维基百科 Tikz 或 Pgfplots 中的 Lipschitz 连续性

Question

有谁知道如何通过 Lipschitz 连续函数优雅地改善以下维基百科（wikimedia）图像质量，使其在 Tikz 或 Pgfplots 中变为矢量？

维基百科 Lipschitz 连续体

可见如下：

感谢您的贡献！

Answer 1

因为完全模仿图像中显示的函数非常具有挑战性，所以我创建了一些其他的东西来帮助你实现同样的目标——这里有一个小脚本，它可以从随机数生成器中创建任意的 Lipschitz 连续函数。

您可以通过调整以下三个参数来定制输出内容：

• \numSamples- 用于创建函数的随机样本数量。样本越多，函数看起来就越参差不齐。
• \K- 函数的 Lipschitz 常数
• \pgfmathsetseed- 设置 Ti 的种子钾z 的随机数生成器

示例输出

---

代码

\documentclass[12pt]{article}
\usepackage[svgnames]{xcolor}
\usepackage{pgfplots}
\usepgfplotslibrary{fillbetween}
\usetikzlibrary{math}

\begin{document}

\def\numSamples{50} 
\def\K{1}
\pgfmathsetseed{123456789}

\def\lipschitzFunc{(0,0)}

\tikzmath{
  \funcValue = 0; % the graph begins at the origin
  for \n in {1,...,\numSamples} { 
    if \n <= \numSamples / 2 then { % first, generate the right side of the graph
      \deltaX = 4 / int(\numSamples / 2);
      \funcValue = \funcValue + rand * \K * \deltaX;
      \xVal = \n * \deltaX;
      {\global\edef\lipschitzFunc{\lipschitzFunc(\xVal,\funcValue)}};
    }
    else { % then, come back and generate the left side of the graph
      if \n == 1 + int(\numSamples / 2) then {
        \funcValue = 0;
      };
      \deltaX = 4 / int((\numSamples + 1)/ 2);
      \funcValue = \funcValue + rand * \K * \deltaX;
      \xVal = (int(\numSamples / 2) - \n) * \deltaX;
      {\global\edef\lipschitzFunc{(\xVal,\funcValue)\lipschitzFunc}};
    };
  };
}

\begin{tikzpicture}
\begin{axis}[
  xmin  = -4,
  xmax  =  4,
  ymin  = -4,
  ymax  =  4,
  xtick = \empty,
  ytick = \empty,
  axis on top,
  %axis line style={draw=none} % uncomment this line to remove surrounding box
]
\addplot[smooth] coordinates {\lipschitzFunc};
\addplot[name path = downup, color = LimeGreen, very thick]{ \K * x};
\addplot[name path = updown, color = LimeGreen, very thick]{-\K * x};
\addplot[fill = PaleGreen]
  fill between
  [
    of = updown and downup,
    soft clip = {domain = 0:4}
  ];
\addplot[fill = PaleGreen]
  fill between
  [
    of = downup and updown,
    soft clip = {domain = -4:0}
  ];
\end{axis}
\end{tikzpicture}

\end{document}