为 PGFplots 定义分段函数

Question

我想定义一个分段函数q(x)，并尝试将解决方案调整为关于使用 pgfmathdeclarefunction 创建单位脉冲函数的这个问题，这样就没问题了。但是，当我尝试绘制时q(x+4)+0.5，得到的图形并不是我所期望的。但是，对上述链接中的单位脉冲函数应用相同的变换就可以了。

那么，有没有更好的方法来定义分段定义函数？

下面的 MWE 产生以下结果。

p(x)请注意，左侧的图表对于和都是符合预期的p(x+4)+0.5。右侧的图表对于是正确的q(x)，但对于是不正确的q(x+4)+0.5。

\documentclass{article}
\usepackage{amsmath}
\usepackage{pgfplots}

\newcommand{\pLabel}{
$p(x)=
\begin{cases}
    1 & 0 < x < 1\\
    0 & \text{otherwise}
\end{cases}$
}
\newcommand{\qLabel}{
$q(x)=
\begin{cases}
    x & 0 < x < 1\\
    0 & \text{otherwise}
\end{cases}$
}

\newcommand{\pShiftedLabel}{$p(x+4)+0.5$}
\newcommand{\qShiftedLabel}{$q(x+4)+0.5$}

\pgfmathdeclarefunction{p}{1}{%
  \pgfmathparse{(and(#1>0, #1<1))}%
}

\pgfmathdeclarefunction{q}{1}{%
  \pgfmathparse{(and(#1>0, #1<1)*x)}%
}

\tikzstyle{MyStyle}=[domain=-5:5, samples=50, ultra thick]
\tikzstyle{pLabelStyle}=[above, yshift=22ex, xshift=-10ex]
\tikzstyle{qLabelStyle}=[below, yshift=-2ex, xshift=-10ex]
\tikzstyle{ShiftedLabelStyle}=[above left, xshift=1ex]


\begin{document}
%------------------ Using \pgfmathdeclarefunction -----------
Plot of $p(x)$ and \pShiftedLabel using PGF Version \pgfversion, followed by a plot of $q(x)$ and \qShiftedLabel

\begin{tikzpicture}
  \begin{axis}
    \addplot[MyStyle, blue]{p(x)} node [pLabelStyle] {\pLabel};
    \addplot[MyStyle, red]{p(x+4)+0.5} node [ShiftedLabelStyle] {\pShiftedLabel};
  \end{axis}
\end{tikzpicture}
%
\begin{tikzpicture}
  \begin{axis}
    \addplot[MyStyle, blue]{q(x)} node [qLabelStyle] {\qLabel};
    \addplot[MyStyle, red]{q(x+4)+0.5} node [ShiftedLabelStyle] {\qShiftedLabel};
  \end{axis}
\end{tikzpicture}

% --------------------- Using "declare function" -------------
Using declare function to define localp(x) and localq(x):

\begin{tikzpicture}
[declare function={localp(\t) =  and(\t > 0, \t < 1);}]
  \begin{axis}
    \addplot[MyStyle, blue]{localp(x)}  node [pLabelStyle] {\pLabel};
    \addplot[MyStyle, red]{localp(x+4)+0.5} node [ShiftedLabelStyle] {\pShiftedLabel};
  \end{axis}
\end{tikzpicture}
%
\begin{tikzpicture}
[declare function={localq(\t) = (and(\t > 0, \t < 1)*x);}]
  \begin{axis}
    \addplot[MyStyle, blue]{localq(x)} node [qLabelStyle] {\qLabel};
    \addplot[MyStyle, red]{localq(x+4)+0.5} node [ShiftedLabelStyle] {\qShiftedLabel};
  \end{axis}
\end{tikzpicture}
\end{document}

Answer 1

定义分段函数的两种方法都可以，但是你应该使用

\pgfmathdeclarefunction{q}{1}{%
  \pgfmathparse{(and(#1>0, #1<1)*#1)}%
}

而不是and(#1>0, #1<1)*x)，并且

[declare function={localq(\t) = (and(\t > 0, \t < 1)*\t);}]

而不是[declare function={localq(\t) = (and(\t > 0, \t < 1)*x);}]，因为您实际上并不希望函数值为x，而是参数的值（x+4在本例中）。

\documentclass{article}
\usepackage{amsmath}
\usepackage{pgfplots}

\newcommand{\pLabel}{
$p(x)=
\begin{cases}
    1 & 0 < x < 1\\
    0 & \text{otherwise}
\end{cases}$
}
\newcommand{\qLabel}{
$q(x)=
\begin{cases}
    x & 0 < x < 1\\
    0 & \text{otherwise}
\end{cases}$
}

\newcommand{\pShiftedLabel}{$p(x+4)+0.5$}
\newcommand{\qShiftedLabel}{$q(x+4)+0.5$}

\pgfmathdeclarefunction{p}{1}{%
  \pgfmathparse{(and(#1>0, #1<1))}%
}

\pgfmathdeclarefunction{q}{1}{%
  \pgfmathparse{(and(#1>0, #1<1)*#1)}%
}

\tikzstyle{MyStyle}=[domain=-5:5, samples=100, ultra thick]
\tikzstyle{pLabelStyle}=[above, yshift=22ex, xshift=-10ex]
\tikzstyle{qLabelStyle}=[below, yshift=-2ex, xshift=-10ex]
\tikzstyle{ShiftedLabelStyle}=[above left, xshift=1ex]


\begin{document}
%------------------ Using \pgfmathdeclarefunction -----------
Plot of $p(x)$ and \pShiftedLabel using PGF Version \pgfversion, followed by a plot of $q(x)$ and \qShiftedLabel

\begin{tikzpicture}
  \begin{axis}
    \addplot[MyStyle, blue]{p(x)} node [pLabelStyle] {\pLabel};
    \addplot[MyStyle, red]{p(x+4)+0.5} node [ShiftedLabelStyle] {\pShiftedLabel};
  \end{axis}
\end{tikzpicture}
%
\begin{tikzpicture}
  \begin{axis}
    \addplot[MyStyle, blue]{q(x)} node [qLabelStyle] {\qLabel};
    \addplot[MyStyle, red]{q(x+4)+0.5} node [ShiftedLabelStyle] {\qShiftedLabel};
  \end{axis}
\end{tikzpicture}

% --------------------- Using "declare function" -------------
Using declare function to define localp(x) and localq(x):

\begin{tikzpicture}
[declare function={localp(\t) =  and(\t > 0, \t < 1);}]
  \begin{axis}
    \addplot[MyStyle, blue]{localp(x)}  node [pLabelStyle] {\pLabel};
    \addplot[MyStyle, red]{localp(x+4)+0.5} node [ShiftedLabelStyle] {\pShiftedLabel};
  \end{axis}
\end{tikzpicture}
%
\begin{tikzpicture}
[declare function={localq(\t) = (and(\t > 0, \t < 1)*\t);}]
  \begin{axis}
    \addplot[MyStyle, blue]{localq(x)} node [qLabelStyle] {\qLabel};
    \addplot[MyStyle, red]{localq(x+4)+0.5} node [ShiftedLabelStyle] {\qShiftedLabel};
  \end{axis}
\end{tikzpicture}
\end{document}