知道圆心和半径位于平面上，如何画圆？

Question 1

这是另一个使用相应建议pic了实验 3dtools图书馆。不能保证它总是有效，但它似乎在这里有效。可以找到一个简单的分析讨论这里。

\documentclass[border=2 mm,12pt,tikz]{standalone} 
\usepackage{tikz-3dplot} 
\usetikzlibrary{3dtools} 
\begin{document} 
\tdplotsetmaincoords{70}{60} 
\begin{tikzpicture}[tdplot_main_coords,scale=1] 
 \pgfmathsetmacro\a{1} 
 \pgfmathsetmacro\h{6} 

 % definitions 
 \path 
 (0,0,0)  coordinate(A)
 (4*\a,0,0)  coordinate (B)
 (4*\a,3*\a,0)  coordinate (C)
 (0,0,\h)  coordinate (S)
 ({4*\h^2*\a/(16*\a^2+\h^2)}, 0, {16*\h*\a^2/(16*\a^2+\h^2)})  coordinate (E)
 ({4*\h^2*\a/(25*\a^2+\h^2)}, {3*\h^2*\a/(25*\a^2+\h^2)}, {25*\h*\a^2/(25*\a^2+\h^2)})
   coordinate (F);
 \begin{scope} 
 \draw[dashed,thick] 
 (A) -- (C) ; 
 \draw[thick] 
 (S) -- (B) (S)-- (A) -- (B)-- (C) -- cycle (A) --(E) (A) --(F); 
 \end{scope} 
 \foreach \point/\position in {A/left,B/below,C/right,S/above,E/right,F/below} 
 { 
 \fill[black] (\point) circle (1.5pt); 
 \node[\position=1.5pt] at (\point) {$\point$}; 
 } 
 \pic[draw=blue,thick]{3d circle through 3 points={A={(E)},B={(B)},C={(C)}}}; 
\end{tikzpicture}
\end{document}

Answer

这是另一个使用相应建议pic了实验 3dtools图书馆。不能保证它总是有效，但它似乎在这里有效。可以找到一个简单的分析讨论这里。

\documentclass[border=2 mm,12pt,tikz]{standalone} 
\usepackage{tikz-3dplot} 
\usetikzlibrary{3dtools} 
\begin{document} 
\tdplotsetmaincoords{70}{60} 
\begin{tikzpicture}[tdplot_main_coords,scale=1] 
 \pgfmathsetmacro\a{1} 
 \pgfmathsetmacro\h{6} 

 % definitions 
 \path 
 (0,0,0)  coordinate(A)
 (4*\a,0,0)  coordinate (B)
 (4*\a,3*\a,0)  coordinate (C)
 (0,0,\h)  coordinate (S)
 ({4*\h^2*\a/(16*\a^2+\h^2)}, 0, {16*\h*\a^2/(16*\a^2+\h^2)})  coordinate (E)
 ({4*\h^2*\a/(25*\a^2+\h^2)}, {3*\h^2*\a/(25*\a^2+\h^2)}, {25*\h*\a^2/(25*\a^2+\h^2)})
   coordinate (F);
 \begin{scope} 
 \draw[dashed,thick] 
 (A) -- (C) ; 
 \draw[thick] 
 (S) -- (B) (S)-- (A) -- (B)-- (C) -- cycle (A) --(E) (A) --(F); 
 \end{scope} 
 \foreach \point/\position in {A/left,B/below,C/right,S/above,E/right,F/below} 
 { 
 \fill[black] (\point) circle (1.5pt); 
 \node[\position=1.5pt] at (\point) {$\point$}; 
 } 
 \pic[draw=blue,thick]{3d circle through 3 points={A={(E)},B={(B)},C={(C)}}}; 
\end{tikzpicture}
\end{document}

Question 2

要绘制圆，我们需要切换到B、C和E（这里还有F）所在的平面，并计算圆的中心和半径。这个答案重点介绍如何转换坐标系（因为您似乎知道中心和半径，而如何获得该坐标系的问题有一个出奇简单的答案）。

