Intersection of a line with a plane, where is wrong in third way?












1















Let SABCD be a pyramid, SA perpendicular to the plane ABC, the base of pyramid is a rectangle,SA=h, AB=a, AD=b. A plane P passing A and perpendicular to the line SC cut the lines SB, SD, SC respectively at E, F, K.
I tried it in three ways.



First way 



documentclass[border=3.14mm,12pt,tikz]{standalone}

usepackage{tikz,tikz-3dplot} 

usepackage{tkz-euclide}

usetkzobj{all}

usetikzlibrary{intersections,calc,backgrounds}

begin{document}

    tdplotsetmaincoords{60}{120}

    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 (a,b,0) 

    coordinate (D) at (0,b,0)                            

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

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

    coordinate (F) at (0, {b*h^2/(b^2+h^2)}, {b^2*h/(b^2+h^2)})

    coordinate (K) at ({a*h^2/(a^2+b^2+h^2)}, {b*h^2/(a^2+b^2+h^2)}, {(a^2+b^2)*h/(a^2+b^2+h^2)});              



begin{scope}

draw [dashed, thick, name path=B--D] (B) -- (D);

draw [dashed, thick, name path=C--A] (C) -- (A);

path [name intersections={of=B--D and C--A,by=O}];

end{scope}



begin{scope}

draw [dashed, thick, name path=S--O] (S) -- (O);

draw [dashed, thick, name path=E--F] (E) -- (F);

path [name intersections={of=S--O and E--F,by=I}];

end{scope}



    begin{scope}

    draw[dashed, thick]

    (A) -- (B)   (D)--(A) (S)--(A);

    draw[dashed, thick]

    (E) --(A)  -- (F);

    draw[ultra thick]

    (S) -- (B) -- (C) -- (D)--cycle (S)--(C) (E) -- (K) --(F);

    draw [thick, dashed] (A) -- (K)  (E) -- (F) ;

    tkzMarkRightAngle(S,A,D)

    tkzMarkRightAngle(S,A,B)

    tkzMarkRightAngle(A,B,C)

    tkzMarkRightAngle(B,A,D)

    tkzMarkRightAngle(A,F,D)

    tkzMarkRightAngle(A,E,B)

    tkzMarkRightAngle(A,K,C)

    end{scope}



    foreach point/position in {A/left,B/below,C/below,S/above,D/right,E/left,D/right,F/right,K/above right,O/below,I/below}

    {

        fill (point) circle (1.5pt);

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

    }

    end{tikzpicture}



end{document}


Second way



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

usepackage{fouriernc}

usepackage{tikz,tikz-3dplot} 

usepackage{tkz-euclide}

usetkzobj{all}



newcounter{smuggle}

DeclareRobustCommandsmuggleone[1]{%

    stepcounter{smuggle}%

    expandafterglobalexpandafterletcsname smuggle@arabic{smuggle}endcsname#1%

    aftergroupletaftergroup#1expandafteraftergroupcsname smuggle@arabic{smuggle}endcsname

}

DeclareRobustCommandsmuggle[2][1]{%

    smuggleone{#2}%

    ifnum#1>1

    aftergroupsmuggleaftergroup[expandafteraftergroupthenumexpr#1-1aftergroup]aftergroup#2%

    fi

}



usetikzlibrary{intersections,calc,backgrounds}

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{60}{120}

 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 (a,b,0) 

 coordinate (D) at (0,b,0)                            

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



 path[projection of point={(0,0,0) on line through (a,0,0) and  (0,0,h)}]

 coordinate (E)

 [projection of point={(0,0,0) on line through (0,0,h) and (0,b,0) }]

 coordinate (F)

 [projection of point={(0,0,0) on line through (0,0,h) and (a,b,0) }]

 coordinate (K);           



 begin{scope}

 draw [dashed, thick, name path=B--D] (B) -- (D);

 draw [dashed, thick, name path=C--A] (C) -- (A);

 path [name intersections={of=B--D and C--A,by=O}];

 end{scope}



 begin{scope}

 draw [dashed, thick, name path=S--O] (S) -- (O);

 draw [dashed, thick, name path=E--F] (E) -- (F);

 path [name intersections={of=S--O and E--F,by=I}];

 end{scope}





 begin{scope}

 draw[dashed, thick]

 (A) -- (B)   (D)--(A) (S)--(A);

 draw[dashed, thick]

 (E) --(A)  -- (F);

 draw[ultra thick]

 (S) -- (B) -- (C) -- (D)--cycle (S)--(C) (E) -- (K) --(F);

  draw [thick, dashed] (A) -- (K)  (E) -- (F) ;



 tkzMarkRightAngle(S,A,D)

 tkzMarkRightAngle(S,A,B)

 tkzMarkRightAngle(A,B,C)

 tkzMarkRightAngle(B,A,D)

 tkzMarkRightAngle(A,F,D)

 tkzMarkRightAngle(A,E,B)

 tkzMarkRightAngle(A,K,C)

 end{scope}



  foreach point/position in {A/left,B/below,C/below,S/above,D/right,E/left,D/right,F/right,K/above right,O/below,I/below}

 {

    fill (point) circle (1.5pt);

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

 }



 end{tikzpicture}

 end{document}


enter image description here



Third way
Base on the answer Is there a command to find coordinates of projection of a point on a plane?



