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TikZ：在两点之间画弧

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TikZ：在两点之间画弧

来自我的帖子TikZ：通过两点绘制椭圆，我尝试为不同的弧复制该过程，但它没有按预期工作。问题是我不太了解下面的代码。

\draw[postaction = decorate] let
    \p0 = ($(B) - (F)$),
    \p1 = ($(A) - (B)$)
  in (B|-A) ++(\x1, 0) arc(45:135:\x1 and \y0);

在做什么(B|-A)？为什么要减分。

我正在尝试将（A）连接到（B）。

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{intersections, decorations.markings, calc, arrows}
\begin{document}
\begin{tikzpicture}[line join = round, line cap = round, >=triangle 45,
    every label/.append style = {font = \scriptsize},
    dot/.style = {inner sep = +0pt, shape = circle,
      draw = black, label = {#1}},
    small dot/.style = {minimum size = .05cm, dot = {#1}},
    big dot/.style = {minimum size = .1cm, dot = {#1}},                                                         
    ]
    \pgfmathsetmacro{\as}{3}
    \pgfmathsetmacro{\bs}{2.25}
    \pgfmathsetmacro{\c}{sqrt(\as^2 - \bs^2)}
    \pgfmathsetmacro{\al}{3.75}
    \pgfmathsetmacro{\bl}{sqrt(\al^2 - \c^2)}

    \coordinate (O) at (0, 0);

    \node[fill = black, big dot = {below left: \(F\)}] (F) at (\c, 0) {};

    \path[name path = line1] (\c, 0) -- ++(45:{\as} and \bs);
    \path[name path = line2] (\c, 0) -- ++(135:5cm and 5cm);

    \path[name path = ell1] (O) ellipse
      (\as cm and \bs cm);

    \path[name path = ell2] (O) ellipse
      (\al cm and \bl cm);

    \path[name intersections = {of = line1 and ell1, by = P1}];
    \node[fill = black, big dot = {above right: \(A\)}] (A) at (P1) {};

    \path[name intersections = {of = line2 and ell2, by = P2}];
    \node[fill = black, big dot = {above right: \(B\)}] (B) at (P2) {};

    \draw[blue] (F) -- (A);
    \draw[red] (F) -- (B) node[scale = .75, fill = white, inner sep = 0cm,
    pos = .5] {\(r_B\)};


    \begin{scope}[decoration = {markings,
        mark = at position 0.5 with {\arrow{>}},
      } ]
      \draw[postaction = decorate] let
        \p0 = ($(B) - (F)$),
        \p1 = ($(A) - (B)$)
      in (B|-A) ++(\x1, 0) arc(45:135:\x1 and \y0);
    \end{scope}
  \end{tikzpicture}
\end{document}

在此处输入图片描述

编辑2：

按照 Qrrbrbirlbel 的解决方案，我得到了正确的数字，但是我收到了错误并且我无法弄清楚：

ERROR: Undefined control sequence.

--- TeX said ---
\tikz@intersect@namedpaths ...name@f'Circle-\solA
                                                  \endcsname {\pgfsyssoftpat...
l.108   {\(\nu_B\)};

--- HELP ---
TeX encountered an unknown command name. You probably misspelled the
name. If this message occurs when a LaTeX command is being processed,
the command is probably in the wrong place---for example, the error
can be produced by an \item command that's not inside a list-making
environment. The error can also be caused by a missing \documentclass
command.

我使用的代码是：

\documentclass[tikz]{standalone}
\usetikzlibrary{intersections, decorations.markings, calc, arrows, backgrounds}
\begin{document}
\tikzset{circle with radius/.style = {shape = circle, inner sep = 0pt,
    outer sep = 0pt, minimum size = {2 * (#1)}}}
\begin{tikzpicture}[line join = round, line cap = round, >=triangle 45,
  every label/.append style = {font = \scriptsize},
  dot/.style = {inner sep = +0pt, shape = circle,
    draw = black, label = {#1}},
  small dot/.style = {minimum size = .05cm, dot = {#1}},
  big dot/.style = {minimum size = .1cm, dot = {#1}},
  ]
  \pgfmathsetmacro{\as}{3}
  \pgfmathsetmacro{\bs}{2.25}
  \pgfmathsetmacro{\c}{sqrt(\as^2 - \bs^2)}
  \pgfmathsetmacro{\al}{3.75}
  \pgfmathsetmacro{\bl}{2.9}
  \pgfmathsetmacro{\cl}{sqrt(\al^2 - \bl^2)}
  \pgfmathsetmacro{\xs}{abs(\c - \cl)}

