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如何使用 3dtools 校正球体上圆的线？

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如何使用 3dtools 校正球体上圆的线？

A'BD我正在尝试使用 在球体上绘制圆圈3dtools。我试过

\documentclass[tikz,border=2mm, 12 pt]{standalone}
\usepackage{tikz-3dplot-circleofsphere}
\usetikzlibrary{3dtools} 
\usepackage{fouriernc}
\begin{document}  
\tdplotsetmaincoords{65}{170}
\begin{tikzpicture}[scale=1,tdplot_main_coords,declare function={a=6;b=6;h=6;r=sqrt(a*a + b*b)/2;R=sqrt(a*a + b*b + h*h)/2;Angle=acos(r/R);
}]
\path (0,0,0) coordinate (O)
(a/2,-b/2,0) coordinate (A)
(a/2,b/2,0) coordinate (B)
(-a/2,b/2,0) coordinate (C)
(-a/2,-b/2,0) coordinate (D)
(0,0,h) coordinate (O')
(a/2,-b/2,h) coordinate (A')
(-a/2,b/2,h) coordinate (C')
(a/2,b/2,h) coordinate (B')
(-a/2,-b/2,h) coordinate (D')
($ (O) !0.5!(O') $) coordinate (I);
\begin{scope}[tdplot_screen_coords]
\draw[thick] (I) circle (R);
\end{scope}
\foreach \p in {A',C',B,D,O,O'}
\draw[fill=black] (\p) circle (1.5pt);
\foreach \p/\g in {A'/90,C'/-90,B/-90,D/-90,O/-90,O'/90}
\path (\p)+(\g:3mm) node{$\p$};
\draw[dashed] (A') -- (B) -- (D) -- (C') -- cycle (A') -- (D) (B) -- (C') (O) -- (O')
;
\begin{scope}[canvas is xy plane at z=0]
\coordinate (M) at (\tdplotmainphi:r);
\coordinate (N) at (\tdplotmainphi+180:r);
%\coordinate (P) at ({r*sin(60)}, {r*cos(60)});
\end{scope}
%
\begin{scope}[canvas is xy plane at z=h]
\coordinate (M') at (\tdplotmainphi:r);
\coordinate (N') at (\tdplotmainphi+180:r);
\end{scope}
\foreach \X in {M,N} \draw[dashed] (\X) -- (\X') (M') -- (N') (M) -- (N);
\pic[draw=blue,dashed]{3d circle through 3 points={A={(A')},B={(B)},C={(D)}}};
\begin{scope}[shift={(I)}]
\tdplotCsDrawLatCircle[tdplotCsFront/.style={thick}]{R}{{-Angle}}
\tdplotCsDrawLatCircle[tdplotCsFront/.style={thick}]{R}{{Angle}}
\end{scope}
\end{tikzpicture}
\end{document}

我有

在此处输入图片描述

我怎样才能得到这样的照片？

在此处输入图片描述

是否存在一种适用于球体上所有圆的通用方法？

答案1

3dtools 库按预期工作，它通过三个点 A'、B 和 D 绘制圆。然而，它没有做，也从未承诺过要做的是区分前侧和后侧路径。您正在加载的另一个非官方包，，正是这样tikz-3dplot-circleofsphere做的。一旦你输入正确的角度，它就可以绘制圆。要确定角度，必须依赖其他工具，具有讽刺意味的是，3d circle through 3 points就是这样的工具。它允许我们计算所有角度，并获得

