Added some labels

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Karma Riuk 2023-06-22 18:18:49 +02:00
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\section{Theoretical Background}
\label{sec:theory}
The theoretical background is everything related to the physics part of the
project. It covers the calculating the inertia of different types of polygons;
different algorithms to detect whether there is a collision between two
@ -326,6 +327,7 @@ 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
@ -537,10 +539,13 @@ $$ \omega \times \vec r = \begin{pmatrix} 0\\0\\\omega \end{pmatrix} \times
We these variables, we can finally define the relative velocities $\vec
v_{p1}$ and $\vec v_{p2}$
\[ \begin{split}
\begin{equation}
\label{eq:vp1}
\begin{split}
\vec v_{p1} = \vec v_{ap1} - \vec v_{bp2}\\
\vec v_{p2} = \vec v_{ap2} - \vec v_{bp2}
\end{split} \]
\end{split}
\end{equation}
If we expand by using \ref{eq:vabp1} and \ref{eq:vabp2}, we get
\begin{equation}
\begin{split}