  \coordinate (O) at (0, 0);

  \node[fill = black, big dot = {below left: \(F\)}] (F) at (\c, 0) {};

  \path[name path global = line1] (\c, 0) -- ++(60:{\as} and \bs);
  \path[name path global = line2] (\c, 0) -- ++(150:6cm);

  \begin{scope}[decoration = {markings,
      mark = at position 0.25 with {\arrow{>}},
      mark = at position 0.375 with {\node[draw, shape = circle,
        inner sep = .08cm, fill = white, scale = .75] {\(1\)};},
      mark = at position 0.75 with {\arrow{>}},
    } ]
    \draw[postaction = decorate, name path global = ell1, blue] (O) ellipse
    (\as cm and \bs cm);
  \end{scope}

  \begin{scope}[decoration = {markings,
      mark = at position 0.25 with {\arrow{>}},
      mark = at position 0.45 with {\node[draw, shape = circle,
        inner sep = .08cm, fill = white, scale = .75] {\(2\)};},
      mark = at position 0.75 with {\arrow{>}},
    },
    on background layer]
    \draw[postaction = decorate, name path global = ell2, red] (-\xs, 0)
    ellipse (\al cm and \bl cm);
  \end{scope}

  \path[name intersections = {of = line1 and ell1, by = P1}];
  \node[fill = black, big dot = {right: \(A\)}] (A) at (P1) {};

  \path[name intersections = {of = line2 and ell2, by = P2}];
  \node[fill = black, big dot = {above: \(B\)}] (B) at (P2) {};

  \draw[blue] (F) -- (A);
  \draw[red] (F) -- (B) node[scale = .75, fill = white, inner sep = 0cm,
  pos = .5] {\(r_B\)};

  \begin{scope}[on background layer]
    \draw[dashed, -latex] ($(-\al , 0) - (1, 0)$) -- ($(\al, 0) + (.5, 0)$)
    coordinate (P3);
    \draw[dashed, -latex] ($(F) - (0, 3)$) -- ($(F) + (0, 3)$);
  \end{scope}

  \begin{scope}[declare function = {doubleA = 7cm;}]
    \begin{pgfinterruptboundingbox}
      \path let
        \p1 = ($(A) - (F)$),
        \p2 = ($(B) - (F)$),
        \n1 = {veclen(\x1, \y1)},
        \n2 = {veclen(\x2, \y2)}
      in
      (A) node[name path global = aCircle, circle with radius = doubleA-\n1] {}
      (B) node[name path global = bCircle, circle with radius = doubleA-\n2] {}
      (F) node[name path global = fCircle, circle with radius = .5 * doubleA]
      {};

      \tikzset{name intersections = {of = aCircle and bCircle, name = F',
          }}
      \foreach \solA in {1, 2} {
        \path ($(F)!.5!(F'-\solA)$) coordinate (C'-\solA)
        ($(C'-\solA)!doubleA/2!(F)$) coordinate (xDir-\solA)
        (F'-\solA) node[name path global = f'Circle-\solA,
        circle with radius = .5 * doubleA] {};
      }                         %!?                                                 
      \foreach \solA in {1, 2} { %!?                                                
        \path[name intersections = {of = fCircle and f'Circle-\solA,
          by = {yDir-\solA}}]
        ($(xDir-\solA)-(C'-\solA)$) coordinate (xDir'-\solA)
        ($(yDir-\solA)-(C'-\solA)$) coordinate (yDir'-\solA)
        ;
      }
    \end{pgfinterruptboundingbox}
    \foreach \solA in {1, 2}
    \draw[x = (xDir'-\solA), y = (yDir'-\solA)] (C'-\solA) circle [radius = 1];
  \end{scope}

