tikz 边界框/裁剪：曲线空间太大

Question 1

这是我尝试获得一种自动化方法。阅读这一页知道如何分割贝塞尔曲线。

limit bb我用两个参数定义新风格：

实际边界框和完美边界框之间的最大距离。
应用于路径的操作（绘制、填充...）。

这种新风格会自动且递归地分割所有贝塞尔曲线，以删除太远的控制点。

\documentclass[tikz]{standalone}
\usetikzlibrary{calc,decorations.pathreplacing}
\tikzset{
  bezier/controls/.code args={(#1) and (#2)}{
    \def\mystartcontrol{#1}
    \def\mytargetcontrol{#2}
  },
  bezier/limit/.store in=\mylimit,
  bezier/limit=1cm,
  bezier/.code={
    \tikzset{bezier/.cd,#1}
    \tikzset{
      to path={
        let
        \p0=(\tikztostart),    \p1=(\mystartcontrol),
        \p2=(\mytargetcontrol), \p3=(\tikztotarget),
        \n0={veclen(\x1-\x0,\y1-\y0)},
        \n1={veclen(\x3-\x2,\y3-\y2)},
        \n2={\mylimit}
        in  \pgfextra{
          \pgfmathtruncatemacro\ok{max((\n0>\n2),(\n1>\n2))}
        }
        \ifnum\ok=1 %
        let
        \p{01}=($(\p0)!.5!(\p1)$), \p{12}=($(\p1)!.5!(\p2)$), \p{23}=($(\p2)!.5!(\p3)$),
        \p{0112}=($(\p{01})!.5!(\p{12})$), \p{1223}=($(\p{12})!.5!(\p{23})$),
        \p{01121223}=($(\p{0112})!.5!(\p{1223})$)
        in
        to[bezier={controls={(\p{01}) and (\p{0112})}}]
        (\p{01121223})
        to[bezier={controls={(\p{1223}) and (\p{23})}}]
        (\p3)
        \else
        [overlay=false] .. controls (\p1) and (\p2) ..  (\p3) [overlay=true]
        \fi
      },
    }%, <-- Comma here results in "Missing character: There is no , in font nullfont!"
  },
  limit bb/.style n args={2}{
    overlay,
    decorate,
    decoration={
      show path construction,
      moveto code={},
      lineto code={\path[#2] (\tikzinputsegmentfirst) -- (\tikzinputsegmentlast);},
      curveto code={
        \path[#2]
        (\tikzinputsegmentfirst)
        to[bezier={limit=#1,controls={(\tikzinputsegmentsupporta) and (\tikzinputsegmentsupportb)}}]
        (\tikzinputsegmentlast);
      },
      closepath code={\path[#2] (\tikzinputsegmentfirst) -- (\tikzinputsegmentlast);},
    },
  },
  limit bb/.default={1mm}{draw},
}

\begin{document}
\begin{tikzpicture}
  \node[draw,circle] (A) at (0,0){A};
  \node[draw,circle] (B) at (3,3){B};
  \draw[limit bb={1mm}{draw=red},bend left=90,looseness=2] (A) to (B);
  \draw[limit bb={1mm}{draw=blue},bend right=90,looseness=2] (A) to (B);
  \draw[green] (current bounding box.south west) rectangle (current bounding box.north east);
\end{tikzpicture}
\end{document}

