我将如何为这个超球面的内圆着色
该图的代码如下：
\documentclass[tikz,border=5mm]{standalone} % <<< added
\usepackage{tikz-3dplot} % <<< added
\tdplotsetmaincoords{60}{115} % <<< added
\begin{document} % <<< added
\begin{tikzpicture}[tdplot_main_coords, scale = 2.5]
\coordinate (P) at ({-1},{0},{0});
\coordinate (P1) at ({sqrt(2)/2},{sqrt(2)/2},0);
\shade[ball color = lightgray, opacity = 0.5] (0,0,0) circle (1cm);
\tdplotsetrotatedcoords{0}{0}{0};
\draw[dashed, tdplot_rotated_coords, gray] (0,0,0) circle (1);
\tdplotsetrotatedcoords{90}{90}{90};
\draw[dashed, tdplot_rotated_coords, gray] (1,0,0) arc (0:180:1);
\tdplotsetrotatedcoords{0}{90}{90};
\draw[dashed, tdplot_rotated_coords, gray] (1,0,0) arc (0:180:1);
\draw[dashed, gray] (0,0,0) -- (-1,0,0);
\draw[dashed, gray] (0,0,0) -- (0,-1,0);
\draw[-stealth] (0,0,0) -- (1.80,0,0) node[below left] {$x$};
\draw[-stealth] (0,0,0) -- (0,1.30,0) node[below right] {$y$};
\draw[-stealth] (0,0,0) -- (0,0,1.30) node[above] {$z$};
\draw[thick, -stealth] (0,0,0) -- (P) node[right] {$P_i$};
\draw[thick, -stealth] (0,0,0) -- (P1) node[right] {$P_f$};
\draw[fill = lightgray!50] (P) circle (.5pt);
\draw[fill = lightgray!50] (P1) circle (.5pt);
\end{tikzpicture}
\end{document} % <<< added
答案1
看起来，你的一些虚线不正确。
\documentclass[tikz,border=1cm]{standalone}
\usetikzlibrary{calc,3dtools}% https://github.com/marmotghost/tikz-3dtools
\begin{document}
\begin{tikzpicture}[line join=round,
line cap=round,3d/install view={phi=120,theta=70},c/.style={circle,fill,inner sep=1pt},
declare function={R=4;h=2;r=sqrt(R*R-h*h;}]
\path
(0,0,0) coordinate (O)
pic{3d/circle on sphere={R=R,P={(O)},n={(0,0,1)}}};
\begin{scope}[canvas is xz plane at y=0]
\draw[3d/hidden] (R,0) arc[start angle=0,end angle=180,radius=R];
\end{scope}
\begin{scope}[canvas is yz plane at x=0]
\draw[3d/hidden] (R,0) arc[start angle=0,end angle=180,radius=R];
\end{scope}
\shade[ball color=white,3d/screen coords,opacity=0.7] (O) circle[radius=R];
\draw[3d/hidden] (0,0,0) -- (0,0,R) (O)--(R,0,0) (O)--(0,R,0);
\draw[3d/visible, -latex] (R,0,0) -- (R + 4,0,0) node[right]{$x$};
\draw[3d/visible, -latex] (0,R,0) -- (0,R + 1,0) node[right]{$y$};
\draw[3d/visible, -latex] (0,0,R) -- (0,0,R + 1) node[above]{$z$};
\path foreach \p/\g in {O/-90}
{(\p)node[c]{}+(\g:2.5mm) node{$\p$}};
\end{tikzpicture}
\begin{tikzpicture}[line join=round,
line cap=round,3d/install view={phi=120,theta=70},c/.style={circle,fill,inner sep=1pt},
declare function={R=4;h=2;r=sqrt(R*R-h*h;}]
\path
pic{3d/circle on sphere={R=R,P={(O)},n={(0,0,1)}}}
pic{3d/circle on sphere={R=R,P={(O)},n={(1,0,0)}}}
pic{3d/circle on sphere={R=R,P={(O)},n={(0,1,0)}}}
;
\shade[ball color=white,3d/screen coords,opacity=0.7] (O) circle[radius=R];
\draw[3d/hidden] (0,0,0) -- (0,0,R) (O)--(R,0,0) (O)--(0,R,0);
