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sections/theoretical_background.tex
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@ -1,4 +1,5 @@
\section{Theoretical Background} \section{Theoretical Background}
\label{sec:theory}
The theoretical background is everything related to the physics part of the The theoretical background is everything related to the physics part of the
project. It covers the calculating 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 two 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. the original objective of the project, which was to have any arbitrary polygon.
\subsubsection{Vertex collisions} \subsubsection{Vertex collisions}
\label{sub:vertex-collision}
The solution that was adopted for the project, after trying SAT, was a more 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 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 We these variables, we can finally define the relative velocities $\vec
v_{p1}$ and $\vec v_{p2}$ v_{p1}$ and $\vec v_{p2}$
\[ \begin{split} \begin{equation}
\label{eq:vp1}
\begin{split}
\vec v_{p1} = \vec v_{ap1} - \vec v_{bp2}\\ \vec v_{p1} = \vec v_{ap1} - \vec v_{bp2}\\
\vec v_{p2} = \vec v_{ap2} - \vec v_{bp2} \vec v_{p2} = \vec v_{ap2} - \vec v_{bp2}
\end{split} \] \end{split}
\end{equation}
If we expand by using \ref{eq:vabp1} and \ref{eq:vabp2}, we get If we expand by using \ref{eq:vabp1} and \ref{eq:vabp2}, we get
\begin{equation} \begin{equation}
\begin{split} \begin{split}