如何反转由椭圆交叉点定义的自定义路径上的裁剪

Question 1

好吧，这不是一个简短的答案。事实上，这将是一个很长的答案，但会以通用的方式解决此类问题。以下是您可以使用此提案进行操作的一些示例。

例 1 例 2

库和包

我们使用了 tikz 的一项功能，称为decorations.markings使用节点重新创建路径。以下是语句：

\tikzset{
    split/.style = {
        , name path = #1
        , /utils/exec={\setcounter{cnt}{0}}
        , postaction = {
            , decorate
            , decoration={
                markings
                , mark = between positions 0 and 1 step 2pt with {
                    \node [
                        circle
                        , minimum size = 0pt
                        , inner sep = .2pt
                        , /utils/exec={\stepcounter{cnt} \setpointsof{#1}{\thecnt}}
                    ] (#1-\thecnt) {};
                }
            }
        }
    }
}

我们还使用该库intersections来查找路径之间的交点。

etoolbox此外，我们使用提供以非常特定的方式存储值的方法并且还具有流控制功能（if，while等）的包。

宏

首先我将解释一下宏。

接下来的两个宏存储并检索两个路径之间的交点数。

% #1: path 1
% #2: path 2
% #3: intersect points
\newcommand{\setintersectpointsof}[3]{\csxdef{intersectpointsof#1and#2}{#3}}

% #1: path 1
% #2: path 2
\newcommand{\getintersectpointsof}[2]{\csuse{intersectpointsof#1and#2}}

接下来的两个宏存储和检索路径的点数。

% #1: path
\newcommand{\pointsof}[1]{\csuse{pointsof#1}}

% #1: path
% #2: intersect points
\newcommand{\setpointsof}[2]{\csnumgdef{pointsof#1}{#2}}

接下来的两个宏存储和检索与索引变量相关的值。类似于编程中的向量。

% #1: name of path
% #2: name of the paths
% #3: index of the data
% #4: data
\newcommand\setstorageof[4]{\csnumgdef{#1in#2#3}{#4}}

% #1: name of the path
% #1: name of the paths
% #2: index of the data
\newcommand\getstorageof[3]{\csuse{#1in#2#3}}

下一个宏标识了每条路径的交点。因此，它也用于微调的误差幅度。

% #1: first path
% #2: second path
% #3: margin of error
\newcommand{\getintersectionsof}[3]{

    \path [
        name intersections = {of = #1 and #2, name = j, total=\t, sort by = #1}
        , /utils/exec = {\xdef\total{\t}}
    ];

    \setintersectpointsof{#1}{#2}{\total}
    \pgfmathsetlengthmacro{\marginoferror}{#3}

    \foreach \q in {1, ..., \getintersectpointsof{#1}{#2}} {
        \pgfextractx{\xref}{\pgfpointanchor{j-\q}{center}}
        \pgfextracty{\yref}{\pgfpointanchor{j-\q}{center}}

        \foreach \p in {#1, #2} {

            \foreach \r in {1, ..., \pointsof{\p}}{
                \pgfextractx{\xone}{\pgfpointanchor{\p-\r}{center}}
                \pgfextracty{\yone}{\pgfpointanchor{\p-\r}{center}}

                \ifboolexpr{
                    test {\ifdimless{\xref - \marginoferror}{\xone}} and test {\ifdimless{\xone}{\xref + \marginoferror}}
                    and
                    test {\ifdimless{\yref - \marginoferror}{\yone}} and test {\ifdimless{\yone}{\yref + \marginoferror}}
                }{
                    \setstorageof{\p}{#1#2}{\q}{\r}
                }{
                }
            }
        }
    }
}

