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\section{Theoretical Background}
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\label{sec:theory}
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The theoretical background is everything related to the physics part of the
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project. It covers the calculating the inertia of different types of polygons;
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different algorithms to detect whether there is a collision between two
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@ -326,6 +327,7 @@ algorithm of our own. Moreover, SAT only supports convex polygons, which limits
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the original objective of the project, which was to have any arbitrary polygon.
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\subsubsection{Vertex collisions}
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\label{sub:vertex-collision}
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The solution that was adopted for the project, after trying SAT, was a more
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intuitive one, developed by Prof. Carzaniga. The idea is simple: check if a
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@ -537,10 +539,13 @@ $$ \omega \times \vec r = \begin{pmatrix} 0\\0\\\omega \end{pmatrix} \times
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We these variables, we can finally define the relative velocities $\vec
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v_{p1}$ and $\vec v_{p2}$
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\[ \begin{split}
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\begin{equation}
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\label{eq:vp1}
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\begin{split}
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\vec v_{p1} = \vec v_{ap1} - \vec v_{bp2}\\
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\vec v_{p2} = \vec v_{ap2} - \vec v_{bp2}
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\end{split} \]
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\end{split}
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\end{equation}
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If we expand by using \ref{eq:vabp1} and \ref{eq:vabp2}, we get
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\begin{equation}
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\begin{split}
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