I defined command defVecMinus(#1,#2,#3)-(#4,#5,#6){(#1-#4,#2-#5,#3-#6)} and tried



documentclass[border=3.14mm,12pt,tikz]{standalone}

usepackage{tikz,tikz-3dplot} 

%% smuggling from https://tex.stackexchange.com/a/470979/121799

newcounter{smuggle}

DeclareRobustCommandsmuggleone[1]{%

    stepcounter{smuggle}%

    expandafterglobalexpandafterletcsname smuggle@arabic{smuggle}endcsname#1%

    aftergroupletaftergroup#1expandafteraftergroupcsname smuggle@arabic{smuggle}endcsname

}

DeclareRobustCommandsmuggle[2][1]{%

    smuggleone{#2}%

    ifnum#1>1

    aftergroupsmuggleaftergroup[expandafteraftergroupthenumexpr#1-1aftergroup]aftergroup#2%

    fi

}



defparsecoord(#1,#2,#3)>(#4,#5,#6){%

    def#4{#1}%

    def#5{#2}%

    def#6{#3}%

    smuggle{#4}%

    smuggle{#5}%

    smuggle{#6}%

}

defSPTD(#1,#2,#3).(#4,#5,#6){#1*#4+#2*#5+#3*#6}

defVPTD(#1,#2,#3)x(#4,#5,#6){(#2*#6-#3*#5,#3*#4-#1*#6,#1*#5-#2*#4)}

defVecMinus(#1,#2,#3)-(#4,#5,#6){(#1-#4,#2-#5,#3-#6)}

defVecAdd(#1,#2,#3)+(#4,#5,#6){(#1+#4,#2+#5,#3+#6)}

tikzset{intersection of line trough/.style args={#1 and #2 with plane

        containing #3 and normal #4}{%

        /utils/exec={pgfmathsetmacro{ltest}{abs(SPTD#2.#4-SPTD#1.#4)}

            parsecoord#1>(myAx,myAy,myAz)

            parsecoord#2>(myBx,myBy,myBz)

            ifdimltest pt<0.01pt

            typeout{Planespace andspace linespace arespace parallel!ltest}

            pgfmathsetmacro{myd}{0}

            else

            pgfmathsetmacro{myd}{(SPTD#3.#4-SPTD#1.#4)/(SPTD#2.#4-SPTD#1.#4)}

            fi

            %typeout{({myAx+myd*(myBx-myAx)},{myAy+myd*(myBy-myAy)},{myAz+myd*(myBz-myAz)})}

            defmyP{({myAx+myd*(myBx-myAx)},{myAy+myd*(myBy-myAy)},{myAz+myd*(myBz-myAz)})}

            smugglemyP},

        insert path={%

            myP}

}}



begin{document}

    tdplotsetmaincoords{75}{110}

    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 (a,b,0) 

    coordinate (D) at (0,b,0)                            

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

    defmynormal{VecMinus(a,b,0)-(0,0,h)}

    typeout{mynormal:(a,b,-h)}

    path[intersection of line trough={(0,0,h) and (a,0,0) with plane containing (0,0,0) and normal (a,b,-h)}]  coordinate (E)

    [intersection of line trough={(0,0,h) and  (0,b,0) with plane containing (0,0,0) and normal (a,b,-h)}]  coordinate (F)

    [intersection of line trough={(0,0,h) and  (a,b,0) with plane containing (0,0,0) and normal (a,b,-h)}]  coordinate (K);





    draw[dashed, thick]

    (A) -- (B)   (D)--(A) (S)--(A);

    draw[dashed, thick]

    (E) --(A)  -- (F);

    draw[ultra thick]

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



    draw [thick, dashed] (A) -- (K) (A) -- (C) (B) -- (D) (E) -- (F) ;

    draw [thick] (E) -- (K) -- (F) ;



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

    {

        fill (point) circle (1.5pt);

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

    }



    end{tikzpicture}

        end{document}


I got
enter image description here






tikz-pgf tikz-3dplot







share|improve this question



























    1















    Let SABCD be a pyramid, SA perpendicular to the plane ABC, the base of pyramid is a rectangle,SA=h, AB=a, AD=b. A plane P passing A and perpendicular to the line SC cut the lines SB, SD, SC respectively at E, F, K.
    I tried it in three ways.