将平面的法向量标准化为n。我们需要找到旋转角度，使得旋转z轴与重合n。然而，旋转z轴只是

 D.(0,0,1)

tikz-3dplot其中旋转矩阵 D 在手册第 7 页给出

所以我们需要解决

   (0)   (nx)
 D |0| = |ny|
   (1)   (nz)

有解决方案

beta  = arccos(nz) 
alpha = arccos(nx/sin(beta))

为了方便用户，可以通过样式插入圆圈

  \draw[red,thick,circle in plane with normal={{\mynormal} with radius {\r} around (I)}];

这种风格也带有修正。答案的早期版本有时会产生错误的输出，原因很简单，只要一个符号反映出cos(x)=cos(-x)（GRRRR）。这里没有计算圆心和半径，我把它们都从我们的聊天。抱歉，标志错误！

\documentclass[tikz,border=3.14mm]{standalone} 
\usepackage{tikz-3dplot} 
\usetikzlibrary{3d} 
\usetikzlibrary{calc} 
\makeatletter 
\newcounter{smuggle} 
\DeclareRobustCommand\smuggleone[1]{% 
\stepcounter{smuggle}% 
\expandafter\global\expandafter\let\csname smuggle@\arabic{smuggle}\endcsname#1% 
\aftergroup\let\aftergroup#1\expandafter\aftergroup\csname smuggle@\arabic{smuggle}\endcsname 
} 
\DeclareRobustCommand\smuggle[2][1]{% 
\smuggleone{#2}% 
\ifnum#1>1 
\aftergroup\smuggle\aftergroup[\expandafter\aftergroup\the\numexpr#1-1\aftergroup]\aftergroup#2% 
\fi 
} 
\makeatother 
\def\parsecoord(#1,#2,#3)>(#4,#5,#6){% 
\def#4{#1}% 
\def#5{#2}% 
\def#6{#3}% 
\smuggle{#4}% 
\smuggle{#5}% 
\smuggle{#6}% 
} 
\def\SPTD(#1,#2,#3).(#4,#5,#6){((#1)*(#4)+1*(#2)*(#5)+1*(#3)*(#6))} 
\def\VPTD(#1,#2,#3)x(#4,#5,#6){((#2)*(#6)-1*(#3)*(#5),(#3)*(#4)-1*(#1)*(#6),(#1)*(#5)-1*(#2)*(#4))} 
\def\VecMinus(#1,#2,#3)-(#4,#5,#6){(#1-1*(#4),#2-1*(#5),#3-1*(#6))} 
\def\VecAdd(#1,#2,#3)+(#4,#5,#6){(#1+1*(#4),#2+1*(#5),#3+1*(#6))} 
\newcommand{\RotationAnglesForPlaneWithNormal}[5]{%\typeout{N=(#1,#2,#3)}
\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
 }}
} 
\tikzset{circle in plane with normal/.style args={#1 with radius #2 around #3}{ 
/utils/exec={\edef\temp{\noexpand\parsecoord#1>(\noexpand\myNx,\noexpand\myNy,\noexpand\myNz)} 
\temp 
\pgfmathsetmacro{\myNx}{\myNx} 
\pgfmathsetmacro{\myNy}{\myNy} 
\pgfmathsetmacro{\myNz}{\myNz} 
\pgfmathsetmacro{\myNormalization}{sqrt(pow(\myNx,2)+pow(\myNy,2)+pow(\myNz,2))} 
\pgfmathsetmacro{\myNx}{\myNx/\myNormalization} 
\pgfmathsetmacro{\myNy}{\myNy/\myNormalization} 
\pgfmathsetmacro{\myNz}{\myNz/\myNormalization} 
% compute the rotation angles that transform us in the corresponding plabe 
\RotationAnglesForPlaneWithNormal{\myNx}{\myNy}{\myNz}{\tmpalpha}{\tmpbeta} 
%\typeout{N=(\myNx,\myNy,\myNz),alpha=\tmpalpha,beta=\tmpbeta,r=#2,#3} 
\tdplotsetrotatedcoords{\tmpalpha}{\tmpbeta}{0}}, 
insert path={[tdplot_rotated_coords,canvas is xy plane at z=0,transform shape] 
#3 circle[radius=#2]} 
}} 



