% edwards-conic2.tex
%
% Group law on an Edwards curve
% Requires \usepackage{nicefrac}
%
% Author: Michael Naehrig
% Public domain

\psset{unit=0.9cm} 
\begin{pspicture}(-5,-5)(5,5) 
\psaxes[labels=none, ticks=none, linewidth=.2pt]{->}(0,0)(-5,-5)(5,5)  
\psplot[plotstyle=curve]{-1}{1}{2 x x mul 2 mul sub 2 x x mul sub div sqrt}
\psplot[plotstyle=curve]{-1}{1}{2 x x mul 2 mul sub 2 x x mul sub div sqrt neg}
\psplot[plotstyle=curve]{-5}{-1.45}{2 x x mul 2 mul sub 2 x x mul sub div sqrt}
\psplot[plotstyle=curve]{-5}{-1.45}{2 x x mul 2 mul sub 2 x x mul sub div sqrt neg}
\psplot[plotstyle=curve]{1.45}{5}{2 x x mul 2 mul sub 2 x x mul sub div sqrt}
\psplot[plotstyle=curve]{1.45}{5}{2 x x mul 2 mul sub 2 x x mul sub div sqrt neg}
\psdots(0,1)(0,-1)(-1.5,3.162)(0.7, 0.822)(-2.690,1.543)(2.690,1.543)
\uput[135](-1.5,3.162){$P_1$}
\uput[-15](0.7,0.822){$P_2$}
\uput[45](-5,2.5){$C$}
\uput[45](3.5,-1.5){$E_{\nicefrac{1}{2}}$}
%\uput[135](0,1){$\mathcal{O}$}
\uput[-45](0,-1){$\mathcal{O}'$}
\psplot[plotstyle=curve, linestyle=dashed, dash=10pt 1pt]{-5}{-0.9}
{2.457411 -11.036607 x mul add 2.457411 5.646423 x mul add div neg}
\psplot[plotstyle=curve, linestyle=dashed, dash=10pt 1pt]{-0.25}{5}
{2.457411 -11.036607 x mul add 2.457411 5.646423 x mul add div neg}
\psline[linestyle=dashed]{-}(-5,1.543)(5,1.543)
\uput[-90](-2.690,1.543){$P_3$}
\uput[90](2.690,1.543){$-P_3$}
\uput[45](0,1.543){$L_{1,P_3}$}
\end{pspicture}