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Fauconnet} diff --git a/sections/appendix.tex b/sections/appendix.tex index 0cf372c..454623f 100644 --- a/sections/appendix.tex +++ b/sections/appendix.tex @@ -44,3 +44,13 @@ it will be useful to simplify the result of the integral. &= \frac{m_Tl^2}{24} \left(1 + 3\cot^2\left(\frac{\theta}{2}\right)\right) \end{split} \end{equation} + +\newpage +\paragraph{Moment of inertia of sub-triangle of arbitrary polygon} +\begin{equation} + \label{eq:subtriangle_arbitrary_moment_long} + \begin{split} + I_{T_i} = \rho \int_0^1 \int_0^1 \vec r^2 hb \alpha \diff \alpha \diff \beta\\ + &= + \end{split} +\end{equation} diff --git a/sections/theoretical_background.tex b/sections/theoretical_background.tex index 0c9fe3d..25233d1 100644 --- a/sections/theoretical_background.tex +++ b/sections/theoretical_background.tex @@ -24,9 +24,12 @@ In the case of this project the axis of rotation is the one along the $z$-axis the polygon. The general formula for the moment of inertia is -$$ I_Q = \int \vec r^2 \rho(\vec r) \diff A $$ +\begin{equation} + \label{eq:moment_general} + I_Q = \int \vec r^2 \rho(\vec r) \diff \mathcal{A} +\end{equation} where $\rho$ is the density of object $Q$ in the point $\vec r$ across the -small pieces of area $A$ of the object. +small pieces of area $\mathcal A$ of the object. In our case, since we are implementing a 2D engine we can use the $\mathbb{R}^2$ coordinate systems, thus the formula becomes @@ -86,6 +89,7 @@ All the steps to compute equation~\ref{eq:rect_moment} can be found in equation \ref{eq:rect_moment_long} in Appendix \ref{appendix:calculations}. \subsubsection{Regular Polygons} +\label{sub:regular_polygons} A regular polygon is a shape that has sides of equal length and angles between those sides of equal measure. A polygon of $n$ sides can be subdivided in $n$ congruent (and isosceles since they are all the radius of the circumscribing @@ -140,8 +144,8 @@ equation \ref{eq:subtriangle_moment_long} in Appendix Now that we have the moment of inertia of the sub-triangle, we can make the link to the overall polygon. Since $$ \theta = \frac{2\pi}{n} \implies \frac{\theta}{2} = \frac{\pi}{n} $$ -and the moment of inertia are additive (as long they are as they are about the same -axis) we can get the moment of inertia with +and the moment of inertia are additive (as long they are as they are about the +same axis) we can get the moment of inertia with $$ I_{\text{regular}} = n I_T $$ and since the mass of the regular polygon $m$ is the sum of the masses of the sub-triangle @@ -152,10 +156,113 @@ we have that I_{\text{regular}} = \frac{ml^2}{24} \left( 1 + 3\cot^2\left(\frac{\pi}{n}\right) \right) \end{equation} - - - \subsubsection{Arbitrary Polygons} + +For arbitrary polygons, we are taking a slightly different approach. Using the +Cartesian coordinate system to solve the equation \ref{eq:moment} revealed to be +more cumbersome than useful. But similarly to regular polygons (c.f. Section +\ref{sub:regular_polygons}), we can use the additive property of the moment +inertia to divide our arbitrary polygon into sub-triangles. As opposed to +regular polygons, these triangles won't be congruent, so we can't just get the +moment of inertia of one of them and multiply it by the number of sides, but we +need to calculate them individually. So given a polygon of $n$ sides, we can +construct $n$ sub-triangles $T_i$, for $i = 1, \dots, n$. So the moment of +inertia $I$ of the polygon will be +\begin{equation} + I = \sum_i I_{T_i} +\end{equation} +\begin{figure}[H] + \centering + \begin{subfigure}[]{.5\textwidth} + \centering + \inputtikz[.7]{arbitrary} + \caption{An arbitrary 6-sided polygon} + \label{fig:arbitrary} + \end{subfigure} + \begin{subfigure}[]{.49\textwidth} + \centering + \inputtikz[.7]{arbitrary_divided} + \caption{Arbitrary polygon divided into 6 sub-triangles} + \label{fig:abitrary_divded} + \end{subfigure} +\end{figure} + +To calculate the moment of inertia $I_{T_i}$, instead of using the classical +$x$- and $y$-axis as we did before, we decided to use the edges of the triangle +as axis and therefore express what we need to integrate in function of those as +can be seen in Figure \ref{fig:abitrary_subtriangle}. +\begin{figure}[H] + \centering + \inputtikz[.7]{arbitrary_subtriangle} + \caption{Sub-triangle of arbitrary polygon} + \label{fig:abitrary_subtriangle} +\end{figure} + +In Figure \ref{fig:abitrary_subtriangle}, $C$ represent the barycenter of the +polygon (as is shown in Figure \ref{fig:abitrary_divded}). The axis we are going +to integrate on are $\vv{CA}$ and $\vv{AB}$. +We can now define +\begin{equation} \label{eq:alpha} + \vv{CP_1} = \alpha \vv{CA}, \qquad \vv{CP_2} = \alpha + \vv{CB}, \qquad \forall \alpha \in [0, 1] +\end{equation} +and +$$ \vv{P_1Q} = \beta \vv{P_1P_2}, \qquad \forall \beta \in [0, 1] $$ + +From \ref{eq:alpha}, it quickly follows that +\[ + \vv{P_1P_2} = \alpha \vv{AB} +\] +therefore +\begin{equation} \label{eq:beta_alpha} + \vv{P_1Q} = \beta \alpha \vv{AB} +\end{equation} +Finally, if we put together equations \ref{eq:alpha} and \ref{eq:beta_alpha}, we +have that +\begin{equation}\label{eq:r} + \vec r = \vv{CP_1} + \vv{P_1Q} = \alpha \vv{CA} + \beta \alpha \vv{AB} +\end{equation} + +Now we got the first part equation \ref{eq:moment_general}. To find the $\diff + \mathcal A$, we +just need to get the area of the square that contains $Q$ in Figure +\ref{fig:abitrary_subtriangle}. Since $\|\vv{AB}\|$ represents the base of the +triangle $T_i$, we can define +$$ b = \| \vv{AB}\| $$ +we consequently have that +\begin{equation} \label{eq:dA} + \diff \mathcal{A} = b \alpha \diff \beta h\diff \alpha +\end{equation} +where $h = \| \vv{CH} \|$ is the height of triangle. +We can now assemble \ref{eq:r} and \ref{eq:dA} +\begin{equation} + \label{eq:subtriangle_arbitrary_moment} + I_{T_i} = \rho \int_0^1 \int_0^1 \vec r^2 hb \alpha \diff \alpha + \diff \beta = \frac{\rho h b}{4} \left(\frac{1}{3} \vv{AB}^2 + + \vv{AB} \cdot \vv{CA} + \vv{CA}^2\right) +\end{equation} + +Since $\frac{\rho h b}{2}$ is the mass of the triangle we can write the result +as +\begin{equation} + I_{T_i} = \frac{m_{T_i}}{2} \left(\frac{1}{3} \vv{AB}^2 + \vv{AB} \cdot \vv{CA} + + \vv{CA}^2\right) +\end{equation} + + +All the steps to compute equation~\ref{eq:subtriangle_arbitrary_moment} can be +found in equation \ref{eq:subtriangle_arbitrary_moment_long} in Appendix +\ref{appendix:calculations}. + +Now that we have the moment of inertia of the sub-triangle, we can make the link +to the overall polygon. +\begin{equation} + I_{\text{arbitrary}} = \sum_i I_{T_i} = \sum_{i=1}^n \frac{m_{T_i}}{2} + \left(\frac{1}{3} \vv{P_iP_{i+1}}^2 + \vv{CP_i} \cdot \vv{P_iP_{i+1}} + + \vv{CP_i}^2\right) +\end{equation} +where, $P_{n+1} = P_1$ in the case of $i = n$. + \subsection{Collision detection} \subsubsection{Separating Axis Theorem} \subsubsection{Vertex collisions}