Collision header file},label={lst:collision}]
-struct collision {
-    bool collides = false;
-    vec2d impact_point;
-    vec2d n;
-    vec2d overlap;
-};
-
-extern collision collides(polygon& p, polygon& q);
-\end{lstlisting}
-
-The \lstinline{collides} function takes in two polygons and checks with
-vertex-collision algorithm whether they collide or not. The result is return
-through an instance of \lstinline{struct collision}. The \lstinline{bool collides}
-and \lstinline{vec2d impact_point} fields is self-explanatory. The
-\lstinline{vec2d n} is the normal vector that pushes \lstinline{polygon& p} away
-from \lstinline{polygon& q}. Finally, \lstinline{vec2d overlap} is a scalar
-multiplication of \lstinline{vec2d n}. Where the latter is a normalized vector,
-the former represents how deep \lstinline{p} is in \lstinline{q}. If we push
-\lstinline{p} the exact amount of \lstinline{overlap}, then \lstinline{p} would
-just be touching \lstinline{q} with point \lstinline{impact_point}, and no
-further overlap would occur.
-
-In reality, we do not simply push \lstinline{p} by \lstinline{overlap}, but we
-push \lstinline{p} and \lstinline{q} away from each other proportionally to
-their mass.
-
-\paragraph{Collision resolution} In \texttt{polygons.cc}, among other functions,
-we apply the collision resolution, which simply uses the physics results found
-in \ref{sub:resolution}.
-
-\subsection{Optimization}
-
-In order to optimise the collision detection and to avoid having to do a check
-whether each vertex of every polygon was colliding with each edge of every other
-polygon we decided to apply the bounding box acceleration structure.
-
-The bounding box of an object is, in two dimensions, the rectangle that contains
-the object and which the sides are aligned with axis of the coordinate system.
-
-\begin{figure}[H]
-	\centering
-	\inputtikz[.7]{bounding_box}
-	\caption{Bounding box example}
-	\label{fig:bounding-box}
-\end{figure}
-
-To get the four coordinates that compose, we just perform a linear scan through
-all the points that compose a polygon, record the minimum and maximum of both
-they $x$ and $y$ coordinate.
-
-Once those points have been found, we just perform some basic axis-aligned
-rectangle collision detection, which is much less computationally expensive than
-checking each vertex-edge pair.
-
-The complexity of the collision detection algorithm went from $\mathcal{O}(n^2)$
-to $\mathcal{O}(n)$, where $n$ is the total number of vertices across all
-polygons in the simulation.
-
-
-\subsection{Known issues}
-
-The simulation has one major flaw, and it is inherent to the last part of
-section \ref{sub:implementation-collision-detection} where we talk about the
-\lstinline{overlap} vector: we said that this vector allows to displace both
-polygons so that and the end of the collision resolution, they are not
-overlapping anymore. This was done to avoid issues where the collision
-resolution had taken place in one frame, but the polygons were still overlapping
-in the following frame, which lead to polygons getting stuck together.
-
-The unfortunate consequence of such a decision is that, when we start playing
-around with the restitution coefficient and all the shapes start falling to the
-ground, the shapes are not able to rest still. Since there is gravity, at each
-frame their speed get updated to simulate its effect, but since they are lying
-on the floor, the collision detection algorithm kicks in right away, resulting
-in the polygon getting pushed completely up, since the ground has infinite mass.
-This results in the polygon bouncing on the floor instead of lying down.

sections/theoretical_background.tex

@@ -1,61 +1,44 @@
 \section{Theoretical Background}
-\label{sec:theory}
 The theoretical background is everything related to the physics part of the
-project. It covers the calculation of the inertia of different types of polygons;
+project. It covers the calculating the inertia of different types of polygons;
-different algorithms to detect whether there is a collision between
+different algorithms to detect whether there is a collision between two
-any two polygons; and the resolution of the collision, that is, finding the final
+polygons; the resolution of the collision, i.e. finding the final
 velocity vectors and angular speed of those polygons.
 