    First way 



    documentclass[border=3.14mm,12pt,tikz]{standalone}
    
usepackage{tikz,tikz-3dplot} 
    
usepackage{tkz-euclide}
    
usetkzobj{all}
    
usetikzlibrary{intersections,calc,backgrounds}
    
begin{document}
    
    tdplotsetmaincoords{60}{120}
    
    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 (a,b,0) 
    
    coordinate (D) at (0,b,0)                            
    
    coordinate (S) at (0,0,h)
    
    coordinate (E) at ({a*h^2/(a^2+h^2)}, 0, {a^2*h/(a^2+h^2)})
    
    coordinate (F) at (0, {b*h^2/(b^2+h^2)}, {b^2*h/(b^2+h^2)})
    
    coordinate (K) at ({a*h^2/(a^2+b^2+h^2)}, {b*h^2/(a^2+b^2+h^2)}, {(a^2+b^2)*h/(a^2+b^2+h^2)});              
    

    
begin{scope}
    
draw [dashed, thick, name path=B--D] (B) -- (D);
    
draw [dashed, thick, name path=C--A] (C) -- (A);
    
path [name intersections={of=B--D and C--A,by=O}];
    
end{scope}
    

    
begin{scope}
    
draw [dashed, thick, name path=S--O] (S) -- (O);
    
draw [dashed, thick, name path=E--F] (E) -- (F);
    
path [name intersections={of=S--O and E--F,by=I}];
    
end{scope}
    

    
    begin{scope}
    
    draw[dashed, thick]
    
    (A) -- (B)   (D)--(A) (S)--(A);
    
    draw[dashed, thick]
    
    (E) --(A)  -- (F);
    
    draw[ultra thick]
    
    (S) -- (B) -- (C) -- (D)--cycle (S)--(C) (E) -- (K) --(F);
    
    draw [thick, dashed] (A) -- (K)  (E) -- (F) ;
    
    tkzMarkRightAngle(S,A,D)
    
    tkzMarkRightAngle(S,A,B)
    
    tkzMarkRightAngle(A,B,C)
    
    tkzMarkRightAngle(B,A,D)
    
    tkzMarkRightAngle(A,F,D)
    
    tkzMarkRightAngle(A,E,B)
    
    tkzMarkRightAngle(A,K,C)
    
    end{scope}
    

    
    foreach point/position in {A/left,B/below,C/below,S/above,D/right,E/left,D/right,F/right,K/above right,O/below,I/below}
    
    {
    
        fill (point) circle (1.5pt);
    
        node[position=3pt] at (point) {$point$};
    
    }
    
    end{tikzpicture}
    

    
end{document}


    Second way



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

    
newcounter{smuggle}
    
DeclareRobustCommandsmuggleone[1]{%
    
    stepcounter{smuggle}%
    
    expandafterglobalexpandafterletcsname smuggle@arabic{smuggle}endcsname#1%
    
    aftergroupletaftergroup#1expandafteraftergroupcsname smuggle@arabic{smuggle}endcsname
    
}
    
DeclareRobustCommandsmuggle[2][1]{%
    
    smuggleone{#2}%
    
    ifnum#1>1
    
    aftergroupsmuggleaftergroup[expandafteraftergroupthenumexpr#1-1aftergroup]aftergroup#2%
    
    fi
    
}
    

    
usetikzlibrary{intersections,calc,backgrounds}
    
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{60}{120}
    
 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 (a,b,0) 
    
 coordinate (D) at (0,b,0)                            
    
 coordinate (S) at (0,0,h);   
    

    
 path[projection of point={(0,0,0) on line through (a,0,0) and  (0,0,h)}]
    
 coordinate (E)
    
 [projection of point={(0,0,0) on line through (0,0,h) and (0,b,0) }]
    
 coordinate (F)
    
 [projection of point={(0,0,0) on line through (0,0,h) and (a,b,0) }]
    
 coordinate (K);           
    

    
 begin{scope}
    
 draw [dashed, thick, name path=B--D] (B) -- (D);
    
 draw [dashed, thick, name path=C--A] (C) -- (A);
    
 path [name intersections={of=B--D and C--A,by=O}];
    
 end{scope}
    

    
 begin{scope}
    
 draw [dashed, thick, name path=S--O] (S) -- (O);
    
 draw [dashed, thick, name path=E--F] (E) -- (F);
    
 path [name intersections={of=S--O and E--F,by=I}];
    
 end{scope}
    

    

    
 begin{scope}
    
 draw[dashed, thick]
    
