Illustrate ECC group

tikz-pgf

I'm trying to get a drawing like this in tikz, illustrating an Elliptic Curve group's binary operator.

I get the curve equation with a single controlling parameter t as (x,y)=(t^2,t(t^2-a)) (here with a=3 and t from -2.1 to 2.1), and can compute t" for P" as (t*t'+a)/(t+t'), with the unmarked point -t".

Update: I kicked me started and get the following, but

I did not manage to use my parametric equation with t the controlling variable
I don't know how to proceed to define my points and draw a line (I could get their (x,y) coordinates but then?). If only I knew where I declare variables, and how I use them to plot!
my node [left=15mm, below=2mm,black] is entirely empirical and super brittle.

Update: t=-3/5, t'=-7/5, t"=-48/25 would do.

\documentclass{amsart}
\usepackage{amssymb}
\usepackage{tikz}
\usepackage{pgfplots}
\begin{document}
\begin{figure}[tb]
\begin{tikzpicture}
    \begin{axis}[
        xmin=0, xmax=4.5,
        ymin=-2.5, ymax=2.5,
        xlabel={$x$}, ylabel={$y$},
        domain=0:4.2, samples=400,
        scale only axis, axis lines=middle,
        smooth, axis equal image=true, clip=true,
    ]
        \addplot [thick,purple] {sqrt((x-3)^2*x)}
            node[left=15mm, below=2mm,black] {$y^2=x(x-3)^2$};
        \addplot [thick,purple] {-sqrt((x-3)^2*x)};
    \end{axis}
\end{tikzpicture}
\caption{What I get}
\label{firstfig}
\end{figure}
\end{document}

Best Answer

With avaiable parameter equations of EEC, I use TikZ for this.

\documentclass[tikz,border=5mm]{standalone}
\begin{document}
\begin{tikzpicture}[>=stealth,declare function={
a=3; 
t1=-1.5;
t2=-.5;
t3=(t1*t2+a)/(t1+t2);       
x(\t)=(\t)^2;         % do not use \t^2
y(\t)=\t*((\t)^2-a);    
}]
\clip (-.5,-2.5) rectangle (4.5,2.5);
\draw[->] (-.5,0)--(4.5,0) node[below left]{$x$};
\draw[->] (0,-2.5)--(0,2.5) node[below left]{$y$};
\draw[magenta,smooth,thick] plot[variable=\t,domain=-2.1:2.1] ({x(\t)},{y(\t)}) node[left=15mm, below=4mm]{$y^2=x(x-3)^2$};

\path 
(0,0) node[below left]{O}
({x(t1)},{y(t1)}) coordinate (P) node[above right]{$P$}
({x(t2)},{y(t2)}) coordinate (Pp) node[below right]{$P'$}
({x(t3)},{y(t3)}) coordinate (Ppp) node[right]{$P''$};

\draw[shorten >=-1cm,shorten <=-2cm] (P)--(Pp);
\draw (Ppp)--+(90:2.5) (Ppp)--+(-90:1.5);
\fill[red] (P) circle(2pt); 
\fill[blue] (Pp) circle(2pt);
\fill (Ppp) circle(2pt);
\end{tikzpicture}
\end{document}

Related Solutions

[Tex/LaTex] Plot to illustrate secant lines

With TikZ it is really easy. I used Plain TeX, so you will need to \input tikz.tex instead of \usepackage{tikz} and instead of \begin{document}...\end{document} you issue \bye at the end of your document.

Typeset the following code with pdftex

\input tikz.tex
\nopagenumbers% for cropping
\usetikzlibrary{arrows,intersections}
\tikzpicture[
        thick,
        >=stealth',
        dot/.style = {
            draw,
            fill=white,
            circle,
            inner sep=0pt,
            minimum size=4pt
        }
    ]
    \coordinate (O) at (0,0);
    \draw[->] (-0.3,0) -- (8,0) coordinate[label={below:$x$}] (xmax);
    \draw[->] (0,-0.3) -- (0,5) coordinate[label={right:$f(x)$}] (ymax);
    \path[name path=x] (0.3,0.5) -- (6.7,4.7);
    \path[name path=y] plot[smooth] coordinates {(-0.3,2) (2,1.5) (4,2.8) (6,5)};
    \scope[name intersections={of=x and y,name=i}]
        \fill[gray!20] (i-1) -- (i-2 |- i-1) -- (i-2) -- cycle;
        \draw (0.3,0.5) -- (6.7,4.7) node[pos=0.8,below right] {Sekante};
        \draw[red] plot[smooth] coordinates {(-0.3,2) (2,1.5) (4,2.8) (6,5)};
        \draw (i-1) node[dot,label={above:$P$}] (i-1) {} -- node[left] {$f(x_0)$} (i-1 |- O) node[dot,label={below:$x_0$}] {};
        \path (i-2) node[dot,label={above:$Q$}] (i-2) {} -- (i-2 |- i-1) node[dot] (i-12) {};
        \draw (i-12) -- (i-12 |- O) node[dot,label={below:$x_0 + \varepsilon$}] {};
        \draw[blue,<->] (i-2) -- node[right] {$f(x_0 + \varepsilon) - f(x_0)$} (i-12);
        \draw[blue,<->] (i-1) -- node[below] {$\varepsilon$} (i-12);
        \path (i-1 |- O) -- node[below] {$\varepsilon$} (i-2 |- O);
        \draw[gray] (i-2) -- (i-2 -| xmax);
        \draw[gray,<->] ([xshift=-0.5cm]i-2 -| xmax) -- node[fill=white] {$f(x_0 + \varepsilon)$}  ([xshift=-0.5cm]xmax);
    \endscope
\endtikzpicture
\bye