-\subsection{Moment of Inertia}
+\subsection{Moment of inertia}
-\label{sub:moment}
 
-The inertia of an object refers to the tendency of an object to resist
+The inertia of an object refers to the tendency of an object to resist a change
-a change of its state of motion or rest, it describes how the object
+of its state of motion or rest, it describes how the object behaves when forces
-behaves when forces are applied to it. In particular, the mass of a
+are applied to it. An object with a lot of inertia requires more force to change
-point object (inertial mass) determines the inertia of that objects.
+its motion, either to make it move if it's at rest or to stop it if it's already
-Thus an object with a large mass requires more force to change its
+moving. On the other hand, an object with less inertia is easier to set in
-motion, either to make it move from an initial state at rest or to
+motion or bring to a halt.
-stop it if it is already moving. On the other hand, an object with a
-smaller mass is easier to set in motion or to bring to a halt.
 
-For objects that are not point-like, the motion includes a rotational
+The moment of inertia is similar but is used in a slightly different context, it
-component.
+specifically refers to the rotational inertia of an object. It measures an
+object's resistance to changes in its rotational motion and how its mass is
+distributed with respect to is axis of rotation.
 
-The moment of inertia is analogous to the mass in determining the
+In the case of this project the axis of rotation is the one along the $z$-axis
-inertia of an object with respect to its rotational motion. In other
+(perpendicular to the plane of the simulation) and placed at the barycenter of
-words, the moment of inertia determines the rotational inertia of an
+the polygon.
-object, that is, the object's resistance to changes in its rotational
-motion.  The moment of inertia of an object is therefore computed with
-respect to a rotation axis, and it is determined by the way the mass
-of the object is distributed with respect to the chosen axis of
-rotation.
 
-
+The general formula for the moment of inertia is
-For the purpose of this project, we are interested in a 2D simulation.
-This means that the axis of rotation is always parallel to the
-$z$-axis, perpendicular to the plane of the simulation.  Furthermore,
-we model the motion of an object using the barycenter of an object as
-its reference point.  We therefore compute the moment of inertia of a
-polygon with respect to a rotation axis that is perpendicular to the
-2D plane and that goes through the center of mass of the polygon.
-
-In general, the moment of inertia about a rotation axis $Z$ of a point
-mass $m$ at distance $r$ from $Z$ is $I_m=mr^2$.  Therefore, for a
-generic 2D object $Q$, we integrate over the whole 2D object:
 \begin{equation}
 	\label{eq:moment_general}
-	I_Q = \int_{\mathcal{A}}\|\vec r\|^2 \rho(\vec r) \diff \mathcal{A}
+	I_Q = \int \vec r^2 \rho(\vec r) \diff \mathcal{A}
 \end{equation}
-where $\rho$ is the density of object $Q$ (mass per unit area) in the
+where $\rho$ is the density of object $Q$ in the point $\vec r$ across the
-point $\vec r$ from the across the small pieces of area $\mathcal A$
+small pieces of area $\mathcal A$ of the object.
-of the object.
 
-In our case, we use 2D Cartesian coordinates centered in the point
+In our case, since we are implementing a 2D engine we can use the $\mathbb{R}^2$
-(axis) about which we compute he moment of inertia.  Furthermore, we
+coordinate systems, thus the formula becomes
-consider polygons of uniform density.  We can therefore compute:
+$$ I_Q = \iint \rho(x, y) \vec r^2 \diff x\diff y$$
+and since the requirements express that the mass of the polygons is spread
+uniformly across its surface, the formula finally becomes
 \begin{equation}
 	\label{eq:moment}
-	I_Q = \rho \iint(x^2 + y^2)\diff x\diff y
+	I_Q = \rho \iint x^2 + y^2 \diff x\diff y
 \end{equation}
 
 The bounds of the integral depend on the shape of the polygon. In the following
@@ -281,7 +264,6 @@ to the overall polygon.
 where, $P_{n+1} = P_1$ in the case of $i = n$.
 