 (A) -- (B)   (D)--(A) (S)--(A);
    
 draw[dashed, thick]
    
 (E) --(A)  -- (F);
    
 draw[ultra thick]
    
 (S) -- (B) -- (C) -- (D)--cycle (S)--(C) (E) -- (K) --(F);
    
  draw [thick, dashed] (A) -- (K)  (E) -- (F) ;
    

    
 tkzMarkRightAngle(S,A,D)
    
 tkzMarkRightAngle(S,A,B)
    
 tkzMarkRightAngle(A,B,C)
    
 tkzMarkRightAngle(B,A,D)
    
 tkzMarkRightAngle(A,F,D)
    
 tkzMarkRightAngle(A,E,B)
    
 tkzMarkRightAngle(A,K,C)
    
 end{scope}
    

    
  foreach point/position in {A/left,B/below,C/below,S/above,D/right,E/left,D/right,F/right,K/above right,O/below,I/below}
    
 {
    
    fill (point) circle (1.5pt);
    
    node[position=3pt] at (point) {$point$};
    
 }
    

    
 end{tikzpicture}
    
 end{document}


    enter image description here



    Third way
    Base on the answer Is there a command to find coordinates of projection of a point on a plane?



    I defined command defVecMinus(#1,#2,#3)-(#4,#5,#6){(#1-#4,#2-#5,#3-#6)} and tried



    documentclass[border=3.14mm,12pt,tikz]{standalone}
    
usepackage{tikz,tikz-3dplot} 
    
%% smuggling from https://tex.stackexchange.com/a/470979/121799
    
newcounter{smuggle}
    
DeclareRobustCommandsmuggleone[1]{%
    
    stepcounter{smuggle}%
    
    expandafterglobalexpandafterletcsname smuggle@arabic{smuggle}endcsname#1%
    
    aftergroupletaftergroup#1expandafteraftergroupcsname smuggle@arabic{smuggle}endcsname
    
}
    
DeclareRobustCommandsmuggle[2][1]{%
    
    smuggleone{#2}%
    
    ifnum#1>1
    
    aftergroupsmuggleaftergroup[expandafteraftergroupthenumexpr#1-1aftergroup]aftergroup#2%
    
    fi
    
}
    

    
defparsecoord(#1,#2,#3)>(#4,#5,#6){%
    
    def#4{#1}%
    
    def#5{#2}%
    
    def#6{#3}%
    
    smuggle{#4}%
    
    smuggle{#5}%
    
    smuggle{#6}%
    
}
    
defSPTD(#1,#2,#3).(#4,#5,#6){#1*#4+#2*#5+#3*#6}
    
defVPTD(#1,#2,#3)x(#4,#5,#6){(#2*#6-#3*#5,#3*#4-#1*#6,#1*#5-#2*#4)}
    
defVecMinus(#1,#2,#3)-(#4,#5,#6){(#1-#4,#2-#5,#3-#6)}
    
defVecAdd(#1,#2,#3)+(#4,#5,#6){(#1+#4,#2+#5,#3+#6)}
    
tikzset{intersection of line trough/.style args={#1 and #2 with plane
    
        containing #3 and normal #4}{%
    
        /utils/exec={pgfmathsetmacro{ltest}{abs(SPTD#2.#4-SPTD#1.#4)}
    
            parsecoord#1>(myAx,myAy,myAz)
    
            parsecoord#2>(myBx,myBy,myBz)
    
            ifdimltest pt<0.01pt
    
            typeout{Planespace andspace linespace arespace parallel!ltest}
    
            pgfmathsetmacro{myd}{0}
    
            else
    
            pgfmathsetmacro{myd}{(SPTD#3.#4-SPTD#1.#4)/(SPTD#2.#4-SPTD#1.#4)}
    
            fi
    
            %typeout{({myAx+myd*(myBx-myAx)},{myAy+myd*(myBy-myAy)},{myAz+myd*(myBz-myAz)})}
    
            defmyP{({myAx+myd*(myBx-myAx)},{myAy+myd*(myBy-myAy)},{myAz+myd*(myBz-myAz)})}
    
            smugglemyP},
    
        insert path={%
    
            myP}
    
}}
    

    
begin{document}
    
    tdplotsetmaincoords{75}{110}
    
    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 (a,b,0) 
    
    coordinate (D) at (0,b,0)                            
    
    coordinate (S) at (0,0,h);                
    
    defmynormal{VecMinus(a,b,0)-(0,0,h)}
    
    typeout{mynormal:(a,b,-h)}
    
    path[intersection of line trough={(0,0,h) and (a,0,0) with plane containing (0,0,0) and normal (a,b,-h)}]  coordinate (E)
    