接下来的两个宏构建交叉点之间的路径。可以直接构建，也可以相对于构建的路径反向构建markings。

% #1: working path
% #2: first element
% #3: last element
% #4: resulting path
% #5: next position of resulting path
\newcommand{\constructreversepath}[5]{
    \pgfmathtruncatemacro{\k}{#2}   
    \csedef{#4}{};
    \unlessboolexpr{test{\ifnumequal{\k}{#3}}}{
        \stepcounter{innercnt}
        \node (#4-\theinnercnt) at (#1-\k.center) {};
        \csxdef{#4}{\csuse{#4} (#4-\theinnercnt)};
        \pgfmathtruncatemacro{\k}{\k - 1}
        \ifnumless{\k}{1}{
            \pgfmathtruncatemacro{\k}{\pointsof{#1} - \k}
        }{
            \ifnumgreater{\k}{\pointsof{#1}}{
                \pgfmathtruncatemacro{\k}{\k - \pointsof{#1}}
            }{
            }
        }
    }
    \stepcounter{innercnt}
    \node (#4-\theinnercnt) at (#1-\k.center) {};
    \csedef{#4}{\csuse{#4} (#4-\theinnercnt)};
}

% #1: working path
% #2: first element
% #3: last element
% #4: resulting path
% #5: next position of resulting path
\newcommand{\constructdirectpath}[5]{
    \pgfmathtruncatemacro{\k}{#2}
    \csedef{#4}{};
    \unlessboolexpr{test{\ifnumequal{\k}{#3}}}{
        \stepcounter{innercnt}
        \node (#4-\theinnercnt) at (#1-\k.center) {};
        \csxdef{#4}{\csuse{#4} (#4-\theinnercnt)};
        \pgfmathtruncatemacro{\k}{\k + 1}
        \ifnumless{\k}{1}{
            \pgfmathtruncatemacro{\k}{\pointsof{#1} - \i}
        }{
            \ifnumgreater{\k}{\pointsof{#1}}{
                \pgfmathtruncatemacro{\k}{\k - \pointsof{#1}}
            }{
            }
        }
    }
    \stepcounter{innercnt}
    \node (#4-\theinnercnt) at (#1-\k.center){};
    \csedef{#4}{\csuse{#4} (#4-\theinnercnt)};
}

下一个宏根据输入选项（直接或反向）使用前两个宏创建存储两个交点之间的完整路径的宏。

% #1: first path 
% #2: direct or reverse first path
% #3: second path
% #4: direct or reverse second path
% #5: index of first point
% #6: index of second point
% #7: name of resulting path
\newcommand{\constructpath}[7]{
    \setcounter{innercnt}{0}

    \pgfmathtruncatemacro{\indexone}{\getstorageof{#1}{#1#3}{#5}}
    \pgfmathtruncatemacro{\indextwo}{\getstorageof{#1}{#1#3}{#6}}

    \ifstrequal{#2}{direct}{
        \constructdirectpath{#1}{\indexone}{\indextwo}{#7}{\theinnercnt}
    }{
        \constructreversepath{#1}{\indexone}{\indextwo}{#7}{\theinnercnt}
    }

    \pgfmathtruncatemacro{\indexone}{\getstorageof{#3}{#1#3}{#5}}
    \pgfmathtruncatemacro{\indextwo}{\getstorageof{#3}{#1#3}{#6}}

    \ifstrequal{#4}{direct}{
        \constructdirectpath{#3}{\indextwo}{\indexone}{#7}{\theinnercnt}
    }{
        \constructreversepath{#3}{\indextwo}{\indexone}{#7}{\theinnercnt}
    }

    \setpointsof{#7}{\theinnercnt}
    \csxdef{#7}{}
    \foreach \k in {1, ..., \theinnercnt}{
        \csxdef{#7}{\csuse{#7} (#7-\k)}
    }
}

就是这样。下面我将逐步展示如何实施针对该问题提出的解决方案。

结构

让我们首先明确文档的基本结构，包括包、库、变量和计数器。所有后续更改都将在环境内进行tikzpicture。

\documentclass{standalone}

\usepackage{etoolbox}

\usepackage{tikz}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{intersections}
\usetikzlibrary{shapes.geometric}

