Is there a command to find coordinates of projection of a point on a plane?












1















Let SABC be a tetrahedron, SA = c, AB = a, AC = b, SA perpendicular to AB, AB perpendicular to AC, and AC perpendicular to SA. I am trying to find the projection H of the point A on the plane SBC. I tried with two ways.



Firt way. With some calculations, I found that H(({b^2*c^2*a/((b^2+c^2)*a^2+b^2*c^2)},{b*c^2*a^2/((b^2+c^2)*a^2+b^2*c^2)},{b^2*c*a^2/((b^2+c^2)*a^2+b^2*c^2)})).



Second way, We can prove that, H is orthocentre of triangle SBC, then I see at
Is there a command to find coordinates of projection of a point on a line in 3D?



documentclass[border=3mm,12pt,tikz]{standalone}
usepackage{fouriernc}
usepackage{tikz,tikz-3dplot}
usepackage{tkz-euclide}
usetkzobj{all}

usetikzlibrary{intersections,calc,backgrounds}
newcommandpgfmathsinandcos[3]{%
pgfmathsetmacro#1{sin(#3)}%
pgfmathsetmacro#2{cos(#3)}%
}
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]
pgfmathsetmacroa{4}
pgfmathsetmacrob{3}
pgfmathsetmacroc{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 (H) at ({b^2*c^2*a/((b^2+c^2)*a^2+b^2*c^2)},{b*c^2*a^2/((b^2+c^2)*a^2+b^2*c^2)},{b^2*c*a^2/((b^2+c^2)*a^2+b^2*c^2)});

begin{scope}
draw[dashed,thick]
(A) -- (B) (A) -- (C) (S)--(A) --(H) ;
draw[thick]
(S) -- (B) -- (C) -- cycle;
end{scope}
foreach point/position in {A/left,B/left,C/below,S/above,H/above}
{
fill (point) circle (1.5pt);
node[position=3pt] at (point) {$point$};
}
end{tikzpicture}

tdplotsetmaincoords{70}{110}
%tdplotsetmaincoords{80}{100}
begin{tikzpicture}[tdplot_main_coords,scale=1.5]
pgfmathsetmacroa{4}
pgfmathsetmacrob{3}
pgfmathsetmacroc{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)
;
path[projection of point={(0,0,0) on line through (a,0,0) and (0,0,a)}]
coordinate (P)
[projection of point={(0,0,0) on line through (0,b,0) and (0,0,a)}]
coordinate (Q)
[projection of point={(0,0,0) on line through (0,b,0) and (a,0,0)}]
coordinate (R);



begin{scope}
draw [very thick] (S) -- (R);
draw [very thick, name path=B--Q] (B) -- (Q);
draw [very thick, name path=C--P] (C) -- (P);
path [name intersections={of=B--Q and C--P,by=H}];
end{scope}

begin{scope}
draw[dashed,thick]
(A) -- (B) (A) -- (C) (S)--(A) (A)--(H) (A)--(R) (A)--(P) (A)--(Q);
draw[thick]
(S) -- (B) -- (C) -- cycle;
end{scope}

tkzMarkRightAngle(S,R,C)
tkzMarkRightAngle(B,P,C)
tkzMarkRightAngle(B,Q,C)
tkzMarkRightAngle(A,R,B)


foreach point/position in {A/left,B/left,C/below,S/above,P/left,Q/above,R/below,H/above}
{
fill (point) circle (1.5pt);
node[position=3pt] at (point) {$point$};
}
end{tikzpicture}
end{document}


enter image description here



enter image description here



Is there a command to find coordinates of projection of a point on a plane?










share|improve this question





























    1















    Let SABC be a tetrahedron, SA = c, AB = a, AC = b, SA perpendicular to AB, AB perpendicular to AC, and AC perpendicular to SA. I am trying to find the projection H of the point A on the plane SBC. I tried with two ways.



    Firt way. With some calculations, I found that H(({b^2*c^2*a/((b^2+c^2)*a^2+b^2*c^2)},{b*c^2*a^2/((b^2+c^2)*a^2+b^2*c^2)},{b^2*c*a^2/((b^2+c^2)*a^2+b^2*c^2)})).



    Second way, We can prove that, H is orthocentre of triangle SBC, then I see at
    Is there a command to find coordinates of projection of a point on a line in 3D?



