How to draw circle passing through three points in 3D?

Question

How to draw circle passing through three points A,P, Q in this code.

\documentclass[border=3mm,12pt,tikz]{standalone}
\usepackage{fouriernc}
\usepackage{tikz,tikz-3dplot} 
\tikzset{projection of point/.style args={(#1,#2,#3) on line through (#4,#5,#6)
        and (#7,#8,#9)}{%
        /utils/exec=\pgfmathsetmacro{\myprefactor}{((#1-#4)*(#7-#4)+(#2-#5)*(#8-#5)+(#3-#6)*(#9-#6))/((#7-#4)*(#7-#4)+(#8-#5)*(#8-#5)+(#9-#6)*(#9-#6))},
        insert path={%
            ({#4+\myprefactor*(#7-#4)},{#5+\myprefactor*(#8-#5)},{#6+\myprefactor*(#9-#6)})}
}}
\begin{document}
    \tdplotsetmaincoords{70}{110}
    \begin{tikzpicture}[tdplot_main_coords]
        \pgfmathsetmacro\a{4}
        \pgfmathsetmacro\b{8}
        \pgfmathsetmacro\h{7}
    \pgfmathsetmacro\r{sqrt(\a*\a + \b*\b)/2}   
        % definitions
        \path
        coordinate(A) at (0,0,0)
        coordinate (B) at (\a,0,0)
        coordinate (C) at (\a,\b,0) 
            coordinate (D) at (0,\b,0)                          
        coordinate (S) at (0,0,\h) 
            coordinate (I) at (\a/2,\b/2,0)   ;            
    
        \path[projection of point={(0,0,0) on line through (\a,0,0) and (0,0,\h)}]
        coordinate  (P)
        [projection of point={(0,0,0) on line through (0,\b,0) and (0,0,\h)}]
        coordinate (Q);
        
        \draw[dashed] (S) -- (A) (P) -- (B) (Q) --(D) (A) -- (C) (B) -- (D)
        (A) -- (B) -- (C) -- (D)--cycle (P) -- (Q);
\draw (S) -- (P) (S) -- (Q) (S) -- (C);
    \draw (I) circle (\r cm); 
\draw[red, thick, dashed]  (A)-- (P) (A)-- (Q) ;
        \foreach \point/\position in {A/left,B/left,C/below,S/above,D/right,I/left,P/left,Q/right}
        {\fill (\point) circle (1.5pt);
            \node[\position=3pt] at (\point) {$\point$};
        }
    \end{tikzpicture}
\end{document} 

Answer 1

You can use 3dtools. The function 3d/circle on sphere can solve your question.

\documentclass[tikz,border=2mm]{standalone}
\usetikzlibrary{3dtools}
\tikzset{reverseclip/.style={insert path={(current bounding box.south west)rectangle (current bounding box.north east)} }}
\begin{document}
    \begin{tikzpicture}[line cap=round,line join=round,c/.style={circle,fill,inner sep=1pt},
        3d/install view={phi=120,theta=65},declare function={a=4;b=8;h=7;}]
        \path 
        (0,0,0) coordinate (A)  
        (a,0,0) coordinate (B)
        (a,b,0) coordinate (C)  
        (0,b,0) coordinate (D)
        (0,0,h) coordinate (S)
        (a/2,b/2,0) coordinate (I);
        \path[3d/line through={(S) and (B) named lSB},3d/line through={(S) and (D) named lSD},3d/line through={(S) and (C) named lSC} ];
        \path [3d/project={(A) on lSB}] coordinate (P)
        [3d/project={(A) on lSD}] coordinate (Q) [3d/project={(A) on lSC}] coordinate (M);
        \pgfmathsetmacro{\myR}{tddistance("(I)","(A)")} 
        \path[save named path=sph,3d/screen coords] (I) circle[radius=\myR];
        \path[3d/circumcircle center={A={(A)},B={(P)},C={(Q)}}] coordinate (T); 
        \pic[red,thick]{3d/circle on sphere={R=\myR,C={(I)}, P={(T)}}}; %circle APQ
        \path pic{3d/circle on sphere={R=\myR,C={(I)}, P={(I)}}}; % circle ABC
        \begin{scope}
            \clip (S) -- (B) -- (D) -- cycle [reverseclip];
            \draw[3d/screen coords,3d/visible]  (I) circle[radius=\myR];
        \end{scope}
        \begin{scope}
            \clip (S) -- (B) -- (D) -- cycle;
            \draw[3d/screen coords,3d/hidden]   (I) circle[radius=\myR];
        \end{scope}
        \path foreach \p/\g in {S/90,A/-90,B/180,C/-90,D/0,I/-90,P/120,Q/45,T/-90}
        {(\p)node[c]{}+(\g:2.5mm) node{$\p$}};
        \draw[3d/hidden] (A) -- (P) (A) -- (Q) (S) -- (A) (A) -- (C) (B) -- (D) (S) -- (I) (A) -- (B) -- (C) -- (D) -- cycle  (P) -- (B) (Q) -- (D) (M) -- (C);
\draw[3d/visible] (S) -- (P) (S) --(Q) (S) -- (M);
    \end{tikzpicture}
    \end{document}