\newcounter{cnt}
\newcounter{innercnt}

\newdimen\xone
\newdimen\yone
\newdimen\xref
\newdimen\yref

%
% here you must put the tikzset and macros.
%

\begin{document}
    \begin{tikzpicture}

        %
        % here you must put all subsequent code
        %

    \end{tikzpicture}
\end{document}

一步步

首先让我们创建三个椭圆作为我们的基础paths。

\path node [ellipse, split = path01, minimum width = 2cm, minimum height = 1cm] {};
\path node [ellipse, split = path02, minimum width = 2cm, minimum height = 1cm, rotate = 60] {};
\path node [ellipse, split = path03, minimum width = 2cm, minimum height = 1cm, rotate = 120] {};

我们将得到：

椭圆底面

因此我们需要椭圆之间的交点。在本例中，我们使用的是第一个水平椭圆和旋转 60 度的椭圆。我放置了每个交点的索引，以显示如何引用每个交点。

交叉口

现在我们需要构建paths在交点之间使用的。在图片中，路径被绘制以突出显示变化。但下面的代码没有。

\constructpath{path01}{direct}{path02}{reverse}{1}{2}{path-1-2-1}
\constructpath{path01}{direct}{path02}{reverse}{2}{3}{path-1-2-2}
\constructpath{path01}{direct}{path02}{reverse}{3}{4}{path-1-2-3}
\constructpath{path01}{direct}{path02}{reverse}{4}{1}{path-1-2-4}

\path [name path = path-1-2-1] plot [smooth cycle] coordinates {\csuse{path-1-2-1}};
\path [name path = path-1-2-2] plot [smooth cycle] coordinates {\csuse{path-1-2-2}};
\path [name path = path-1-2-3] plot [smooth cycle] coordinates {\csuse{path-1-2-3}};
\path [name path = path-1-2-4] plot [smooth cycle] coordinates {\csuse{path-1-2-4}};

交点之间的路径

我们对剩下的椭圆也做同样的操作。

\getintersectionsof{path01}{path03}{0.8pt}

\constructpath{path01}{direct}{path03}{reverse}{1}{2}{path-1-3-1}
\constructpath{path01}{direct}{path03}{reverse}{2}{3}{path-1-3-2}
\constructpath{path01}{direct}{path03}{reverse}{3}{4}{path-1-3-3}
\constructpath{path01}{direct}{path03}{reverse}{4}{1}{path-1-3-4}

\path [name path = path-1-3-1] plot [smooth cycle] coordinates {\csuse{path-1-3-1}};
\path [name path = path-1-3-2] plot [smooth cycle] coordinates {\csuse{path-1-3-2}};
\path [name path = path-1-3-3] plot [smooth cycle] coordinates {\csuse{path-1-3-3}};
\path [name path = path-1-3-4] plot [smooth cycle] coordinates {\csuse{path-1-3-4}};

paths现在我们使用paths之前准备好的来定义最终版本。

\getintersectionsof{path02}{path-1-3-1}{0.8pt}
\getintersectionsof{path02}{path-1-3-3}{0.8pt}
\constructpath{path02}{direct}{path-1-3-1}{direct}{1}{2}{path-1-2-3-3}
\constructpath{path02}{direct}{path-1-3-3}{direct}{1}{2}{path-1-2-3-4}

\getintersectionsof{path03}{path-1-2-2}{0.8pt}
\getintersectionsof{path03}{path-1-2-4}{0.8pt}
\constructpath{path03}{direct}{path-1-2-2}{reverse}{1}{2}{path-1-2-3-5}
\constructpath{path03}{direct}{path-1-2-4}{reverse}{1}{2}{path-1-2-3-6}

现在只需画出感兴趣的区域。

\fill [orange!80] plot [smooth cycle] coordinates {\csuse{path-1-2-3-1}};
\fill [orange!80] plot [smooth cycle] coordinates {\csuse{path-1-2-3-4}};
\fill [orange!80] plot [smooth cycle] coordinates {\csuse{path-1-2-3-5}};

然后 TAH-DAH！！！

最后结果

Answer