 \subsection{Collision detection}
-\label{sub:collision-detection}
 
 Collision detection, as the name suggests, are the algorithms used to detect
 whether two polygons are colliding. The result of this procedure must be an
@@ -342,7 +324,6 @@ algorithm of our own. Moreover, SAT only supports convex polygons, which limits
 the original objective of the project, which was to have any arbitrary polygon.
 
 \subsubsection{Vertex collisions}
-\label{sub:vertex-collision}
 
 The solution that was adopted for the project, after trying SAT, was a more
 intuitive one, developed by Prof. Carzaniga. The idea is simple: check if a
@@ -481,164 +462,3 @@ find the normal. The results look realistic enough to be accepted.
 
 \subsection{Collision resolution}
 \label{sub:resolution}
-
-The collision resolution is the last step in the processing of the collision.
-Algorithmically, it is much less heavy than collision detection, since, once the
-simulation has two colliding polygons, a point of impact and a normal vector,
-it's just a case of applying the rigid body physics formulas to the polygons
-that are colliding. This part has been helped a lot by the works of Erik Neumann
-\cite{collision:resolution-site} and Chris Hecker
-\cite{collision:resolution-paper}.
-
-\begin{figure}[H]
-	\centering
-	\inputtikz[.5]{collision_resolution}
-	\caption{Collision resolution between polygons $A$ and $B$}
-	\label{fig:collision_resolution}
-\end{figure}
-
-\paragraph{Variable definition} Before getting into any maths, let's define some
-variables that we are going to use
-
-\begin{multicols}{2}
-	\begin{itemize}
-		\item $m_a, m_b = $ mass of the bodies $A$ and $B$
-		\item $\vec r_{ap} = $ distance vector from center of mass of body $A$ to
-		      point $P$
-		\item $\vec r_{bp} = $ distance vector from center of mass of body $A$ to
-		      point $P$
-		\item $\omega_{a1}, \omega_{b1} = $ initial angular velocity of bodies
-		      $A, B$
-		\item $\omega_{a2}, \omega_{b2} = $ final angular velocity of bodies
-		      $A, B$
-		\item $\vec v_{a1}, \vec v_{b1} =$ initial velocities of center of mass
-		      bodies $A, B$
-		\item $\vec v_{a2}, \vec v_{b2} =$ final velocities of center of mass
-		      bodies $A, B$
-		\item $\vec v_{ap1}=$ initial velocity of impact point $P$ on body $A$
-		\item $\vec v_{bp1}=$ initial velocity of impact point $P$ on body $B$
-		\item $\vec v_{p1}=$ initial relative velocity of impact points on body $A, B$
-		\item $\vec v_{p2}=$ final relative velocity of impact points on body $A, B$
-		\item $\vec n=$ normal vector
-		\item $ e=$ elastic coefficient (0 = inelastic, 1 = perfectly elastic)
-	\end{itemize}
-\end{multicols}
-
-Some of those variables, like the ones in the left column, are already given by
-the simulation, some of them have been computed thanks to the algorithms in
-section \ref{sub:collision-detection} (normal vector and position of point $P$),
-and some have still to be defined mathematically, such as $\vec v_{ap1},\vec
-	v_{bp1},\vec v_{p1}$ and $\vec v_{p2}$. So the velocity vectors of impact
-point $P$ on both bodies, before the collision, are
-\begin{equation} \label{eq:vabp1}
-	\begin{split}
-		\vec v_{ap1} = \vec v_{a1} + \omega_{a1} \times \vec r_{ap} \\
-		\vec v_{bp1} = \vec v_{b1} + \omega_{b1} \times \vec r_{bp}
-	\end{split}
-\end{equation}
-
-Similarly, the final velocities are
-\begin{equation} \label{eq:vabp2}
-	\begin{split}
-		\vec v_{ap2} = \vec v_{a2} + \omega_{a2} \times \vec r_{ap}\\
-		\vec v_{bp2} = \vec v_{b2} + \omega_{b2} \times \vec r_{bp}
-	\end{split}
-\end{equation}
-Here we are regarding the angular velocity as a 3-dimensional vector
-perpendicular to the plane, so that the cross product is calculated as
-
-$$ \omega \times \vec r = \begin{pmatrix} 0\\0\\\omega \end{pmatrix} \times
-	\begin{pmatrix} r_x\\r_y\\0 \end{pmatrix} = \begin{pmatrix} -\omega r_y \\
-		\omega r_x  \\0\end{pmatrix}  $$
-
-We these variables, we can finally define the relative velocities $\vec