\draw[3d/visible, -latex] (R,0,0) -- (R + 4,0,0) node[right]{$x$};
\draw[3d/visible, -latex] (0,R,0) -- (0,R + 1,0) node[right]{$y$};
\draw[3d/visible, -latex] (0,0,R) -- (0,0,R + 1) node[above]{$z$};
\path foreach \p/\g in {O/-90}
{(\p)node[c]{}+(\g:2.5mm) node{$\p$}};
\end{tikzpicture}
\end{document}
我懂了这个答案在球体上绘制圆弧。
\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{3dtools}% https://github.com/marmotghost/tikz-3dtools
\makeatletter
\tikzset{3d/point on sphere/.code={\tikzset{3d/.cd,#1,
/tikz/insert path={
({\pgfkeysvalueof{/tikz/3d/R}*cos(\pgfkeysvalueof{/tikz/3d/longitude})*cos(\pgfkeysvalueof{/tikz/3d/latitude})},
{\pgfkeysvalueof{/tikz/3d/R}*sin(\pgfkeysvalueof{/tikz/3d/longitude})*cos(\pgfkeysvalueof{/tikz/3d/latitude})},
{\pgfkeysvalueof{/tikz/3d/R}*sin(\pgfkeysvalueof{/tikz/3d/latitude})})}}},
3d/latitude/.initial=0,3d/longitude/.initial=0,
3d/arcs/O/.initial={(0,0,0)},
3d/arcs/n/.initial={(0,0,1)},
3d/arcs/A/.initial={(1,0,0)},
3d/arcs/B/.initial={(0,1,0)},
pics/3d/great circle arc/.style={code={%
\tikzset{3d/arcs/.cd,#1}%
\def\pv##1{\pgfkeysvalueof{/tikz/3d/arcs/##1}}%
\pgfmathsetmacro{\pgfutil@tmpa}{tddistance("\pgfkeysvalueof{/tikz/3d/arcs/A}","\pgfkeysvalueof{/tikz/3d/arcs/O}")}%
\pgfmathsetmacro{\pgfutil@tmpb}{tddistance("\pgfkeysvalueof{/tikz/3d/arcs/B}","\pgfkeysvalueof{/tikz/3d/arcs/O}")}%
\pgfmathtruncatemacro{\pgfutil@tmpi}{abs(\pgfutil@tmpa-\pgfutil@tmpb)<0.01?1:0}%
\ifnum\pgfutil@tmpi=0\relax
\PackageWarning{3dtools}{The points \pv{A} and \pv{B} do not sit on the
same sphere around \pv{O}.}%
\else
\pgfmathsetmacro{\pgfutil@tmpA}{TD("\pgfkeysvalueof{/tikz/3d/arcs/A}")}%
\pgfmathsetmacro{\pgfutil@tmpB}{TD("\pgfkeysvalueof{/tikz/3d/arcs/B}")}%
\pgfmathsetmacro{\pgfutil@tmpO}{TD("\pgfkeysvalueof{/tikz/3d/arcs/O}")}%
\pgfmathtruncatemacro{\pgfutil@tmpV}{sign(screendepth(\pgfutil@tmpA)-screendepth(\pgfutil@tmpO))}%
\pgfmathtruncatemacro{\pgfutil@tmpW}{sign(screendepth(\pgfutil@tmpB)-screendepth(\pgfutil@tmpO))}%
\pgfmathsetmacro{\pgfutil@tmpr}{sqrt(\pgfutil@tmpa*\pgfutil@tmpb)}%
\pgfmathsetmacro{\pgfutil@tmpc}{Mod(360+acos(TD("\pv{A}-\pv{O}o\pv{B}-\pv{O}")/\pgfutil@tmpa/\pgfutil@tmpb),360)}%
\pgfmathsetmacro{\pgfutil@tmpi}{(tddistance("\pgfkeysvalueof{/tikz/3d/arcs/B}","\pgfkeysvalueof{/tikz/3d/arcs/A}")
-\pgfutil@tmpa-\pgfutil@tmpb<0.1?1:0)}%
\begingroup
\ifnum\pgfutil@tmpi=0\relax
\pgfmathsetmacro{\pgfutil@tmpC}{TD("\pgfkeysvalueof{/tikz/3d/arcs/A}x\pgfkeysvalueof{/tikz/3d/arcs/n}")}%
\tikzset{3d/define orthonormal dreibein={A={\pv{O}},B={\pv{A}},C={(\pgfutil@tmpC)}}}%
\typeout{May not properly work yet.}%
\else
\tikzset{3d/define orthonormal dreibein={A={\pv{O}},B={\pv{A}},C={\pv{B}}}}
\fi
\endgroup
\edef\pgfutil@tmpp{0}%
\tikzset{3d/arc/draw arc}
\fi
}},
pics/3d/circle arc/.style={code={%