Answer

这是我尝试获得一种自动化方法。阅读这一页知道如何分割贝塞尔曲线。

limit bb我用两个参数定义新风格：

实际边界框和完美边界框之间的最大距离。
应用于路径的操作（绘制、填充...）。

这种新风格会自动且递归地分割所有贝塞尔曲线，以删除太远的控制点。

\documentclass[tikz]{standalone}
\usetikzlibrary{calc,decorations.pathreplacing}
\tikzset{
  bezier/controls/.code args={(#1) and (#2)}{
    \def\mystartcontrol{#1}
    \def\mytargetcontrol{#2}
  },
  bezier/limit/.store in=\mylimit,
  bezier/limit=1cm,
  bezier/.code={
    \tikzset{bezier/.cd,#1}
    \tikzset{
      to path={
        let
        \p0=(\tikztostart),    \p1=(\mystartcontrol),
        \p2=(\mytargetcontrol), \p3=(\tikztotarget),
        \n0={veclen(\x1-\x0,\y1-\y0)},
        \n1={veclen(\x3-\x2,\y3-\y2)},
        \n2={\mylimit}
        in  \pgfextra{
          \pgfmathtruncatemacro\ok{max((\n0>\n2),(\n1>\n2))}
        }
        \ifnum\ok=1 %
        let
        \p{01}=($(\p0)!.5!(\p1)$), \p{12}=($(\p1)!.5!(\p2)$), \p{23}=($(\p2)!.5!(\p3)$),
        \p{0112}=($(\p{01})!.5!(\p{12})$), \p{1223}=($(\p{12})!.5!(\p{23})$),
        \p{01121223}=($(\p{0112})!.5!(\p{1223})$)
        in
        to[bezier={controls={(\p{01}) and (\p{0112})}}]
        (\p{01121223})
        to[bezier={controls={(\p{1223}) and (\p{23})}}]
        (\p3)
        \else
        [overlay=false] .. controls (\p1) and (\p2) ..  (\p3) [overlay=true]
        \fi
      },
    }%, <-- Comma here results in "Missing character: There is no , in font nullfont!"
  },
  limit bb/.style n args={2}{
    overlay,
    decorate,
    decoration={
      show path construction,
      moveto code={},
      lineto code={\path[#2] (\tikzinputsegmentfirst) -- (\tikzinputsegmentlast);},
      curveto code={
        \path[#2]
        (\tikzinputsegmentfirst)
        to[bezier={limit=#1,controls={(\tikzinputsegmentsupporta) and (\tikzinputsegmentsupportb)}}]
        (\tikzinputsegmentlast);
      },
      closepath code={\path[#2] (\tikzinputsegmentfirst) -- (\tikzinputsegmentlast);},
    },
  },
  limit bb/.default={1mm}{draw},
}

\begin{document}
\begin{tikzpicture}
  \node[draw,circle] (A) at (0,0){A};
  \node[draw,circle] (B) at (3,3){B};
  \draw[limit bb={1mm}{draw=red},bend left=90,looseness=2] (A) to (B);
  \draw[limit bb={1mm}{draw=blue},bend right=90,looseness=2] (A) to (B);
  \draw[green] (current bounding box.south west) rectangle (current bounding box.north east);
\end{tikzpicture}
\end{document}

Question 2

默认情况下，边界框的一部分来自

\def\pgf@lt@moveto#1#2{%
  \pgf@protocolsizes{#1}{#2}%
  \pgfsyssoftpath@moveto{\the#1}{\the#2}%
}
\def\pgf@lt@lineto#1#2{%
  \pgf@protocolsizes{#1}{#2}%
  \pgfsyssoftpath@lineto{\the#1}{\the#2}%
}
\def\pgf@lt@curveto#1#2#3#4#5#6{%
  \pgf@protocolsizes{#1}{#2}%
  \pgf@protocolsizes{#3}{#4}%
  \pgf@protocolsizes{#5}{#6}%
  \pgfsyssoftpath@curveto{\the#1}{\the#2}{\the#3}{\the#4}{\the#5}{\the#6}%
}

这就是我们看到的原因所有控制点均参与：控制点直接传递给边界框计算（\pgf@protocolsizes）。要解决这个问题，只能在里面进行数学运算\pgf@lt@curveto。如果算上 TeX 以外的编程语言，这个主题肯定是重复的。例如一种查找封闭贝塞尔曲线边界框的算法？在堆栈溢出中。

但在 TeX 中做数学运算很困难。但仍然可以牺牲一些效率来获得一个相当可接受的结果。例如：因为 3(1-t)t^2,3(1-t)^2t≤4/9，我们知道

xmax≤max(xA，xD)+4/9|xB-max(xA，xD)|+4/9|xC-max(xA，xD)|

所以右边可以改进计算。

Answer