    documentclass[border=3mm,12pt,tikz]{standalone}
    usepackage{fouriernc}
    usepackage{tikz,tikz-3dplot}
    usepackage{tkz-euclide}
    usetkzobj{all}

    usetikzlibrary{intersections,calc,backgrounds}
    newcommandpgfmathsinandcos[3]{%
    pgfmathsetmacro#1{sin(#3)}%
    pgfmathsetmacro#2{cos(#3)}%
    }
    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]
    pgfmathsetmacroa{4}
    pgfmathsetmacrob{3}
    pgfmathsetmacroc{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 (H) at ({b^2*c^2*a/((b^2+c^2)*a^2+b^2*c^2)},{b*c^2*a^2/((b^2+c^2)*a^2+b^2*c^2)},{b^2*c*a^2/((b^2+c^2)*a^2+b^2*c^2)});

    begin{scope}
    draw[dashed,thick]
    (A) -- (B) (A) -- (C) (S)--(A) --(H) ;
    draw[thick]
    (S) -- (B) -- (C) -- cycle;
    end{scope}
    foreach point/position in {A/left,B/left,C/below,S/above,H/above}
    {
    fill (point) circle (1.5pt);
    node[position=3pt] at (point) {$point$};
    }
    end{tikzpicture}

    tdplotsetmaincoords{70}{110}
    %tdplotsetmaincoords{80}{100}
    begin{tikzpicture}[tdplot_main_coords,scale=1.5]
    pgfmathsetmacroa{4}
    pgfmathsetmacrob{3}
    pgfmathsetmacroc{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)
    ;
    path[projection of point={(0,0,0) on line through (a,0,0) and (0,0,a)}]
    coordinate (P)
    [projection of point={(0,0,0) on line through (0,b,0) and (0,0,a)}]
    coordinate (Q)
    [projection of point={(0,0,0) on line through (0,b,0) and (a,0,0)}]
    coordinate (R);



    begin{scope}
    draw [very thick] (S) -- (R);
    draw [very thick, name path=B--Q] (B) -- (Q);
    draw [very thick, name path=C--P] (C) -- (P);
    path [name intersections={of=B--Q and C--P,by=H}];
    end{scope}

    begin{scope}
    draw[dashed,thick]
    (A) -- (B) (A) -- (C) (S)--(A) (A)--(H) (A)--(R) (A)--(P) (A)--(Q);
    draw[thick]
    (S) -- (B) -- (C) -- cycle;
    end{scope}

    tkzMarkRightAngle(S,R,C)
    tkzMarkRightAngle(B,P,C)
    tkzMarkRightAngle(B,Q,C)
    tkzMarkRightAngle(A,R,B)


    foreach point/position in {A/left,B/left,C/below,S/above,P/left,Q/above,R/below,H/above}
    {
    fill (point) circle (1.5pt);
    node[position=3pt] at (point) {$point$};
    }
    end{tikzpicture}
    end{document}


    enter image description here



    enter image description here



    Is there a command to find coordinates of projection of a point on a plane?










    share|improve this question



























      1












      1








      1








      Let SABC be a tetrahedron, SA = c, AB = a, AC = b, SA perpendicular to AB, AB perpendicular to AC, and AC perpendicular to SA. I am trying to find the projection H of the point A on the plane SBC. I tried with two ways.



      Firt way. With some calculations, I found that H(({b^2*c^2*a/((b^2+c^2)*a^2+b^2*c^2)},{b*c^2*a^2/((b^2+c^2)*a^2+b^2*c^2)},{b^2*c*a^2/((b^2+c^2)*a^2+b^2*c^2)})).



      Second way, We can prove that, H is orthocentre of triangle SBC, then I see at
      Is there a command to find coordinates of projection of a point on a line in 3D?



      documentclass[border=3mm,12pt,tikz]{standalone}
      usepackage{fouriernc}
      usepackage{tikz,tikz-3dplot}
      usepackage{tkz-euclide}
      usetkzobj{all}

      usetikzlibrary{intersections,calc,backgrounds}
      newcommandpgfmathsinandcos[3]{%
      pgfmathsetmacro#1{sin(#3)}%
      pgfmathsetmacro#2{cos(#3)}%
      }
      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]
      pgfmathsetmacroa{4}
      pgfmathsetmacrob{3}
      pgfmathsetmacroc{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 (H) at ({b^2*c^2*a/((b^2+c^2)*a^2+b^2*c^2)},{b*c^2*a^2/((b^2+c^2)*a^2+b^2*c^2)},{b^2*c*a^2/((b^2+c^2)*a^2+b^2*c^2)});

      begin{scope}
      draw[dashed,thick]
      (A) -- (B) (A) -- (C) (S)--(A) --(H) ;
      draw[thick]
      (S) -- (B) -- (C) -- cycle;
      end{scope}
      foreach point/position in {A/left,B/left,C/below,S/above,H/above}
      {
      fill (point) circle (1.5pt);
      node[position=3pt] at (point) {$point$};
      }
      end{tikzpicture}

      tdplotsetmaincoords{70}{110}
      %tdplotsetmaincoords{80}{100}
      begin{tikzpicture}[tdplot_main_coords,scale=1.5]
      pgfmathsetmacroa{4}
      pgfmathsetmacrob{3}
      pgfmathsetmacroc{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)
      ;
      path[projection of point={(0,0,0) on line through (a,0,0) and (0,0,a)}]
      coordinate (P)
      [projection of point={(0,0,0) on line through (0,b,0) and (0,0,a)}]
      coordinate (Q)
      [projection of point={(0,0,0) on line through (0,b,0) and (a,0,0)}]
      coordinate (R);



