Is there a command to find coordinates of projection of a point on a line in 3D?

Question

Sometimes, I need determine coordinates of projection of a point on a line in 3D. I know The Syntax of Projection in 2D at 13.5.5 The Syntax of Projection Modifiers at Manual for Version 3.1. This syntex is not correct in 3D. In my example. In triangle, SA=AB, then projection must be midpoint of segment SB, point E. With syntax coordinate (J) at ($(B)!(A)!(S)$) is not correct. Is there a command to find coordinates of projection of a point on a line in 3D?

My code

\documentclass[border=3mm,12pt]{standalone}
\usepackage{fouriernc}
\usepackage{tikz,tikz-3dplot} 
\usepackage{tkz-euclide}
\usetkzobj{all}
  \begin{document}
 \tdplotsetmaincoords{70}{110}
  %\tdplotsetmaincoords{80}{100}
 \begin{tikzpicture}[tdplot_main_coords,scale=1.5]
  \pgfmathsetmacro\a{4}
 \pgfmathsetmacro\b{3}
 \pgfmathsetmacro\h{4}

 % 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,\a)                
coordinate (E) at ($(B)!0.5!(S)$)
coordinate (J) at ($(B)!(A)!(S)$);
 \draw[dashed,thick]
       (A) -- (B)  (A) -- (C)  (A) -- (E)  (S)--(A)  ;
       \draw[thick]
       (S) -- (B) -- (C) -- cycle;
  \draw[red, thick, dashed]  (A)-- ($(B)!(A)!(S)$);
 \foreach \point/\position in {A/left,B/left,C/below,S/above,E/left,J/left}
 {
   \fill (\point) circle (.8pt);
   \node[\position=3pt] at (\point) {$\point$};
 }
 \end{tikzpicture}
     \end{document} 

Answer 1

TikZ does not store the 3d components of coordinates AFAIK, so the best I can offer is a style that computes the projection for points for which you specify the 3d coordinates explicitly. This is what the line

\path[projection of point={(0,0,0) on line through (\a,0,0) and (0,0,\a)}]
 coordinate[label=above left:$P$] (P)
 [projection of point={(0,0,0) on line through (0,\b,0) and (0,0,\a)}]
 coordinate[label=above right:$Q$] (Q)
 [projection of point={(0,0,0) on line through (0,\b,0) and (\a,0,0)}]
 coordinate[label=below:$R$] (R);

does in

\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}
  %\tdplotsetmaincoords{80}{100}
 \begin{tikzpicture}[tdplot_main_coords,scale=1.5]
  \pgfmathsetmacro\a{4}
 \pgfmathsetmacro\b{3}
 \pgfmathsetmacro\h{4}

 % 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,\a)                
%coordinate (E) at ($(B)!0.5!(S)$)
coordinate (J) at ($(B)!(A)!(S)$);
 \path[projection of point={(0,0,0) on line through (\a,0,0) and (0,0,\a)}]
 coordinate[label=above left:$P$] (P)
 [projection of point={(0,0,0) on line through (0,\b,0) and (0,0,\a)}]
 coordinate[label=above right:$Q$] (Q)
 [projection of point={(0,0,0) on line through (0,\b,0) and (\a,0,0)}]
 coordinate[label=below:$R$] (R);
 \draw[dashed,thick]
       (A) -- (B)  (A) -- (C)  (S)--(A)  ;
       \draw[thick]
       (S) -- (B) -- (C) -- cycle;
  %\draw[red, thick, dashed]  (A)-- ($(B)!(A)!(S)$);
  \draw[red, thick, dashed]  (A)-- (P) (A)-- (Q) (A)-- (R);
 \foreach \point/\position in {A/left,B/left,C/below,S/above,J/left}
 {
   \fill (\point) circle (.8pt);
   \node[\position=3pt] at (\point) {$\point$};
 }
 \end{tikzpicture}
\end{document} 

An animation to illustrate this.

\documentclass[border=3mm,12pt,tikz]{standalone}
\usepackage{fouriernc}
\usepackage{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}
\foreach \X in {5,15,...,355}  
{\tdplotsetmaincoords{70}{\X}
  %\tdplotsetmaincoords{80}{100}
 \begin{tikzpicture}[tdplot_main_coords,scale=1.5]
 \path[tdplot_screen_coords,use as bounding box] (-5,-2) rectangle (5,5);
 \pgfmathsetmacro\a{4}
 \pgfmathsetmacro\b{3}
 \pgfmathsetmacro\h{4}

 % 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,\a)                
%coordinate (E) at ($(B)!0.5!(S)$)
coordinate (J) at ($(B)!(A)!(S)$);
 \path[projection of point={(0,0,0) on line through (\a,0,0) and (0,0,\a)}]
 coordinate[label=above left:$P$] (P)
 [projection of point={(0,0,0) on line through (0,\b,0) and (0,0,\a)}]
 coordinate[label=above right:$Q$] (Q)
 [projection of point={(0,0,0) on line through (0,\b,0) and (\a,0,0)}]
 coordinate[label=below:$R$] (R);
 \draw[dashed,thick]
       (A) -- (B)  (A) -- (C)  (S)--(A)  ;
       \draw[thick]
       (S) -- (B) -- (C) -- cycle;
  %\draw[red, thick, dashed]  (A)-- ($(B)!(A)!(S)$);
  \draw[red, thick, dashed]  (A)-- (P) (A)-- (Q) (A)-- (R);
 \foreach \point/\position in {A/left,B/left,C/below,S/above,J/left}
 {
   \fill (\point) circle (.8pt);
   \node[\position=3pt] at (\point) {$\point$};
 }
 \end{tikzpicture}}
\end{document}