    [intersection of line trough={(0,0,h) and  (0,b,0) with plane containing (0,0,0) and normal (a,b,-h)}]  coordinate (F)
    
    [intersection of line trough={(0,0,h) and  (a,b,0) with plane containing (0,0,0) and normal (a,b,-h)}]  coordinate (K);
    

    

    
    draw[dashed, thick]
    
    (A) -- (B)   (D)--(A) (S)--(A);
    
    draw[dashed, thick]
    
    (E) --(A)  -- (F);
    
    draw[ultra thick]
    
    (S) -- (B) -- (C) -- (D)--cycle (S)--(C);
    

    
    draw [thick, dashed] (A) -- (K) (A) -- (C) (B) -- (D) (E) -- (F) ;
    
    draw [thick] (E) -- (K) -- (F) ;
    

    
    foreach point/position in {A/left,B/below,C/below,S/above,D/right,E/left,D/right,F/right,K/above right}
    
    {
    
        fill (point) circle (1.5pt);
    
        node[position=3pt] at (point) {$point$};
    
    }
    

    
    end{tikzpicture}
    
        end{document}


    I got
    enter image description here






    tikz-pgf tikz-3dplot







    share|improve this question

























      1












      1








      1








      Let SABCD be a pyramid, SA perpendicular to the plane ABC, the base of pyramid is a rectangle,SA=h, AB=a, AD=b. A plane P passing A and perpendicular to the line SC cut the lines SB, SD, SC respectively at E, F, K.
      I tried it in three ways.



      First way 



      documentclass[border=3.14mm,12pt,tikz]{standalone}
      
usepackage{tikz,tikz-3dplot} 
      
usepackage{tkz-euclide}
      
usetkzobj{all}
      
usetikzlibrary{intersections,calc,backgrounds}
      
begin{document}
      
    tdplotsetmaincoords{60}{120}
      
    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 (a,b,0) 
      
    coordinate (D) at (0,b,0)                            
      
    coordinate (S) at (0,0,h)
      
    coordinate (E) at ({a*h^2/(a^2+h^2)}, 0, {a^2*h/(a^2+h^2)})
      
    coordinate (F) at (0, {b*h^2/(b^2+h^2)}, {b^2*h/(b^2+h^2)})
      
    coordinate (K) at ({a*h^2/(a^2+b^2+h^2)}, {b*h^2/(a^2+b^2+h^2)}, {(a^2+b^2)*h/(a^2+b^2+h^2)});              
      

      
begin{scope}
      
draw [dashed, thick, name path=B--D] (B) -- (D);
      
draw [dashed, thick, name path=C--A] (C) -- (A);
      
path [name intersections={of=B--D and C--A,by=O}];
      
end{scope}
      

      
begin{scope}
      
draw [dashed, thick, name path=S--O] (S) -- (O);
      
draw [dashed, thick, name path=E--F] (E) -- (F);
      
path [name intersections={of=S--O and E--F,by=I}];
      
end{scope}
      

      
    begin{scope}
      
    draw[dashed, thick]
      
    (A) -- (B)   (D)--(A) (S)--(A);
      
    draw[dashed, thick]
      
    (E) --(A)  -- (F);
      
    draw[ultra thick]
      
    (S) -- (B) -- (C) -- (D)--cycle (S)--(C) (E) -- (K) --(F);
      
    draw [thick, dashed] (A) -- (K)  (E) -- (F) ;
      
    tkzMarkRightAngle(S,A,D)
      
    tkzMarkRightAngle(S,A,B)
      
    tkzMarkRightAngle(A,B,C)
      
    tkzMarkRightAngle(B,A,D)
      
    tkzMarkRightAngle(A,F,D)
      
    tkzMarkRightAngle(A,E,B)
      
    tkzMarkRightAngle(A,K,C)
      
    end{scope}
      

      
    foreach point/position in {A/left,B/below,C/below,S/above,D/right,E/left,D/right,F/right,K/above right,O/below,I/below}
      
    {
      
        fill (point) circle (1.5pt);
      
        node[position=3pt] at (point) {$point$};
      
    }
      
    end{tikzpicture}
      

      
end{document}


      Second way



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

      
newcounter{smuggle}
      
DeclareRobustCommandsmuggleone[1]{%
      
    stepcounter{smuggle}%
      
    expandafterglobalexpandafterletcsname smuggle@arabic{smuggle}endcsname#1%
      
    aftergroupletaftergroup#1expandafteraftergroupcsname smuggle@arabic{smuggle}endcsname
      
