Dtyjlui

I am trying to draw a circle (sphere) passing through four points B, C, E, F like this picture

I tried with tikz-3dplot and my code

documentclass[border=3mm,12pt]{standalone}

usepackage{fouriernc}

usepackage{tikz,tikz-3dplot} 

usepackage{tkz-euclide}

usetkzobj{all}



 usetikzlibrary{intersections,calc,backgrounds}



 begin{document}



 tdplotsetmaincoords{70}{110}

  %tdplotsetmaincoords{80}{100}

 begin{tikzpicture}[tdplot_main_coords,scale=1.5]

 pgfmathsetmacroa{3}

 pgfmathsetmacrob{4}

 pgfmathsetmacroh{5}



 % definitions

 path

 coordinate(A) at (0,0,0)

coordinate (B) at (a,0,0)

coordinate (C) at (0,b,0)                           

coordinate (S) at (0,0,h)                

coordinate (E) at  ({a*h^2/(a*a + h*h)},0,{(a*a*h)/(a*a + h*h)})

coordinate (F) at  (0,{(b*h*h)/(b*b + h*h)},{(b*b*h)/(b*b + h*h)});

 draw[dashed,thick]

       (A) -- (B)  (A) -- (C)  (A) -- (E)  (S)--(A)  (F)--(A);

       draw[thick]

       (S) -- (B) -- (C) -- cycle;

       draw[thick] 

       (F) -- (B) (C)--(E) (F)--(E);

tkzMarkRightAngle(S,E,A);

tkzMarkRightAngle(S,F,A);



 foreach point/position in {A/below,B/left,C/below,S/above,E/left,F/above}

 {

   fill (point) circle (.8pt);

   node[position=3pt] at (point) {$point$};

 }



 end{tikzpicture}

 end{document} 

and got

How can I draw a circle (sphere) passing through four points B, C, E, F?

A circle is determined by 3 points. A sphere, of course, needs at least 4 points on its boundary to be determined. However, the projection of the sphere, i.e. the circle, won't necessarily run through the projections of these points. (Actually, if the sphere is uniquely determined by these points, the boundary circle, i.e. the projection of the sphere on the screen coordinates, will never run through all projections of the points because for this to happen, the points need to lie in a plane, but then they no longer uniquely determine the circle.)

This shows two ways to construct circles that run through some of the points:

1. The dotted circle runs through F, E and C. It is fixed by this requirement. As a consequence it misses B by a small amount.


2. The red dashed circle runs through the midpoint of BC and through these points. It misses F and E by small amounts.

documentclass[border=3mm,12pt]{standalone}

usepackage{fouriernc}

usepackage{tikz,tikz-3dplot} 

usepackage{tkz-euclide}

usetkzobj{all}

usetikzlibrary{calc,through}

tikzset{circle through 3 points/.style n args={3}{%

insert path={let    p1=($(#1)!0.5!(#2)$),

                    p2=($(#1)!0.5!(#3)$),

                    p3=($(#1)!0.5!(#2)!1!-90:(#2)$),

                    p4=($(#1)!0.5!(#3)!1!90:(#3)$),

                    p5=(intersection of p1--p3 and p2--p4)

                    in },

at={(p5)},

circle through= {(#1)}

}}



 usetikzlibrary{intersections,calc,backgrounds}



 begin{document}



 tdplotsetmaincoords{70}{110}

  %tdplotsetmaincoords{80}{100}

 begin{tikzpicture}[tdplot_main_coords,scale=1.5]

 pgfmathsetmacroa{3}

 pgfmathsetmacrob{4}

 pgfmathsetmacroh{5}



 % definitions

 path

 coordinate(A) at (0,0,0)

coordinate (B) at (a,0,0)

coordinate (C) at (0,b,0)                           

coordinate (S) at (0,0,h)                

coordinate (E) at  ({a*h^2/(a*a + h*h)},0,{(a*a*h)/(a*a + h*h)})

coordinate (F) at  (0,{(b*h*h)/(b*b + h*h)},{(b*b*h)/(b*b + h*h)});

 draw[dashed,thick]

       (A) -- (B)  (A) -- (C)  (A) -- (E)  (S)--(A)  (F)--(A);

       draw[thick]

       (S) -- (B) -- (C) -- cycle;

       draw[thick] 

       (F) -- (B) (C)--(E) (F)--(E);

tkzMarkRightAngle(S,E,A);

tkzMarkRightAngle(S,F,A);



 foreach point/position in {A/below,B/left,C/below,S/above,E/left,F/above}

 {

   fill (point) circle (.8pt);

   node[position=3pt] at (point) {$point$};

 }

  node[circle through 3 points={F}{E}{C},draw=blue,dotted]{};

  draw[red,dashed] 

  let p1=($(B)-(C)$), n1={veclen(x1,y1)/2} in ($(B)!0.5!(C)$) circle (n1);

end{tikzpicture}

end{document}

It will be possible to construct the sphere as well. However, as mentioned its boundary may not run through any of the points.

ADDENDUM: In your setup, the four points do not determine a unique sphere because they all lie in a plane. Using Mathematica I was able to express F as a linear combination

 F = x B + y C + z E

where

So in this setup it is not possible to draw a unique sphere.