  \draw[on background layer, red] let
    \p0 = (F),
    \p1 = (B),
    \p2 = (P3),
    \n1 = {atan2(\x1 - \x0, \y1 - \y0)},
    \n2 = {atan2(\x2 - \x0, \y2 - \y0)},
    \n3 = {.75cm},
    \n4 = {(\n2 + \n1) / 2}
  in (F) +(\n1:\n3) arc[radius = \n3, start angle = \n1, end angle = \n2]
  node[font = \tiny, fill = white, inner sep = 0cm] at ([shift = (F)] \n4:\n3)
  {\(\nu_B\)};

  \draw[blue] let
    \p0 = (F),
    \p1 = (A),
    \p2 = (P3),
    \n1 = {atan2(\x1 - \x0, \y1 - \y0)},
    \n2 = {atan2(\x2 - \x0, \y2 - \y0)},
    \n3 = {.5cm},
    \n4 = {(\n2 + \n1) / 2}
  in (F) +(\n1:\n3) arc[radius = \n3, start angle = \n1, end angle = \n2]
  node[font = \tiny, fill = white, inner sep = 0cm] at
  ([shift = (F)] \n4:.75cm) {\(\nu_A\)};
\end{tikzpicture}
\end{document}

在此处输入图片描述

答案1

这里有一个想法，可以得到一个焦点F经过点A和的椭圆B。我冒昧地假设了
  2A= 7 厘米
，其中A是椭圆长轴的长度。

根据我记得的关于椭圆的信息，这也是椭圆上一个点与两个焦点之间的距离之和。我们可以用这个来得到两个可能的焦点F'1和F'2。

为了在 TikZ 中做到这一点，我使用calc语法let … in来找到给定焦点和点之间的长度，然后我在这些点周围放置节点，并使用这两个点的其余部分A。这intersection cs有助于找到焦点。椭圆弧的中心就在这两个焦点的中间，我们发现C “”。

从这个中心我们可以找到X，大轴指向的点（也是X旋转椭圆的轴）；再次calc来救援：($(C')!doubleA/2!(F)$)。

现在我们可以做一些有趣的计算A以及线性偏心率b，小轴的长度，但这是数学 PGF 不能没有帮助。但如果我没记错的话，坐标是小轴指向与两个焦点距离相等的点；即A。那么，让我们画出圆圈F和F'并intersection cs再次使用。

该坐标X和是xDir在下面的 TikZ 解决方案中，和被定义yDir。我们可以使用它们来安装一个局部坐标系，该坐标系的轴恰好与椭圆的轴相同。半径为 1 的圆就是我们搜索的椭圆（或您寻找其部分的整个圆弧）。

由于圆形辅助节点确实改变了边界框，我将这些计算放在pgfinterruptboundingbox环境中并将向量保存在xDir'和中yDir'。

然后可以简单地绘制一个椭圆

\draw[x=(xDir'),y=(yDir')] (C') circle [radius=1];

我在这里使用了intersection cs，因为它更容易维护（并且你可以在一个路径上完成所有操作。使用intersections库也可以实现同样的效果。

C'-2但是，出于某种原因，当您想要命名节点路径并在同一个循环中找到它的交点时，TikZ 会弄乱第二个中心 () 的位置（注释掉!?带有标记的行）。但是，将其分成两个循环将产生正确的结果：

代码（intersections库）

\begin{scope}[declare function={doubleA=7cm;}]
  \begin{pgfinterruptboundingbox}
    \path let \p1=($(A)-(F)$), \p2=($(B)-(F)$), \n1={veclen(\x1,\y1)},\n2={veclen(\x2,\y2)} in
      (A) node[name path=aCircle, circle with radius=doubleA-\n1] {}
      (B) node[name path=bCircle, circle with radius=doubleA-\n2] {}
      (F) node[name path=fCircle, circle with radius=.5*doubleA]  {};