\begin{document} 
\foreach \X in  {5,15,...,355} %  {50}% 
{\tdplotsetmaincoords{70}{\X} 
\begin{tikzpicture}[tdplot_main_coords,scale=1] 
\path [tdplot_screen_coords,use as bounding box] (-7,-3) rectangle (7,7); 
\pgfmathsetmacro\a{1} 
\pgfmathsetmacro\h{7} 
\pgfmathsetmacro\rprime{5*sqrt(\a^2*\h^2/(25*\a^2+\h^2))*(1/2))} 
\pgfmathsetmacro\r{(1/2)*sqrt((400*\a^4+9*\a^2*\h^2)/(16*\a^2+\h^2))}
% definitions 
\path 
coordinate(A) at (0,0,0) 
coordinate (B) at (4*\a,0,0) 
coordinate (C) at (4*\a,3*\a,0) 
coordinate (S) at (0,0,\h) 
coordinate (E) at ({4*\h^2*\a/(16*\a^2+\h^2)}, 0, {16*\h*\a^2/(16*\a^2+\h^2)}) 
coordinate (F) at ({4*\h^2*\a/(25*\a^2+\h^2)}, {3*\h^2*\a/(25*\a^2+\h^2)},{25*\h*\a^2/(25*\a^2+\h^2)}) 
coordinate (I') at ($(F)!0.5!(A) $) 
coordinate (I) at ($(C)!0.5!(E) $);
\begin{scope} 
\draw[dashed,thick] 
(A) -- (C) ; 
\draw[thick] 
(S) -- (B) (S)-- (A) -- (B)-- (C) -- cycle (A) --(E) (A) --(F); 
\end{scope} 
\foreach \point/\position in {A/left,B/below,C/right,S/above,E/right,F/below} 
{ 
\fill[black] (\point) circle (1.5pt); 
\node[\position=1.5pt] at (\point) {$\point$}; 
} 

% % store the coordinates of E, A and F in marcros 
\parsecoord({4*\h^2*\a/(16*\a^2+\h^2)},0,{16*\h*\a^2/(16*\a^2+\h^2)})>(\myEx,\myEy,\myEz) 
\parsecoord(0,0,0)>(\myAx,\myAy,\myAz) 
\parsecoord({4*\h^2*\a/(25*\a^2+\h^2)},{3*\h^2*\a/(25*\a^2+\h^2)},{25*\h*\a^2/(25*\a^2+\h^2)})>(\myFx,\myFy,\myFz) 
\parsecoord(4*\a,0,0)>(\myBx,\myBy,\myBz)
\parsecoord(4*\a,3*\a,0)>(\myCx,\myCy,\myCz)
% % compute the normal of the plane in which E, B and C sit 
\def\mynormal{\VPTD({\myEx-\myAx},{\myEy-\myAy},{\myEz-\myAz})x({\myFx-\myAx},{\myFy-\myAy},{\myFz-\myAz})} 
\edef\temp{\noexpand\parsecoord\mynormal>(\noexpand\myNx,\noexpand\myNy,\noexpand\myNz)} 
\draw[blue,thick,circle in plane with normal={{\mynormal} with radius {\rprime} 
around (I')}]; 
\def\mynormal{\VPTD({\myEx-\myBx},{\myEy-\myBy},{\myEz-\myBz})x({\myCx-\myBx},{\myCy-\myBy},{\myCz-\myBz})}
\draw[red,thick,circle in plane with normal={{\mynormal} with radius {\r} 
around (I)}]; 
\node[anchor=north east] at (current bounding box.north east) {$\theta=\tdplotmaintheta^\circ,\phi=\tdplotmainphi^\circ$}; 
\end{tikzpicture}} 
\end{document}

注：（i）对我而言，这个技巧可能对物体隐藏部分和可见部分的分析区分很重要。如果有一天我重新发现了这一点，我至少可以说我以前就知道了。（ii）无论谁试图进一步发展这个非常好的答案可能会发现这很有用。

Answer