AreaAproximation
Author:
Javi Pérez
Last Updated:
4 anni fa
License:
Creative Commons CC BY 4.0
Abstract:
Area calculation aproximation: rectangles + triangles
\begin
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\begin
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\documentclass[compress,9pt]{beamer}
\usepackage{tikz} %TikZ is required for this to work. Make sure this exists before the next line
\begin{document}
\section{Area aproximation: Rectangles + Triangles }
\begin{frame}[fragile]{Area $\approxeq$ Rectangles + Triangles}
\begin{figure}[h]
\centering
\pgfmathsetmacro{\circleRadius}{4}
\resizebox{0.5\textwidth}{!}{
\begin{tikzpicture}
% Draw X axis
\draw[thick,->] (0,0) -- (\circleRadius+0.3,0) node[anchor=north west]{$t$};
% Draw X axis labels
\foreach \x in {0,...,\circleRadius}
{
\draw (\x,0) -- (\x,-3pt) node[anchor=north] {$t_\x$};
}
% Draw Y axis
\draw[thick,->] (0,0) -- (0,\circleRadius+0.3) node[anchor=south east]{$\omega$};
% Draw a quarter of circle
\draw[red,thick] (0,0\circleRadius) arc[start angle=90, end angle=0, radius=\circleRadius];
% Draw trapeziums
\foreach \thisX in {1,...,\circleRadius}
{
% Calculate rectangle points
\pgfmathsetmacro{\thisAngle}{acos((\thisX)/\circleRadius)}
\pgfmathsetmacro{\thisY}{\circleRadius*sin(\thisAngle)}
% Calculate triangle points
\pgfmathsetmacro{\previousAngle}{acos((\thisX-1)/\circleRadius)}
\pgfmathsetmacro{\previousY}{\circleRadius*sin(\previousAngle)}
% Draw triangle
\draw[dashed, fill=blue!30] (\thisX-1,\previousY) -- (\thisX-1,\thisY) -- (\thisX,\thisY) -- cycle;
% Draw rectangle
\draw[fill=gray!30] (\thisX-1,0) rectangle (\thisX,\thisY);
% draw y axis labels
\draw (0,\thisY) -- (-3pt,\thisY) node[anchor=east] {$\omega_\thisX$};
}
\end{tikzpicture}
}
\end{figure}
\begin{equation}\label{eq:omega2phiTrapezIni}
\boldsymbol{\theta}_k \approx \boldsymbol{\theta}_{k-1} + (t_k - t_{k-1}) \boldsymbol{\omega}_{k-1} + \frac{1}{2} (t_k - t_{k-1}) (\boldsymbol{\omega}_{k} - \boldsymbol{\omega}_{k-1}),\quad k \ge 1
\end{equation}
\end{frame}
\end{document}