\documentclass[tikz,border=2mm, 12 pt]{standalone}

\usepackage{tikz-3dplot-circleofsphere}
\usetikzlibrary{3dtools} 
\usepackage{fouriernc}
\newcommand{\RotationAnglesForPlaneWithNormal}[5]{%\typeout{N=(#1,#2,#3)}
    \pgfmathtruncatemacro{\itest}{ifthenelse(abs(#3)==1,0,1)}
    \ifnum\itest=0
        \pgfmathtruncatemacro{\jtest}{sign(#1)}
        \ifnum\jtest=1
            \xdef#4{0}   
            \xdef#5{0}
        \else
            \xdef#4{180}   
            \xdef#5{0}      
        \fi 
    \else
    \foreach \XS in {1,-1}
    {\foreach \YS in {1,-1}
        {\pgfmathsetmacro{\mybeta}{\XS*acos(#3)} 
            \pgfmathsetmacro{\myalpha}{\YS*acos(#1/sin(\mybeta))} 
            \pgfmathsetmacro{\ntest}{abs(cos(\myalpha)*sin(\mybeta)-#1)%
                +abs(sin(\myalpha)*sin(\mybeta)-#2)+abs(cos(\mybeta)-#3)}
            \ifdim\ntest pt<0.1pt
            \xdef#4{\myalpha}   
            \xdef#5{\mybeta}
            \fi
    }}
    \fi
} 
\begin{document}
     \tdplotsetmaincoords{70}{100}
\begin{tikzpicture}[scale=1,tdplot_main_coords,declare function={a=6;b=6;h=6;r=sqrt(a*a + b*b)/2;R=sqrt(a*a + b*b + h*h)/2;Angle=acos(r/R);
}]
\path (0,0,0) coordinate (O)
(a/2,-b/2,0) coordinate (A)
(a/2,b/2,0) coordinate (B)
(-a/2,b/2,0) coordinate (C)
(-a/2,-b/2,0) coordinate (D)
(0,0,h) coordinate (O')
(a/2,-b/2,h) coordinate (A')
(-a/2,b/2,h) coordinate (C')
(a/2,b/2,h) coordinate (B')
(-a/2,-b/2,h) coordinate (D')
($ (O) !0.5!(O') $) coordinate (I);
\begin{scope}[tdplot_screen_coords]
\draw[thick] (I) circle (R);
\end{scope}
\foreach \p in {A',C',B,D,O,O'}
\draw[fill=black] (\p) circle (1.5pt);
\foreach \p/\g in {A'/90,C'/-90,B/-90,D/-90,O/-90,O'/90}
\path (\p)+(\g:3mm) node{$\p$};
\draw[dashed] (A') -- (B) -- (D) -- (C') -- cycle (A') -- (D) (B) -- (C') (O) -- (O')
;
\begin{scope}[canvas is xy plane at z=0]
\coordinate (M) at (\tdplotmainphi:r);
\coordinate (N) at (\tdplotmainphi+180:r);
%\coordinate (P) at ({r*sin(60)}, {r*cos(60)});
\end{scope}
%
\begin{scope}[canvas is xy plane at z=h]
\coordinate (M') at (\tdplotmainphi:r);
\coordinate (N') at (\tdplotmainphi+180:r);
\end{scope}
\foreach \X in {M,N} \draw[dashed] (\X) -- (\X') (M') -- (N') (M) -- (N);
\pic[draw=blue,dashed]{3d circle through 3 points={A={(A')},B={(B)},C={(D)}}};
\begin{scope}[shift={(I)}]
\tdplotCsDrawLatCircle[tdplotCsFront/.style={thick}]{R}{{-Angle}}
\tdplotCsDrawLatCircle[tdplotCsFront/.style={thick}]{R}{{Angle}}
\path[overlay] [3d coordinate={(A'-B)=(A')-(B)},
3d coordinate={(D-B)=(D)-(B)},
3d coordinate={(myn)=(A'-B)x(D-B)},
3d coordinate={(A'-M)=(A')-(M)}];
\pgfmathsetmacro{\mynormal}{1/sqrt(TD("(myn)o(myn)"))}
\pgfmathsetmacro{\mynormal}{TD("\mynormal*(myn)")}
\pgfmathsetmacro{\mynormalx}{xcomp3(\mynormal)}
\pgfmathsetmacro{\mynormaly}{ycomp3(\mynormal)}
\pgfmathsetmacro{\mynormalz}{zcomp3(\mynormal)}
\pgfmathsetmacro{\mygamma}{acos(sqrt(TD("(A'-M)o(A'-M)"))/R)}
\RotationAnglesForPlaneWithNormal{\mynormalx}{\mynormaly}{\mynormalz}{\myalpha}{\mybeta}
\tdplotCsDrawCircle[tdplotCsFront/.style={thick,red}]{R}{\myalpha}{\mybeta}{\mygamma} 
\end{scope}
\end{tikzpicture}
\end{document}