}
      
DeclareRobustCommandsmuggle[2][1]{%
      
    smuggleone{#2}%
      
    ifnum#1>1
      
    aftergroupsmuggleaftergroup[expandafteraftergroupthenumexpr#1-1aftergroup]aftergroup#2%
      
    fi
      
}
      

      
usetikzlibrary{intersections,calc,backgrounds}
      
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{60}{120}
      
 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 (a,b,0) 
      
 coordinate (D) at (0,b,0)                            
      
 coordinate (S) at (0,0,h);   
      

      
 path[projection of point={(0,0,0) on line through (a,0,0) and  (0,0,h)}]
      
 coordinate (E)
      
 [projection of point={(0,0,0) on line through (0,0,h) and (0,b,0) }]
      
 coordinate (F)
      
 [projection of point={(0,0,0) on line through (0,0,h) and (a,b,0) }]
      
 coordinate (K);           
      

      
 begin{scope}
      
 draw [dashed, thick, name path=B--D] (B) -- (D);
      
 draw [dashed, thick, name path=C--A] (C) -- (A);
      
 path [name intersections={of=B--D and C--A,by=O}];
      
 end{scope}
      

      
 begin{scope}
      
 draw [dashed, thick, name path=S--O] (S) -- (O);
      
 draw [dashed, thick, name path=E--F] (E) -- (F);
      
 path [name intersections={of=S--O and E--F,by=I}];
      
 end{scope}
      

      

      
 begin{scope}
      
 draw[dashed, thick]
      
 (A) -- (B)   (D)--(A) (S)--(A);
      
 draw[dashed, thick]
      
 (E) --(A)  -- (F);
      
 draw[ultra thick]
      
 (S) -- (B) -- (C) -- (D)--cycle (S)--(C) (E) -- (K) --(F);
      
  draw [thick, dashed] (A) -- (K)  (E) -- (F) ;
      

      
 tkzMarkRightAngle(S,A,D)
      
 tkzMarkRightAngle(S,A,B)
      
 tkzMarkRightAngle(A,B,C)
      
 tkzMarkRightAngle(B,A,D)
      
 tkzMarkRightAngle(A,F,D)
      
 tkzMarkRightAngle(A,E,B)
      
 tkzMarkRightAngle(A,K,C)
      
 end{scope}
      

      
  foreach point/position in {A/left,B/below,C/below,S/above,D/right,E/left,D/right,F/right,K/above right,O/below,I/below}
      
 {
      
    fill (point) circle (1.5pt);
      
    node[position=3pt] at (point) {$point$};
      
 }
      

      
 end{tikzpicture}
      
 end{document}


      enter image description here



      Third way
      Base on the answer Is there a command to find coordinates of projection of a point on a plane?



      I defined command defVecMinus(#1,#2,#3)-(#4,#5,#6){(#1-#4,#2-#5,#3-#6)} and tried



      documentclass[border=3.14mm,12pt,tikz]{standalone}
      
usepackage{tikz,tikz-3dplot} 
      
%% smuggling from https://tex.stackexchange.com/a/470979/121799
      
newcounter{smuggle}
      
DeclareRobustCommandsmuggleone[1]{%
      
    stepcounter{smuggle}%
      
    expandafterglobalexpandafterletcsname smuggle@arabic{smuggle}endcsname#1%
      
    aftergroupletaftergroup#1expandafteraftergroupcsname smuggle@arabic{smuggle}endcsname
      
}
      
DeclareRobustCommandsmuggle[2][1]{%
      
    smuggleone{#2}%
      
    ifnum#1>1
      
    aftergroupsmuggleaftergroup[expandafteraftergroupthenumexpr#1-1aftergroup]aftergroup#2%
      
    fi
      
}
      

      
defparsecoord(#1,#2,#3)>(#4,#5,#6){%
      
    def#4{#1}%
      
    def#5{#2}%
      
    def#6{#3}%
      
    smuggle{#4}%
      
    smuggle{#5}%
      
    smuggle{#6}%
      
}
      
defSPTD(#1,#2,#3).(#4,#5,#6){#1*#4+#2*#5+#3*#6}
      
defVPTD(#1,#2,#3)x(#4,#5,#6){(#2*#6-#3*#5,#3*#4-#1*#6,#1*#5-#2*#4)}
      
defVecMinus(#1,#2,#3)-(#4,#5,#6){(#1-#4,#2-#5,#3-#6)}
      
defVecAdd(#1,#2,#3)+(#4,#5,#6){(#1+#4,#2+#5,#3+#6)}
      
tikzset{intersection of line trough/.style args={#1 and #2 with plane
      
        containing #3 and normal #4}{%
      
        /utils/exec={pgfmathsetmacro{ltest}{abs(SPTD#2.#4-SPTD#1.#4)}
      
            parsecoord#1>(myAx,myAy,myAz)
      