\tikzset{3d/arcs/.cd,#1}%
\def\pv##1{\pgfkeysvalueof{/tikz/3d/arcs/##1}}%
\pgfmathsetmacro{\pgfutil@tmpa}{tddistance("\pv{A}","\pv{O}")}%
\pgfmathsetmacro{\pgfutil@tmpb}{tddistance("\pv{B}","\pv{O}")}%
\pgfmathtruncatemacro{\pgfutil@tmpi}{abs(\pgfutil@tmpa-\pgfutil@tmpb)<0.01?1:0}%
\ifnum\pgfutil@tmpi=0\relax
\PackageWarning{3dtools}{The points \pv{A} and \pv{B} do not sit on the
same sphere around \pv{O}.}%
\else
\pgfmathsetmacro{\pgfutil@tmpA}{TD("\pv{A}")}%
\pgfmathsetmacro{\pgfutil@tmpB}{TD("\pv{B}")}%
\pgfmathsetmacro{\pgfutil@tmpO}{TD("\pv{O}")}%
\pgfmathsetmacro{\pgfutil@tmpn}{TDunit("\pv{n}")}%
\pgfmathsetmacro{\pgfutil@tmpp}{TD("(\pgfutil@tmpn)o\pv{A}")}%
\pgfmathsetmacro{\pgfutil@tmpq}{TD("(\pgfutil@tmpn)o\pv{B}")}%
\pgfmathtruncatemacro{\pgfutil@tmpi}{abs(\pgfutil@tmpp-\pgfutil@tmpq)<0.01?1:0}%
\ifnum\pgfutil@tmpi=0\relax
\PackageWarning{3dtools}{The points \pv{A} and \pv{B} are not located on
the sphere around \pv{O} in such a way that the normal \pv{n} can be used
to define an arc.}%
\else
\pgfmathsetmacro{\pgfutil@tmpX}{TD("\pv{A}-\pv{O}x\pv{B}-\pv{O}")}%
\pgfmathsetmacro{\pgfutil@tmpx}{TD("(\pgfutil@tmpX)o(\pgfutil@tmpX)")}%
\pgfmathtruncatemacro{\pgfutil@tmpi}{(\pgfutil@tmpx<0.02?0:1)}%
\ifnum\pgfutil@tmpi=0\relax
\pgfmathsetmacro{\pgfutil@tmpC}{TD("\pv{A}-\pv{O}x(\pgfutil@tmpn)")}%
\else
\let\pgfutil@tmpC\pgfutil@tmpB
\fi
\pgfmathsetmacro{\pgfutil@tmpD}{TD("\pv{O}+\pgfutil@tmpp*(\pgfutil@tmpn)")}%
%\typeout{\pgfutil@tmpp,\pgfutil@tmpq,(D)=(\pgfutil@tmpD)}%
\pgfmathtruncatemacro{\pgfutil@tmpV}{sign(screendepth(\pgfutil@tmpA)-screendepth(\pgfutil@tmpO))}%
\pgfmathtruncatemacro{\pgfutil@tmpW}{sign(screendepth(\pgfutil@tmpB)-screendepth(\pgfutil@tmpO))}%
\pgfmathsetmacro{\pgfutil@tmpr}{sqrt(TD("\pv{A}-(\pgfutil@tmpD)o\pv{A}-(\pgfutil@tmpD)"))}%
\pgfmathsetmacro{\pgfutil@tmpc}{Mod(360+acos(TD("\pv{A}-(\pgfutil@tmpD)o\pv{B}-(\pgfutil@tmpD)")/\pgfutil@tmpr/\pgfutil@tmpr),360)}%
%\typeout{A=(\pgfutil@tmpA),B=(\pgfutil@tmpB),D=(\pgfutil@tmpD)}%
\begingroup
\tikzset{3d/define orthonormal dreibein={B={\pv{A}},
A={(\pgfutil@tmpD)},C/.expanded={(\pgfutil@tmpC)}}}%
\endgroup
\pgfmathsetmacro{\pgfutil@tmpp}{TD("\pgfkeysvalueof{/tikz/3d/aux keys/ez}o(\pgfutil@tmpA)")}
\tikzset{3d/arc/draw arc}
\fi
\fi
}},
/tikz/3d/arc/draw arc/.code={\begin{scope}[shift={(\pgfutil@tmpO)},
x/.expanded={\pgfkeysvalueof{/tikz/3d/aux keys/ex}},
y/.expanded={\pgfkeysvalueof{/tikz/3d/aux keys/ey}},
z/.expanded={\pgfkeysvalueof{/tikz/3d/aux keys/ez}}]
\pgfmathsetmacro{\pgfutil@tmpt}{min(Mod(360+atan2(-1*nscreenx,nscreeny),360),Mod(540+atan2(-1*nscreenx,nscreeny),360))}%
\pgfmathsetmacro{\pgfutil@tmpu}{max(Mod(360+atan2(-1*nscreenx,nscreeny),360),Mod(540+atan2(-1*nscreenx,nscreeny),360))}%