在此处输入图片描述

让我们详细说明一下：说完之后

 \pic[draw=blue,dashed]{3d circle through 3 points={A={(A')},B={(B)},C={(D)}}};

我们知道圆心，它默认存储在坐标中(M)（当然可以更改）。然后我们可以将法线计算为((A')-(B))x((D)-(B))。如果我们对其进行标准化，我们将获得两个角度，可以使用\RotationAnglesForPlaneWithNormal上一个答案中提供的命令计算这两个角度。它找到将 z 轴旋转到给定法线的 3d 旋转角度。最后一个角度确定圆与大圆的距离，它由给出acos(r/R)，其中r是圆的半径和R球体的半径。在这里我们知道圆心就派上用场了(M)，所以确定r很简单。

一个稍微简短的版本是

\documentclass[tikz,border=2mm, 12 pt]{standalone}
\usepackage{tikz-3dplot-circleofsphere}
\usetikzlibrary{3dtools} 
\usepackage{fouriernc}
\begin{document}
\tdplotsetmaincoords{70}{100}
\begin{tikzpicture}[scale=1,tdplot_main_coords,declare function={a=6;b=6;h=6;r=sqrt(a*a + b*b)/2;R=sqrt(a*a + b*b + h*h)/2;Angle=acos(r/R);
}]
\path (0,0,0) coordinate (O)
(a/2,-b/2,0) coordinate (A)
(a/2,b/2,0) coordinate (B)
(-a/2,b/2,0) coordinate (C)
(-a/2,-b/2,0) coordinate (D)
(0,0,h) coordinate (O')
(a/2,-b/2,h) coordinate (A')
(-a/2,b/2,h) coordinate (C')
(a/2,b/2,h) coordinate (B')
(-a/2,-b/2,h) coordinate (D')
($ (O) !0.5!(O') $) coordinate (I);
\begin{scope}[tdplot_screen_coords]
\draw[thick] (I) circle (R);
\end{scope}
\foreach \p in {A',C',B,D,O,O'}
\draw[fill=black] (\p) circle (1.5pt);
\foreach \p/\g in {A'/90,C'/-90,B/-90,D/-90,O/-90,O'/90}
\path (\p)+(\g:3mm) node{$\p$};
\draw[dashed] (A') -- (B) -- (D) -- (C') -- cycle (A') -- (D) (B) -- (C') (O) -- (O')
;
\begin{scope}[canvas is xy plane at z=0]
\coordinate (M) at (\tdplotmainphi:r);
\coordinate (N) at (\tdplotmainphi+180:r);
%\coordinate (P) at ({r*sin(60)}, {r*cos(60)});
\end{scope}
%
\begin{scope}[canvas is xy plane at z=h]
\coordinate (M') at (\tdplotmainphi:r);
\coordinate (N') at (\tdplotmainphi+180:r);
\end{scope}
\foreach \X in {M,N} \draw[dashed] (\X) -- (\X') (M') -- (N') (M) -- (N);
\pic[draw=none]{3d circle through 3 points={A={(A')},B={(B)},C={(D)}}};
\begin{scope}[shift={(I)}]
\tdplotCsDrawLatCircle[tdplotCsFront/.style={thick}]{R}{{-Angle}}
\tdplotCsDrawLatCircle[tdplotCsFront/.style={thick}]{R}{{Angle}}
\path[overlay] [3d coordinate={(myn)=(A')-(B)x(D)-(B)},
3d coordinate={(A'-M)=(A')-(M)}];
\pgfmathsetmacro{\myaxisangles}{axisangles("(myn)")}
\pgfmathsetmacro{\myalpha}{{\myaxisangles}[0]}
\pgfmathsetmacro{\mybeta}{{\myaxisangles}[1]}
\pgfmathsetmacro{\mygamma}{acos(sqrt(TD("(A'-M)o(A'-M)"))/R)}
\tdplotCsDrawCircle[tdplotCsFront/.style={thick,red}]{R}{\myalpha}{\mybeta}{\mygamma} 
\end{scope}
\end{tikzpicture}
\end{document}

在此处输入图片描述

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