            parsecoord#2>(myBx,myBy,myBz)
      
            ifdimltest pt<0.01pt
      
            typeout{Planespace andspace linespace arespace parallel!ltest}
      
            pgfmathsetmacro{myd}{0}
      
            else
      
            pgfmathsetmacro{myd}{(SPTD#3.#4-SPTD#1.#4)/(SPTD#2.#4-SPTD#1.#4)}
      
            fi
      
            %typeout{({myAx+myd*(myBx-myAx)},{myAy+myd*(myBy-myAy)},{myAz+myd*(myBz-myAz)})}
      
            defmyP{({myAx+myd*(myBx-myAx)},{myAy+myd*(myBy-myAy)},{myAz+myd*(myBz-myAz)})}
      
            smugglemyP},
      
        insert path={%
      
            myP}
      
}}
      

      
begin{document}
      
    tdplotsetmaincoords{75}{110}
      
    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 (a,b,0) 
      
    coordinate (D) at (0,b,0)                            
      
    coordinate (S) at (0,0,h);                
      
    defmynormal{VecMinus(a,b,0)-(0,0,h)}
      
    typeout{mynormal:(a,b,-h)}
      
    path[intersection of line trough={(0,0,h) and (a,0,0) with plane containing (0,0,0) and normal (a,b,-h)}]  coordinate (E)
      
    [intersection of line trough={(0,0,h) and  (0,b,0) with plane containing (0,0,0) and normal (a,b,-h)}]  coordinate (F)
      
    [intersection of line trough={(0,0,h) and  (a,b,0) with plane containing (0,0,0) and normal (a,b,-h)}]  coordinate (K);
      

      

      
    draw[dashed, thick]
      
    (A) -- (B)   (D)--(A) (S)--(A);
      
    draw[dashed, thick]
      
    (E) --(A)  -- (F);
      
    draw[ultra thick]
      
    (S) -- (B) -- (C) -- (D)--cycle (S)--(C);
      

      
    draw [thick, dashed] (A) -- (K) (A) -- (C) (B) -- (D) (E) -- (F) ;
      
    draw [thick] (E) -- (K) -- (F) ;
      

      
    foreach point/position in {A/left,B/below,C/below,S/above,D/right,E/left,D/right,F/right,K/above right}
      
    {
      
        fill (point) circle (1.5pt);
      
        node[position=3pt] at (point) {$point$};
      
    }
      

      
    end{tikzpicture}
      
        end{document}


      I got
      enter image description here






      tikz-pgf tikz-3dplot







      share|improve this question














      Let SABCD be a pyramid, SA perpendicular to the plane ABC, the base of pyramid is a rectangle,SA=h, AB=a, AD=b. A plane P passing A and perpendicular to the line SC cut the lines SB, SD, SC respectively at E, F, K.
      I tried it in three ways.



      First way 



      documentclass[border=3.14mm,12pt,tikz]{standalone}
      
usepackage{tikz,tikz-3dplot} 
      
usepackage{tkz-euclide}
      
usetkzobj{all}
      
usetikzlibrary{intersections,calc,backgrounds}
      
begin{document}
      
    tdplotsetmaincoords{60}{120}
      
    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 (a,b,0) 
      
    coordinate (D) at (0,b,0)                            
      
    coordinate (S) at (0,0,h)
      
    coordinate (E) at ({a*h^2/(a^2+h^2)}, 0, {a^2*h/(a^2+h^2)})
      
    coordinate (F) at (0, {b*h^2/(b^2+h^2)}, {b^2*h/(b^2+h^2)})
      
    coordinate (K) at ({a*h^2/(a^2+b^2+h^2)}, {b*h^2/(a^2+b^2+h^2)}, {(a^2+b^2)*h/(a^2+b^2+h^2)});              
      

      
begin{scope}
      
draw [dashed, thick, name path=B--D] (B) -- (D);
      
draw [dashed, thick, name path=C--A] (C) -- (A);
      
path [name intersections={of=B--D and C--A,by=O}];
      
end{scope}
      

      
begin{scope}
      
draw [dashed, thick, name path=S--O] (S) -- (O);
      
draw [dashed, thick, name path=E--F] (E) -- (F);
      
path [name intersections={of=S--O and E--F,by=I}];
      
end{scope}
      

      
    begin{scope}
      
    draw[dashed, thick]
      
