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geom-hyperbola+geometry+physics.tex
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geom-hyperbola+geometry+physics.tex
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% https://wiki.physik.uzh.ch/cms/latex:tikz:hyperbola
% Author: Izaak Neutelings (July, 2017)
\documentclass{article}
\usepackage{amsmath} % for \dfrac
\usepackage{tikz}
\tikzset{>=latex} % for LaTeX arrow head
\usepackage{pgfplots} % for the axis environment
\usetikzlibrary{calc} % to do arithmetic with coordinates
\usetikzlibrary{angles,quotes} % for pic
% colors
\definecolor{mylightgrey}{RGB}{230,230,230}
\definecolor{mygrey}{RGB}{190,190,190}
\definecolor{mydarkgrey}{RGB}{110,110,110}
\definecolor{mygreen}{RGB}{120,220,160}
\definecolor{mydarkgreen}{RGB}{60,120,60}
\definecolor{myverydarkgreen}{RGB}{20,60,20}
\definecolor{mydarkred}{RGB}{140,40,40}
% mark right angle
\newcommand{\MarkRightAngle}[4][1.5mm]
{\coordinate (tempa) at ($(#3)!#1!(#2)$);
\coordinate (tempb) at ($(#3)!#1!(#4)$);
\coordinate (tempc) at ($(tempa)!0.5!(tempb)$);%midpoint
\draw (tempa) -- ($(#3)!2!(tempc)$) -- (tempb);
}
% split figures into pages
\usepackage[active,tightpage]{preview}
\PreviewEnvironment{tikzpicture}
\setlength\PreviewBorder{1pt}%
\begin{document}
% RUTHERFORD SCATTERING - hyperbola orbit
\begin{tikzpicture}[scale=1]
% limits & parameters
\def\xa{-2.4}
\def\xb{ 4}
\def\ya{-4}
\def\yb{ 4}
\def\tmax{2.1}
\def\a{1.3}
\def\b{1}
\def\c{{sqrt(\a^2+\b^2)}}
\def\N{100} % number of points
% coordinates
\coordinate (O) at ( 0, 0 );
\coordinate (A) at ( \a, 0 );
\coordinate (F1) at ( {sqrt(\a^2+\b^2)}, 0 );
\coordinate (F2) at ( -{sqrt(\a^2+\b^2)}, 0 );
\coordinate (P) at ( -{\a^2/sqrt(\a^2+\b^2)}, -{\a*\b/sqrt(\a^2+\b^2)} );
\coordinate (P1) at (\xb*\a, \yb*\b);
\coordinate (P2) at (\xb*\a,-\yb*\b);
\coordinate (yshift) at (0,0.4);
% axes & asymptotes
\draw[mygrey] % x axis
(\xa*\a,0) -- (\xb*\a,0);
\draw[dashed,mydarkgrey]
(-\xb*\a*0.45, \ya*\b*0.45) -- (\xb*\a, \yb*\b)
(-\xb*\a*0.45,-\ya*\b*0.45) -- (\xb*\a,-\yb*\b);
% arrows
\def\vtheta{30}
\def\vradius{0.8}
\draw[->,myverydarkgreen,shift=($(P1)-(yshift)$),scale=0.6]
(0,0) -- (-\a,-\b) node[midway,below right=0pt] {${v}_i$};
\draw[->,myverydarkgreen,shift=($(P2)+(yshift)$),scale=0.6]
(-\a,\b) -- (0,0) node[midway,above right=-2pt] {${v}_f$};
\draw[->,myverydarkgreen]
(\a+0.35,{\vradius*sin(\vtheta)}) arc (180-\vtheta:180+\vtheta+10:\vradius)
node[above right=0pt] {$v^*$};
% angles
\MarkRightAngle{F2}{P}{O}
\pic [draw,myverydarkgreen,"$\theta$",angle radius=12,angle eccentricity=1.4] {angle = F2--O--P};
\pic [draw,below,angle radius=16,angle eccentricity=1.4] {angle = P2--O--P1};
\node[right=1pt,below=-2pt,myverydarkgreen] at ($(O)!0.5!(A)$) {\small$\phi$};
% hyperbola
\draw[color=mylightgrey,line width=0.5,samples=\N,variable=\t,domain=-\tmax*0.58:\tmax*0.58] % left
plot({-\a*cosh(\t)},{\b*sinh(\t)});
\draw[color=mydarkgreen,line width=1,samples=\N,variable=\t,domain=-\tmax:\tmax] % right
plot({ \a*cosh(\t)},{\b*sinh(\t)}); % {exp(\y)+exp(-\y)
% nodes
\draw[myverydarkgreen]
(F2) -- (P) node[midway,below left=1pt,] {$b$};
\fill[radius=1.5pt]
(A) circle node[above left=0pt] {A}
(O) circle node[above=2pt] {O};
\fill[radius=2.5pt,mydarkred]
(F2) circle node[above=2pt] {N};
\draw[]
%(O) -- (F2)
node[left=1pt,above=0pt] at ($(F2)!0.5!(O)$) {$c$}
node[below right] at ($(P)!0.5!(O)$) {$a$};
% alpha particle
\draw[radius=1pt,mydarkgreen,fill]
({ \a*cosh(\tmax*1.02)},{\b*sinh(\tmax*1.02)}) circle node[above right=0pt] {$\alpha$};
\end{tikzpicture}
% RUTHERFORD SCATTERING - hyperbolic orbits with different impact parameters
\begin{tikzpicture}[scale=1]
% limits & parameters
\def\xa{-35}
\def\xb{ 55}
\def\ya{ -1}
\def\yb{ 55}
\def\tmax{4.5}
\def\N{50} % number of points
% use axes to get square box which cuts of the long curves
\begin{axis}[ xmin=\xa,xmax=\xb,
ymin=\ya,ymax=\yb,
hide x axis, hide y axis,
xticklabels={,,},yticklabels={,,}
axis line style={draw=none}, tick style={draw=none}
]
% loop over multiples \u of impact parameters \b=\u*0.25
\def\a{1}
\foreach \u in {1,3,6,10,15,21,28,38}{
\def\b{\u*0.25}
\def\c{sqrt(\a^2+\b^2)}
% hyperbola
\addplot[color=mydarkgreen,line width=0.5,samples=\N,smooth,variable=\t,domain=-\tmax:\tmax]
({ \a/\c*(-\a*cosh(\t)-\c) + \b/\c*\b*sinh(\t) },
{ -\b/\c*(-\a*cosh(\t)-\c) + \a/\c*\b*sinh(\t) });
}
% nucleus
\addplot[mydarkred,mark=*,mark size=2pt,mark options=solid] coordinates {(0,0)};
\end{axis}
\end{tikzpicture}
\end{document}