Gap when plotting Elliptic Curve

Question

I try to plot a elliptic curve with this code:

\documentclass{article}
\usepackage{amssymb}
\usepackage{tikz}
\usepackage{pgfplots}
\begin{document}
\begin{figure}[htp]
    \centering
    \begin{tikzpicture}
        \begin{axis}[
            xmin=-2, xmax=2,
            ymin=-2, ymax=2,
            axis lines=center,
            xlabel={$x$},
            ylabel={$y$},
            samples=200]
            \addplot [blue] {sqrt(x^3 - x + 1};
            \addplot [blue] {-sqrt(x^3 - x + 1)};
        \end{axis}
    \end{tikzpicture}
    \caption{Elliptische Kurve in der Form $x^3-x+1$ im $\mathbb{R}2$}
    \label{fig:Elliptische Kurve}
\end{figure}
\end{document}

My problem is that in the plot is a visible gap. Does exists a solution for this issue?

Welcome to TeX.SX! Please extend your code to a minimal working example (MWE). — dexteritas, Jun 19 '23 at 13:58

score 14 · Accepted Answer · answered Jun 19 '23 at 14:12

Adaptations

Use samples at to define specific values that should be calculated and different sampling rates for different areas.

Code

\documentclass{article}
\usepackage{amssymb}
\usepackage{tikz}
\usepackage{pgfplots}
\begin{document}
\begin{figure}[htp]
    \centering
    \begin{tikzpicture}
        \begin{axis}[
            xmin=-2, xmax=2,
            ymin=-2, ymax=2,
            axis lines=center,
            xlabel={$x$},
            ylabel={$y$},
            samples at={-1.3247, -1.32, -1.31, ..., -1.1, -1.05, -1, ..., 2.0},
            ]
            \addplot [blue] {sqrt(x^3 - x + 1};
            \addplot [blue] {-sqrt(x^3 - x + 1)};
        \end{axis}
    \end{tikzpicture}
    \caption{Elliptische Kurve in der Form $x^3-x+1$ im $\mathbb{R}2$}
    \label{fig:Elliptische Kurve}
\end{figure}
\end{document}

Result

Thank you, it's working flawlessly! – oliver1707 Jun 21 '23 at 11:06 — oliver1707, Jun 21 '23 at 11:06

score 6 · Answer 2 · answered Jun 19 '23 at 20:54

With a short iterative method, we see that (the nonnegative) branch of your cubic is defined for x ≥ −1.3247179572…

Using the method proposed in https://tex.stackexchange.com/a/639053/4427

\documentclass{article}
\usepackage{pgfplots}
\usepackage{pgfmath-xfp}
\pgfplotsset{compat=1.18}
\begin{document}
\begin{tikzpicture}
  \pgfmxfpdeclarefunction{cubic}{1}{sqrt((#1)^3-(#1)+1)}
  \begin{axis}[
    xmin=-2,
    xmax=2,
    ymin=-2,
    ymax=2,
    xlabel={$x$},
    ylabel={$y$},
    scale only axis,
    axis lines=middle,
    domain=-1.3247179572:1.5,
    samples=200,
    smooth,
    % to avoid that the "plot node" is clipped (partially)
    clip=false,
    % use same unit vectors on the axis
    axis equal image=true,
  ]
    \addplot [red] {cubic(x)};
    \addplot [red] {-cubic(x)};
  \end{axis}
\end{tikzpicture}
\end{document}

No gap.

Thank you for the explanation. It helps me understand why the plot wasn't working the way I wanted it to. — oliver1707, Jun 21 '23 at 11:09

Gap when plotting Elliptic Curve

2 Answers2

Adaptations

Code

Result