From 9879a04efea29292f2dcf286e23cc8368ccb9229 Mon Sep 17 00:00:00 2001 From: Karma Riuk Date: Thu, 22 Jun 2023 18:18:49 +0200 Subject: [PATCH] Added some labels --- sections/theoretical_background.tex | 9 +++++++-- 1 file changed, 7 insertions(+), 2 deletions(-) diff --git a/sections/theoretical_background.tex b/sections/theoretical_background.tex index c612586..0e25022 100644 --- a/sections/theoretical_background.tex +++ b/sections/theoretical_background.tex @@ -1,4 +1,5 @@ \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}