如何在 PGF/TikZ 中查找与椭圆的交点

Question

我试图在 PGF/TikZ 中显示一个球体来说明大圆的概念。

我当前结果的代码是：

\begin{tikzpicture}

\tikzfading[name=fade right,
left color=transparent!20,
right color=transparent!90]

\tikzfading[name=fade out,
inner color=transparent!100,
outer color=transparent!10]

\tikzfading[name=fade right gc,
left color=transparent!0,
right color=transparent!70]

\draw [<->, dashed] (0,-5) -- (0,5); % y-axis
\draw [->, dashed] (0, 0) -- (20:5); % x-axis
\draw [->, dashed] (0, 0) -- (200:5); % x-axis
\draw [->, dashed] (0, 0) -- (340:5); % z-axis
\draw [->, dashed] (0, 0) -- (160:5); % z-axis

\fill [color=cyan, opacity=0.15, path fading=fade out] (0,0) circle (4cm); % bounding circle
\fill [color=cyan, opacity=0.25, path fading=fade right, fading angle=90] (0,0) ellipse (4cm and 1cm); % x-y-axis area

% great circle 1
\draw [rotate=-40, color=red, path fading=fade right gc, fading angle=40] (0,0) ellipse (4cm and 1cm);

% great circle 2
\draw[rotate=5, color=red, path fading=fade right gc, fading angle=5] (0,0) ellipse (1.5cm and 4cm);

\end{tikzpicture}

如何

1. 找到两个红色椭圆（注释为大圆1和2）的两个交点，
2. 找到一条线的交点（始于中心（0,0） ）和一个椭圆，并
3. 在那里放置一个小圆形或矩形？

放置一个小圆圈或矩形没有问题。 非常感谢！

Answer 1

查看第 4.1.4 节。 TikZ 和 PGF 手册，标题为“圆的交点”。您需要使用 intersections 库，它允许您使用 name junctions 键，如 \path [name junctions={of=path 1 和 path 2}];。要使用此功能，您需要使用 name path 键，如 \draw [name path = y axis, <->, dashed] (0,-5) -- (0,5) ; % y 轴。不同版本的交叉路口访问方式似乎有所不同；我本地的手册副本与我链接到的手册的说明不同。但是，至少在我的版本中，您可以使用 (intersection-1)、(intersection-2) 等访问交叉点。要在每个交叉点处获取圆圈，例如，那么，我会将您的代码更改为如下所示：

\begin{tikzpicture}
  \tikzfading[ name        = fade right
             , left color  = transparent!20
             , right color = transparent!90 ]

  \tikzfading[name         = fade out
             , inner color = transparent!100
             , outer color = transparent!10 ]

  \tikzfading[name         = fade right gc
             , left color  = transparent!0
             , right color = transparent!70]

  \draw [name path = y  axis, <->, dashed] (0,-5) -- (0,5)   ; % y-axis
  \draw [name path = x- axis,  ->, dashed] (0, 0) -- (20:5)  ; % x-axis
  \draw [name path = x+ axis,  ->, dashed] (0, 0) -- (200:5) ; % x-axis
  \draw [name path = z+ axis,  ->, dashed] (0, 0) -- (340:5) ; % z-axis
  \draw [name path = z- axis,  ->, dashed] (0, 0) -- (160:5) ; % z-axis

  % bounding circle
  \fill [color=cyan, opacity=0.15, path fading=fade out]
        (0,0) circle (4cm) ;

  % x-y-axis area
  \fill [color=cyan, opacity=0.25, path fading=fade right, fading angle=90]
        (0,0) ellipse (4cm and 1cm);

  % great circle 1
  \draw [ name path    = great circle 1
        , rotate       = -40
        , color        = red
        , path fading  = fade right gc
        , fading angle = 40]
        (0,0) ellipse (4cm and 1cm);

  % great circle 2
  \draw [ name path    = great circle 2
        , rotate       = 5
        , color        = red
        , path fading  = fade right gc
        , fading angle = 5]
        (0,0) ellipse (1.5cm and 4cm);

  % Intersections
  \path [name intersections={of=great circle 1 and great circle 2}] ;
  \foreach \i in {1,...,4}
    \fill [color=red] (intersection-\i) circle (2pt) ;

  \path [name intersections={of=y axis and great circle 1}] ;
  \fill (intersection-1) circle (2pt) ;
  \fill (intersection-2) circle (2pt) ;
  \path [name intersections={of=y axis and great circle 2}] ;
  \fill (intersection-1) circle (2pt) ;
  \fill (intersection-2) circle (2pt) ;

  \foreach \a in {x,z} {
    \foreach \ss in {+,-} {
      \def\s.{\ss} % Otherwise the space in `\a\s axis` would get gobbled.
      \path [name intersections={of=\a\s. axis and great circle 1}] ;
      \fill (intersection-1) circle (2pt) ;
      \path [name intersections={of=\a\s. axis and great circle 2}] ;
      \fill (intersection-1) circle (2pt) ;
    }
  }
\end{tikzpicture}

除了重新格式化（以避免水平滚动条）之外，我对现有代码所做的所有更改都是添加 name path 键到你的轴和大圆。然后我添加了交叉点代码，这应该是相对不言自明的。请记住首先使用 \usetikzlibrary{intersections} ，一切都应该正常。