在图表中越来越紧密地插值的最简单方法是什么？

Question

我正在尝试创建一个如下所示的图表：

我尝试拟合具有越来越多节点的函数，但看起来不正确（在图表上显示所有 3 条曲线以更好地说明）：

\documentclass[10pt,landscape,a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage[UKenglish]{babel}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture} [scale = 0.5]

\foreach \p in {-5,...,5} \node[circle,fill=green] at (\p,2*rand)  (\p) {};
\draw [cyan, xshift=4cm] plot [smooth, tension=1] coordinates { (-5) (-4) (-3) (-2) (-1) (0) (1) (2) (3) (4) (5) };
\draw [red, xshift=4cm] plot [smooth, tension=1] coordinates { (-5)  (-3)  (-1)  (1)  (3)  (5) };
\draw [blue, xshift=4cm] plot [smooth, tension=1] coordinates { (-5)  (5)  };
\end{tikzpicture}

\end{document}

我也尝试过创建这样的节点（即使用定义函数 x^2+x+1 +“noise”来代替）：

\foreach \p in {-5,...,5} \node[circle,fill=green] at (\p, \p * \p + \p + 1 + rand) (\p) {};

但这根本不起作用。

无论如何，我的代码迫使我手动列出节点，这不是很好。

我想我需要一种使用越来越高次数的多项式进行插值的方法？但我不确定该怎么做。

Answer 1

您可以定义“插值”坐标并将它们与平滑的图连接起来。

\documentclass[10pt,landscape,a4paper]{article}
\usepackage{tikz}
\begin{document}

\begin{tikzpicture} [scale = 0.5]
\pgfmathtruncatemacro{\imin}{-5}
\pgfmathtruncatemacro{\imax}{5}
\foreach \p in {\imin,...,\imax} \node[circle,fill=green] at (\p,2*rand)  (p\p) {};
\foreach \X [evaluate=\X as \Xm using {int(\X-1)},evaluate=\X as \Xp using
{int(\X+1)}] in {\imin,...,\imax}
{\ifnum\X=\imin
  \path (p\X) coordinate (i\X);
 \else
  \ifnum\X=\imax
   \path (p\X) coordinate (i\X);
  \else
   \path (barycentric cs:p\X=0.6,p\Xm=0.2,p\Xp=0.2) coordinate (i\X);
  \fi 
 \fi
}
\draw [blue] plot [smooth, tension=1,variable=\X,samples at={\imin,...,\imax}] 
(i\X);
\end{tikzpicture}
\end{document}

当然，除了张力之外，你还可以改变重量。

\documentclass[tikz,border=3.14mm]{standalone}
\begin{document}
\foreach \X in {1,1.5,...,10,9.5,9,...,1.5}
{\begin{tikzpicture}
\pgfmathtruncatemacro{\imin}{-5}
\pgfmathtruncatemacro{\imax}{5}
\pgfmathsetmacro{\mainweight}{\X}
\pgfmathsetseed{27}
\node[anchor=north west,font=\sffamily] at (\imin,-2)
{weight is \pgfmathprintnumber{\mainweight}};
\foreach \p in {\imin,...,\imax} \node[circle,fill=green] at (\p,2*rand)  (p\p) {};
\foreach \X [evaluate=\X as \Xm using {int(\X-1)},evaluate=\X as \Xp using
{int(\X+1)}] in {\imin,...,\imax}
{\ifnum\X=\imin
  \path (p\X) coordinate (i\X);
 \else
  \ifnum\X=\imax
   \path (p\X) coordinate (i\X);
  \else
   \path (barycentric cs:p\X=\mainweight,p\Xm=1,p\Xp=1) coordinate (i\X);
  \fi 
 \fi
}
\draw [blue] plot [smooth, tension=0.5,variable=\X,samples at={\imin,...,\imax}] 
(i\X);
\end{tikzpicture}}
\end{document}