This will produce the following output (cropped)

As intersections are computed by the intersections library this solution is adaptive. In this code, not actually the point Q is moved, but one of the points on the red line, which causes Q to move (if you look closely you can see, that the red line gets an ugly bump while Q moves right).

My workflow for creating animations is the following:

Modify the source file, such that for each variation a seperate page in the output is created (in most cases using the PGF \foreach loop, like here)
Crop the resulting PDF using Heiko Oberdiek's pdfcrop.
Import the cropped PDF in GIMP.
In GIMP: Reverse the layer order and export as .gif with the option As Animation checked and a delay of 200 milliseconds (otherwise it is too fast for me).

The following contains the code used to create the animation. I marked the extra and modified lines needed in contrast to the above code.

\input tikz.tex
\nopagenumbers% for cropping
\usetikzlibrary{arrows,intersections}
\foreach \Q in {4,4.1,4.2,...,5,4.9,4.8,...,4.1} {%<-- added
\tikzpicture[
        thick,
        >=stealth',
        dot/.style = {
            draw,
            fill=white,
            circle,
            inner sep=0pt,
            minimum size=4pt
        }
    ]
    \coordinate (O) at (0,0);
    \draw[->] (-0.3,0) -- (8,0) coordinate[label={below:$x$}] (xmax);
    \draw[->] (0,-0.3) -- (0,5) coordinate[label={right:$f(x)$}] (ymax);
    \path[name path=x] (0.3,0.5) -- (6.7,4.7);
    \path[name path=y] plot[smooth] coordinates {(-0.3,2) (2,1.5) (\Q,2.8) (6,5)};%<-- modified
    \scope[name intersections={of=x and y,name=i}]
        \fill[gray!20] (i-1) -- (i-2 |- i-1) -- (i-2) -- cycle;
        \draw (0.3,0.5) -- (6.7,4.7) node[pos=0.8,below right] {Sekante};
        \draw[red] plot[smooth] coordinates {(-0.3,2) (2,1.5) (\Q,2.8) (6,5)};%<-- modified
        \draw (i-1) node[dot,label={above:$P$}] (i-1) {} -- node[left] {$f(x_0)$} (i-1 |- O) node[dot,label={below:$x_0$}] {};
        \path (i-2) node[dot,label={above:$Q$}] (i-2) {} -- (i-2 |- i-1) node[dot] (i-12) {};
        \draw (i-12) -- (i-12 |- O) node[dot,label={below:$x_0 + \varepsilon$}] {};
        \draw[blue,<->] (i-2) -- node[right] {$f(x_0 + \varepsilon) - f(x_0)$} (i-12);
        \draw[blue,<->] (i-1) -- node[below] {$\varepsilon$} (i-12);
        \path (i-1 |- O) -- node[below] {$\varepsilon$} (i-2 |- O);
        \draw[gray] (i-2) -- (i-2 -| xmax);
        \draw[gray,<->] ([xshift=-0.5cm]i-2 -| xmax) -- node[fill=white] {$f(x_0 + \varepsilon)$}  ([xshift=-0.5cm]xmax);
    \endscope
\endtikzpicture
\eject%<-- added
}%<-- added
\bye

[Tex/LaTex] How to get a pigeon to illustrate the pigeonhole principle

\documentclass[pstricks,margin=5mm]{standalone}
\usepackage{pst-fun}
\begin{document}
\pspicture[dimen=m](12,12)
\multirput(0,0)(4,0){3}{\multirput(0,0)(0,4){3}{\psframe(4,4)\rput(0,1){\psBird}}}
\endpspicture
\end{document}

Best Answer

Related Solutions

[Tex/LaTex] Plot to illustrate secant lines

[Tex/LaTex] How to get a pigeon to illustrate the pigeonhole principle

Related Question