%==========================================
%
%   Appendices related to angular corrections
%
%==========================================

\vspace*{-2.cm}\section{Angular parametrization}\label{app:AngCorr}
\vspace{-1.0cm}
\begin{figure}[hb!]
	\centering\vspace{-5pt}
	\includegraphics[width=0.30\textwidth]{./Angular/projections/ctkeff0_KplusPi0Resolved_Run1.eps}
	\includegraphics[width=0.30\textwidth]{./Angular/projections/ctkeff1_KplusPi0Resolved_Run1.eps}
	\includegraphics[width=0.30\textwidth]{./Angular/projections/ctkeff2_KplusPi0Resolved_Run1.eps}\\  \vspace{-1pt}
	\includegraphics[width=0.30\textwidth]{./Angular/projections/ctkeff3_KplusPi0Resolved_Run1.eps}
	\includegraphics[width=0.30\textwidth]{./Angular/projections/ctkeff4_KplusPi0Resolved_Run1.eps}
	\includegraphics[width=0.30\textwidth]{./Angular/projections/ctkeff5_KplusPi0Resolved_Run1.eps}\\  \vspace{-1pt}
	\includegraphics[width=0.30\textwidth]{./Angular/projections/ctkeff6_KplusPi0Resolved_Run1.eps}
	\includegraphics[width=0.30\textwidth]{./Angular/projections/ctkeff7_KplusPi0Resolved_Run1.eps}
	\includegraphics[width=0.30\textwidth]{./Angular/projections/ctkeff8_KplusPi0Resolved_Run1.eps}\\  \vspace{-1pt}
	\includegraphics[width=0.30\textwidth]{./Angular/projections/ctkeff9_KplusPi0Resolved_Run1.eps}
	\includegraphics[width=0.30\textwidth]{./Angular/projections/ctkeff10_KplusPi0Resolved_Run1.eps} 
	\includegraphics[width=0.30\textwidth]{./Angular/projections/ctkeff11_KplusPi0Resolved_Run1.eps}\\ \vspace{-1pt}
	\includegraphics[width=0.30\textwidth]{./Angular/projections/ctkeff12_KplusPi0Resolved_Run1.eps}
	\includegraphics[width=0.30\textwidth]{./Angular/projections/ctkeff13_KplusPi0Resolved_Run1.eps}
	\includegraphics[width=0.30\textwidth]{./Angular/projections/ctkeff14_KplusPi0Resolved_Run1.eps}\\ \vspace{-1pt}
	\includegraphics[width=0.30\textwidth]{./Angular/projections/ctkeff15_KplusPi0Resolved_Run1.eps}
	\includegraphics[width=0.30\textwidth]{./Angular/projections/ctkeff16_KplusPi0Resolved_Run1.eps}
	\includegraphics[width=0.30\textwidth]{./Angular/projections/ctkeff17_KplusPi0Resolved_Run1.eps}\\ \vspace{-5pt}
	\captionof{figure}[Angular acceptance parametrization projections in \ctk for \runI.]{One-dimensional projections of the angular acceptance in the dimension of \ctk in 18 bins of \qsq. The data points are \runI PHSP MC sample, the solid curve is the four dimensional Legendre-polynomial parametrization. \label{fig:app_angProj_ctk_Run1}}
\end{figure}
\clearpage

\include{Chapters/Acceptance/app_angProj_Run1}
\include{Chapters/Acceptance/app_angProj_Run2}


%Both Legendre and Chebyshev are classical orthogonal polynomials
%\begin{equation}\label{eq:legendre}
%\int_{-1}^{1}P_m(x)P_n(x)dx =0 \quad\text{if\,} n \neq m\,.
%\end{equation}
%\begin{equation}\label{eq:OGlegendre}
%P_0(x)=1\,,\\
%P_1(x)=x\,,\\	
%\left(n+1\right)P_{n+1}(x) = \left(2n+1\right)xP_{n}(x)-nP_{n-1}(x)\,. 
%\end{equation}
%
%\begin{equation}\label{eq:OGcebysev}
%T_0(x)=1\,,\\
%T_1(x)=x\,,\\	
%T_{n+1}(x) = 2xT_{n}(x)-T_{n-1}(x)\,. 	
%\end{equation}
%
%p value:  greater than 0.05 $\Rightarrow$ believe the variables are independent
%\begin{equation}\label{eq:chi2}
%\chisq = \left[ \frac{(O-E)^2}{E}\right]\,\quad\text{O=observed, E = expected} 
%\end{equation}
%