-	v_{p1}$ and $\vec v_{p2}$
-
-\begin{equation}
-	\label{eq:vp1}
-	\begin{split}
-		\vec v_{p1} = \vec v_{ap1} - \vec v_{bp2}\\
-		\vec v_{p2} = \vec v_{ap2} - \vec v_{bp2}
-	\end{split}
-\end{equation}
-If we expand by using \ref{eq:vabp1} and \ref{eq:vabp2}, we get
-\begin{equation}
-	\begin{split}
-		\vec v_{p1} = \vec v_{a1} + \omega_{a1} \times \vec r_{ap} - \vec v_{b1}
-		+ \omega_{b1} \times \vec r_{bp}\\
-		\vec v_{p2} = \vec v_{a2} + \omega_{a2} \times \vec r_{ap} - \vec v_{b2} + \omega_{b2} \times \vec r_{bp}
-	\end{split}
-\end{equation}
-
-The relative velocity of point $P$ along the normal vector $\vec n$ is
-$$ \vec v_{p1} \cdot \vec n $$
-Note that for a collision to occur this relative normal velocity must be
-negative (that is, the objects must be approaching each other). Let e be the
-elasticity of the collision, having a value between 0 (inelastic) and 1
-(perfectly elastic). We now make an important assumption in the form of the
-following relation
-\begin{equation}
-	\label{eq:vp2n}
-	\vec v_{p2} \cdot \vec n = - e \vec v_{p1} \cdot \vec n
-\end{equation}
-This says that the velocity at which the objects fly apart is proportional to
-the velocity with which they were coming together. The proportionality factor is
-the elasticity $e$.
-
-\paragraph{Collision Impulse} In simple terms, collision impulse refers to the
-change in momentum experienced by an object during a collision. It is a measure
-of the force applied to an object over a short period of time. We imagine that
-during the collision there is a very large force acting for a very brief period
-of time. If you integrate (sum) that force over that brief time, you get the
-impulse.
-
-In our simulation, we are assuming that no friction is happening during the
-collision, so that the only force we have to consider is the one of along the
-normal vector $\vec n$. The friction would create a force perpendicular to the
-normal, and it would make things a little too complicated for the scope of this
-project.
-
-Since the only force we consider is the normal one, we can consider the net
-impulse to be $j \vec n$, where $j$ is the impulse parameter. The body $A$ will
-experience the net impulse $j \vec n$ and body $B$ will experience it's negation
-$- j \vec n$ since the force that $B$ experience is equal an opposite to the one
-experienced by $A$. The impulse is a change in momentum. Momentum has units of
-velocity times mass, so if we divide the impulse by the mass we get the change
-in velocity. We can relate initial and final velocities as
-
-\begin{equation}
-	\label{eq:va2}
-	\vec v_{a2} = \vec v_{a1} +  \frac{j\vec n}{m_a}
-\end{equation}
-\begin{equation}
-	\label{eq:vb2}
-	\vec v_{b2} = \vec v_{b1} -  \frac{j\vec n}{m_b}
-\end{equation}
-
-
-For the final angular speed, it's change from the impulse $j\vec n$ is given by
-$\vec r_{ap} \times j \vec n$. We can divide the result by the moment of
-inertia, which was calculated in section \ref{sub:moment}, to get convert the
-change in angular momentum into change in angular speed. Similarly as above, we
-can relate the initial and final angular velocity as
-
-\begin{equation}
-	\label{eq:omega_a2}
-	\omega_{a2} = \omega_{a1} + \frac{\vec r_{ap} \times j\vec n}{I_a}
-\end{equation}
-\begin{equation}
-	\label{eq:omega_b2}
-	\omega_{b2} = \omega_{b1} - \frac{\vec r_{bp} \times j\vec n}{I_b}
-\end{equation}
-
-\paragraph{Solving for the impulse parameter}
-Now we have everything we need to solve for the impulse parameter $j$. The bulk
-of the calculations can be found in section \ref{app:impulse_long} of the
-appendix \ref{appendix:calculations}. But basically we start with equation
-\ref{eq:vp2n} and we end up with
-\begin{equation}
-	\label{eq:j}
-	j = \frac{ - (1+e) \cdot \vec v_{ap1} \cdot \vec n }{\frac{1}{m_a} + \frac{1}{m_b} +
-		\frac{\left( \vec r_{ap} \times \vec n \right)^2}{I_a} + \frac{\left( \vec
-			r_{bp} \times \vec n \right)^2}{I_b}}
-\end{equation}