\pgfmathtruncatemacro{\pgfutil@tmpi}{(\pgfutil@tmpt>0&&\pgfutil@tmpt<\pgfutil@tmpc)+(\pgfutil@tmpu>0&&\pgfutil@tmpu<\pgfutil@tmpc)}%
%\typeout{i=\pgfutil@tmpi, t=\pgfutil@tmpt, u=\pgfutil@tmpu,
% c=\pgfutil@tmpc, p=\pgfutil@tmpp, V=\pgfutil@tmpV,W=\pgfutil@tmpW}
\begin{scope}[canvas is xy plane at z=\pgfutil@tmpp]
\ifcase\pgfutil@tmpi
\ifnum\pgfutil@tmpV<0
\path[3d/hidden] (0:\pgfutil@tmpr) arc[start angle=0,end angle=\pgfutil@tmpc,radius=\pgfutil@tmpr];
\else
\path[3d/visible] (0:\pgfutil@tmpr) arc[start angle=0,end angle=\pgfutil@tmpc,radius=\pgfutil@tmpr];
\fi
\or
\ifnum\pgfutil@tmpV<0
\path[3d/hidden] (0:\pgfutil@tmpr)
arc[start angle=0,end angle=\pgfutil@tmpt,radius=\pgfutil@tmpr];
\path[3d/visible] (\pgfutil@tmpt:\pgfutil@tmpr)
arc[start angle=\pgfutil@tmpt,end angle=\pgfutil@tmpc,radius=\pgfutil@tmpr];
\else
\path[3d/visible] (0:\pgfutil@tmpr)
arc[start angle=0,end angle=\pgfutil@tmpt,radius=\pgfutil@tmpr];
\path[3d/hidden] (\pgfutil@tmpt:\pgfutil@tmpr)
arc[start angle=\pgfutil@tmpt,end angle=\pgfutil@tmpc,radius=\pgfutil@tmpr];
\fi
\or
\ifnum\pgfutil@tmpV<0
\path[3d/hidden] (0:\pgfutil@tmpr)
arc[start angle=0,end angle=\pgfutil@tmpt,radius=\pgfutil@tmpr];
\path[3d/visible] (\pgfutil@tmpt:\pgfutil@tmpr)
arc[start angle=\pgfutil@tmpt,end angle=\pgfutil@tmpu,radius=\pgfutil@tmpr];
\path[3d/hidden] (\pgfutil@tmpu:\pgfutil@tmpr)
arc[start angle=\pgfutil@tmpu,end angle=\pgfutil@tmpc,radius=\pgfutil@tmpr];
\else
\path[3d/visible] (0:\pgfutil@tmpr)
arc[start angle=0,end angle=\pgfutil@tmpt,radius=\pgfutil@tmpr];
\path[3d/hidden] (\pgfutil@tmpt:\pgfutil@tmpr)
arc[start angle=\pgfutil@tmpt,end angle=\pgfutil@tmpu,radius=\pgfutil@tmpr];
\path[3d/visible] (\pgfutil@tmpu:\pgfutil@tmpr)
arc[start angle=\pgfutil@tmpu,end angle=\pgfutil@tmpc,radius=\pgfutil@tmpr];
\fi
\fi
\end{scope}
\end{scope}}}
\makeatother
\begin{document}
\begin{tikzpicture}[3d/install view={phi=140,psi=0,theta=70},
line join = round, line cap = round,
declare function={R=3;}]
\path (0,0,0) coordinate (O);
\shade[ball color=white,3d/screen coords,opacity=0.7] (O) circle[radius=R];
\path
(0,0,0) coordinate (O)
pic[green]{3d/circle on sphere={R=R,P={(O)},n={(0,0,1)}}};
\path (R,0,0) coordinate (M)
(-R,0,0) coordinate (N)
(O) pic[blue]{3d/circle arc={A={(M)},B={(N)},O={(O)},n={(0,1,0)}}};
\path (0,R,0) coordinate (E)
(0,-R, 0) coordinate (F)
(O) pic[red]{3d/circle arc={A={(E)},B={(F)},O={(O)},n={(-1,0,0)}}};
%\path foreach \X in {M,N,E,F} {(\X)node[circle,fill,inner sep=1pt]{}};
\draw[3d/hidden] (0,0,0) -- (0,0,R) (-R,0,0)--(R,0,0) (0,-R,0)--(0,R,0);
\draw[3d/visible, -latex] (R,0,0) -- (R + 2,0,0) node[above]{$x$};
\draw[3d/visible, -latex] (0,R,0) -- (0,R + 2,0) node[right]{$y$};
\draw[3d/visible, -latex] (0,0,R) -- (0,0,R + 1) node[above]{$z$};
\end{tikzpicture}
\end{document}