    \tikzset{name intersections={of=aCircle and bCircle,name=F',by={[big dot=below:$F'_1$],[big dot=below:$F'_2$]}}}
    \foreach \solA in {1,2} {
      \path ($(F)!.5!(F'-\solA)$) coordinate[big dot=$C'_\solA$] (C'-\solA)
            ($(C'-\solA)!doubleA/2!(F)$) coordinate[big dot=right:$x_\solA$] (xDir-\solA)
            (F'-\solA) node[name path global=f'Circle-\solA, circle with radius=.5*doubleA] {};
    }                         %!?
    \foreach \solA in {1,2} { %!?
      \path[name intersections={of=fCircle and f'Circle-\solA,by={[big dot=above right:$y_\solA$]yDir-\solA}}]
                ($(xDir-\solA)-(C'-\solA)$) coordinate (xDir'-\solA)
                ($(yDir-\solA)-(C'-\solA)$) coordinate (yDir'-\solA)
      ;
    }
 \end{pgfinterruptboundingbox}
 \foreach \solA in {1,2}
   \draw[x=(xDir'-\solA),y=(yDir'-\solA)] (C'-\solA) circle [radius=1];
\end{scope}

代码

\documentclass[tikz]{standalone}
\usetikzlibrary{intersections, decorations.markings, calc, arrows}
\tikzset{circle with radius/.style={shape=circle, inner sep=+0pt, outer sep=+0pt, minimum size={2*(#1)}}}
\begin{document}
\begin{tikzpicture}[line join = round, line cap = round, >=triangle 45,
    every label/.append style = {font = \scriptsize},
    dot/.style = {inner sep = +0pt, shape = circle, draw = black, label = {#1}},
    small dot/.style = {minimum size = .05cm, dot = {#1}},
    big dot/.style = {minimum size = .1cm, dot = {#1}},                                                         
    ]
    \pgfmathsetmacro{\as}{3} \pgfmathsetmacro{\bs}{2.25}
    \pgfmathsetmacro{\c}{sqrt(\as^2 - \bs^2)}
    \pgfmathsetmacro{\al}{3.75}
    \pgfmathsetmacro{\bl}{sqrt(\al^2 - \c^2)}
    \coordinate (O) at (0,0); \node[fill = black, big dot = {below left: \(F\)}] (F) at (\c, 0) {};

    \path[name path = line1] (\c, 0) -- ++(45:{\as} and \bs);
    \path[name path = line2] (\c, 0) -- ++(135:5cm and 5cm);
    \path[name path = ell1] (O) ellipse (\as cm and \bs cm);
    \path[name path = ell2] (O) ellipse (\al cm and \bl cm);

    \path[name intersections = {of = line1 and ell1, by = P1}];
    \node[fill = black, big dot = {above right: \(A\)}] (A) at (P1) {};
    \path[name intersections = {of = line2 and ell2, by = P2}];
    \node[fill = black, big dot = {above right: \(B\)}] (B) at (P2) {};

    \draw[blue] (F) -- (A);
    \draw[red]  (F) -- (B) node[scale = .75, fill = white, inner sep = 0cm, pos = .5] {\(r_B\)};

    \begin{scope}[declare function={doubleA=7cm;}]
      \foreach \solA in {1,2} {
        \begin{pgfinterruptboundingbox}
        \path let \p1=($(A)-(F)$), \p2=($(B)-(F)$), \n1={veclen(\x1,\y1)},\n2={veclen(\x2,\y2)} in
          (A) node[circle with radius=doubleA-\n1] (aCircle) {}
          (B) node[circle with radius=doubleA-\n2] (bCircle) {}
          (intersection cs: first node=aCircle, second node=bCircle, solution=\solA) coordinate[big dot=below:$F'_\solA$] (F')
          ($(F)!.5!(F')$) coordinate[big dot=$C'_\solA$] (C')% the center of the arc
          ($(C')!doubleA/2!(F)$) coordinate[big dot=right:$x_\solA$] (xDir)
          (F) node[circle with radius=.5*doubleA] (fCircle) {}
          (F') node[circle with radius=.5*doubleA] (f'Circle) {}
          (intersection cs: first node=fCircle, second node=f'Circle, solution=1) coordinate[big dot=above right:$y_\solA$] (yDir)
          ($(xDir)-(C')$) coordinate (xDir')
          ($(yDir)-(C')$) coordinate (yDir')
        ;
        \end{pgfinterruptboundingbox}
        \draw[x=(xDir'),y=(yDir')] (C') circle [radius=1];
      }
    \end{scope}
  \end{tikzpicture}
\end{document}

输出

在此处输入图片描述

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