tikzs/arbitrary_subtriangle.tikz

@@ -37,7 +37,7 @@
 		\draw [style=none] (15.center) to (14.center);
 		\draw [style=none] (14.center) to (13.center);
 		\draw [style=none] (13.center) to (12.center);
-		\draw [style=Vector] (0.center) to (16);
+		\draw [style=Axis] (0.center) to (16);
 		\draw [style=Dotted] (0.center) to (21);
 		\draw (23.center) to (26.center);
 		\draw (26.center) to (25.center);

tikzs/bounding_box.tikz

@@ -1,71 +0,0 @@
-\begin{tikzpicture}
-	\begin{pgfonlayer}{nodelayer}
-		\node [style=none] (0) at (0, -1) {};
-		\node [style=none] (1) at (0, 8) {};
-		\node [style=none] (2) at (-1, 0) {};
-		\node [style=none] (3) at (15, 0) {};
-		\node [style=none] (4) at (4, 4) {};
-		\node [style=none] (5) at (2, 1) {};
-		\node [style=none] (6) at (6, 3) {};
-		\node [style=none] (7) at (9, 3) {};
-		\node [style=none] (8) at (11, 2) {};
-		\node [style=none] (9) at (10.5, 6.5) {};
-		\node [style=none] (10) at (12.75, 5.75) {};
-		\node [style=none] (11) at (13.5, 3.5) {};
-		\node [style=none] (12) at (8, 5) {};
-		\node [style=none] (13) at (2, 4) {};
-		\node [style=none] (14) at (6, 1) {};
-		\node [style=none] (15) at (6, 4) {};
-		\node [style=none] (16) at (8, 2) {};
-		\node [style=none] (17) at (8, 6.5) {};
-		\node [style=none] (18) at (13.5, 6.5) {};
-		\node [style=none] (19) at (13.5, 2) {};
-		\node [style=none] (20) at (2, 0) {};
-		\node [style=none] (21) at (6, 0) {};
-		\node [style=none] (22) at (0, 4) {};
-		\node [style=none] (23) at (0, 1) {};
-		\node [style=none] (24) at (-0.25, 1) {$A_{y_{\min}}$};
-		\node [style=none] (25) at (-0.25, 4) {$A_{y_{\max}}$};
-		\node [style=none] (26) at (2, -0.25) {$A_{x_{\min}}$};
-		\node [style=none] (27) at (6, -0.25) {$A_{x_{\max}}$};
-		\node [style=none] (28) at (8, 0) {};
-		\node [style=none] (29) at (8, -0.25) {$B_{x_{\min}}$};
-		\node [style=none] (30) at (13.5, 0) {};
-		\node [style=none] (31) at (13.5, -0.25) {$B_{x_{\max}}$};
-		\node [style=none] (32) at (0, 2) {};
-		\node [style=none] (33) at (-0.25, 2) {$B_{y_{\min}}$};
-		\node [style=none] (34) at (0, 6.5) {};
-		\node [style=none] (35) at (-0.25, 6.5) {$B_{y_{\max}}$};
-		\node [style=none] (36) at (4, 2.75) {$A$};
-		\node [style=none] (37) at (10.5, 4.5) {$B$};
-	\end{pgfonlayer}
-	\begin{pgfonlayer}{edgelayer}
-		\draw [style=Vector] (0.center) to (1.center);
-		\draw [style=Vector] (2.center) to (3.center);
-		\draw (5.center) to (4.center);
-		\draw (4.center) to (6.center);
-		\draw (6.center) to (5.center);
-		\draw (12.center) to (9.center);
-		\draw (9.center) to (10.center);
-		\draw (10.center) to (11.center);
-		\draw (11.center) to (8.center);
-		\draw (8.center) to (7.center);
-		\draw (7.center) to (12.center);
-		\draw [style=Dashed] (13.center) to (5.center);
-		\draw [style=Dashed] (5.center) to (14.center);
-		\draw [style=Dashed] (14.center) to (15.center);
-		\draw [style=Dashed] (15.center) to (13.center);
-		\draw [style=Dashed] (16.center) to (17.center);
-		\draw [style=Dashed] (17.center) to (18.center);
-		\draw [style=Dashed] (18.center) to (19.center);
-		\draw [style=Dashed] (19.center) to (16.center);
-		\draw [style=Dotted] (19.center) to (30.center);
-		\draw [style=Dotted] (16.center) to (28.center);
-		\draw [style=Dotted] (16.center) to (32.center);
-		\draw [style=Dotted] (13.center) to (22.center);
-		\draw [style=Dotted] (17.center) to (34.center);
-		\draw [style=Dotted] (5.center) to (20.center);
-		\draw [style=Dotted] (5.center) to (23.center);
-		\draw [style=Dotted] (14.center) to (21.center);
-	\end{pgfonlayer}
-\end{tikzpicture}