      begin{scope}
      draw [very thick] (S) -- (R);
      draw [very thick, name path=B--Q] (B) -- (Q);
      draw [very thick, name path=C--P] (C) -- (P);
      path [name intersections={of=B--Q and C--P,by=H}];
      end{scope}

      begin{scope}
      draw[dashed,thick]
      (A) -- (B) (A) -- (C) (S)--(A) (A)--(H) (A)--(R) (A)--(P) (A)--(Q);
      draw[thick]
      (S) -- (B) -- (C) -- cycle;
      end{scope}

      tkzMarkRightAngle(S,R,C)
      tkzMarkRightAngle(B,P,C)
      tkzMarkRightAngle(B,Q,C)
      tkzMarkRightAngle(A,R,B)


      foreach point/position in {A/left,B/left,C/below,S/above,P/left,Q/above,R/below,H/above}
      {
      fill (point) circle (1.5pt);
      node[position=3pt] at (point) {$point$};
      }
      end{tikzpicture}
      end{document}


      enter image description here



      enter image description here



      Is there a command to find coordinates of projection of a point on a plane?










      share|improve this question
















      Let SABC be a tetrahedron, SA = c, AB = a, AC = b, SA perpendicular to AB, AB perpendicular to AC, and AC perpendicular to SA. I am trying to find the projection H of the point A on the plane SBC. I tried with two ways.



      Firt way. With some calculations, I found that H(({b^2*c^2*a/((b^2+c^2)*a^2+b^2*c^2)},{b*c^2*a^2/((b^2+c^2)*a^2+b^2*c^2)},{b^2*c*a^2/((b^2+c^2)*a^2+b^2*c^2)})).



      Second way, We can prove that, H is orthocentre of triangle SBC, then I see at
      Is there a command to find coordinates of projection of a point on a line in 3D?



      documentclass[border=3mm,12pt,tikz]{standalone}
      usepackage{fouriernc}
      usepackage{tikz,tikz-3dplot}
      usepackage{tkz-euclide}
      usetkzobj{all}

      usetikzlibrary{intersections,calc,backgrounds}
      newcommandpgfmathsinandcos[3]{%
      pgfmathsetmacro#1{sin(#3)}%
      pgfmathsetmacro#2{cos(#3)}%
      }
      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]
      pgfmathsetmacroa{4}
      pgfmathsetmacrob{3}
      pgfmathsetmacroc{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 (H) at ({b^2*c^2*a/((b^2+c^2)*a^2+b^2*c^2)},{b*c^2*a^2/((b^2+c^2)*a^2+b^2*c^2)},{b^2*c*a^2/((b^2+c^2)*a^2+b^2*c^2)});

      begin{scope}
      draw[dashed,thick]
      (A) -- (B) (A) -- (C) (S)--(A) --(H) ;
      draw[thick]
      (S) -- (B) -- (C) -- cycle;
      end{scope}
      foreach point/position in {A/left,B/left,C/below,S/above,H/above}
      {
      fill (point) circle (1.5pt);
      node[position=3pt] at (point) {$point$};
      }
      end{tikzpicture}

      tdplotsetmaincoords{70}{110}
      %tdplotsetmaincoords{80}{100}
      begin{tikzpicture}[tdplot_main_coords,scale=1.5]
      pgfmathsetmacroa{4}
      pgfmathsetmacrob{3}
      pgfmathsetmacroc{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)
      ;
      path[projection of point={(0,0,0) on line through (a,0,0) and (0,0,a)}]
      coordinate (P)
      [projection of point={(0,0,0) on line through (0,b,0) and (0,0,a)}]
      coordinate (Q)
      [projection of point={(0,0,0) on line through (0,b,0) and (a,0,0)}]
      coordinate (R);



      begin{scope}
      draw [very thick] (S) -- (R);
      draw [very thick, name path=B--Q] (B) -- (Q);
      draw [very thick, name path=C--P] (C) -- (P);
      path [name intersections={of=B--Q and C--P,by=H}];
      end{scope}

      begin{scope}
      draw[dashed,thick]
      (A) -- (B) (A) -- (C) (S)--(A) (A)--(H) (A)--(R) (A)--(P) (A)--(Q);
      draw[thick]
      (S) -- (B) -- (C) -- cycle;
      end{scope}

      tkzMarkRightAngle(S,R,C)
      tkzMarkRightAngle(B,P,C)
      tkzMarkRightAngle(B,Q,C)
      tkzMarkRightAngle(A,R,B)


      foreach point/position in {A/left,B/left,C/below,S/above,P/left,Q/above,R/below,H/above}
      {
      fill (point) circle (1.5pt);
      node[position=3pt] at (point) {$point$};
      }
      end{tikzpicture}
      end{document}


      enter image description here



      enter image description here



      Is there a command to find coordinates of projection of a point on a plane?







      tikz-pgf 3d tikz-3dplot






      share|improve this question















      share|improve this question













      share|improve this question




      share|improve this question








      edited 19 mins ago







      minhthien_2016

















      asked 25 mins ago









      minhthien_2016minhthien_2016

      1,163816




      1,163816






















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