tikzs/isosceles.tikz

@@ -17,8 +17,8 @@
 		\node [style=none] (14) at (2, .5) {\Large$\frac{\theta}{2}$};
 	\end{pgfonlayer}
 	\begin{pgfonlayer}{edgelayer}
-		\draw [style=Vector] (0.center) to (1.center);
+		\draw [style=Axis] (0.center) to (1.center);
-		\draw [style=Vector] (2.center) to (3.center);
+		\draw [style=Axis] (2.center) to (3.center);
 		\draw (4.center) to (5.center);
 		\draw (5.center) to (6.center);
 		\draw (6.center) to (4.center);

tikzs/rectangle_inertia2d.tikz

@@ -25,7 +25,7 @@
 		\draw (1.center) to (2.center);
 		\draw (2.center) to (3.center);
 		\draw (3.center) to (0.center);
-		\draw [style=Vector] (16.center) to (11.center);
+		\draw [style=Axis] (16.center) to (11.center);
-		\draw [style=Vector] (15.center) to (12.center);
+		\draw [style=Axis] (15.center) to (12.center);
 	\end{pgfonlayer}
 \end{tikzpicture}

tikzs/rectangle_inertia3d.tikz

@@ -25,8 +25,8 @@
 		\draw (1.center) to (2.center);
 		\draw (2.center) to (3.center);
 		\draw (3.center) to (0.center);
-		\draw [style=Vector] (16.center) to (11.center);
+		\draw [style=Axis] (16.center) to (11.center);
-		\draw [style=Vector] (15.center) to (12.center);
+		\draw [style=Axis] (15.center) to (12.center);
-		\draw [style=Vector] (21.center) to (22.center);
+		\draw [style=Axis] (21.center) to (22.center);
 	\end{pgfonlayer}
 \end{tikzpicture}