    (A) -- (B)   (D)--(A) (S)--(A);
      
    draw[dashed, thick]
      
    (E) --(A)  -- (F);
      
    draw[ultra thick]
      
    (S) -- (B) -- (C) -- (D)--cycle (S)--(C) (E) -- (K) --(F);
      
    draw [thick, dashed] (A) -- (K)  (E) -- (F) ;
      
    tkzMarkRightAngle(S,A,D)
      
    tkzMarkRightAngle(S,A,B)
      
    tkzMarkRightAngle(A,B,C)
      
    tkzMarkRightAngle(B,A,D)
      
    tkzMarkRightAngle(A,F,D)
      
    tkzMarkRightAngle(A,E,B)
      
    tkzMarkRightAngle(A,K,C)
      
    end{scope}
      

      
    foreach point/position in {A/left,B/below,C/below,S/above,D/right,E/left,D/right,F/right,K/above right,O/below,I/below}
      
    {
      
        fill (point) circle (1.5pt);
      
        node[position=3pt] at (point) {$point$};
      
    }
      
    end{tikzpicture}
      

      
end{document}


      Second way



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

      
newcounter{smuggle}
      
DeclareRobustCommandsmuggleone[1]{%
      
    stepcounter{smuggle}%
      
    expandafterglobalexpandafterletcsname smuggle@arabic{smuggle}endcsname#1%
      
    aftergroupletaftergroup#1expandafteraftergroupcsname smuggle@arabic{smuggle}endcsname
      
}
      
DeclareRobustCommandsmuggle[2][1]{%
      
    smuggleone{#2}%
      
    ifnum#1>1
      
    aftergroupsmuggleaftergroup[expandafteraftergroupthenumexpr#1-1aftergroup]aftergroup#2%
      
    fi
      
}
      

      
usetikzlibrary{intersections,calc,backgrounds}
      
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{60}{120}
      
 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 (a,b,0) 
      
 coordinate (D) at (0,b,0)                            
      
 coordinate (S) at (0,0,h);   
      

      
 path[projection of point={(0,0,0) on line through (a,0,0) and  (0,0,h)}]
      
 coordinate (E)
      
 [projection of point={(0,0,0) on line through (0,0,h) and (0,b,0) }]
      
 coordinate (F)
      
 [projection of point={(0,0,0) on line through (0,0,h) and (a,b,0) }]
      
 coordinate (K);           
      

      
 begin{scope}
      
 draw [dashed, thick, name path=B--D] (B) -- (D);
      
 draw [dashed, thick, name path=C--A] (C) -- (A);
      
 path [name intersections={of=B--D and C--A,by=O}];
      
 end{scope}
      

      
 begin{scope}
      
 draw [dashed, thick, name path=S--O] (S) -- (O);
      
 draw [dashed, thick, name path=E--F] (E) -- (F);
      
 path [name intersections={of=S--O and E--F,by=I}];
      
 end{scope}
      

      

      
 begin{scope}
      
 draw[dashed, thick]
      
 (A) -- (B)   (D)--(A) (S)--(A);
      
 draw[dashed, thick]
      
 (E) --(A)  -- (F);
      
 draw[ultra thick]
      
 (S) -- (B) -- (C) -- (D)--cycle (S)--(C) (E) -- (K) --(F);
      
  draw [thick, dashed] (A) -- (K)  (E) -- (F) ;
      

      
 tkzMarkRightAngle(S,A,D)
      
 tkzMarkRightAngle(S,A,B)
      
 tkzMarkRightAngle(A,B,C)
      
 tkzMarkRightAngle(B,A,D)
      
 tkzMarkRightAngle(A,F,D)
      
 tkzMarkRightAngle(A,E,B)
      
 tkzMarkRightAngle(A,K,C)
      
 end{scope}
      

      
  foreach point/position in {A/left,B/below,C/below,S/above,D/right,E/left,D/right,F/right,K/above right,O/below,I/below}
      
 {
      
    fill (point) circle (1.5pt);
      
    node[position=3pt] at (point) {$point$};
      
 }
      

      
 end{tikzpicture}
      
 end{document}


      enter image description here



      Third way
      Base on the answer Is there a command to find coordinates of projection of a point on a plane?



      I defined command defVecMinus(#1,#2,#3)-(#4,#5,#6){(#1-#4,#2-#5,#3-#6)} and tried



      documentclass[border=3.14mm,12pt,tikz]{standalone}
      
usepackage{tikz,tikz-3dplot} 
      
%% smuggling from https://tex.stackexchange.com/a/470979/121799
      
newcounter{smuggle}
      
DeclareRobustCommandsmuggleone[1]{%
      
    stepcounter{smuggle}%
      
    expandafterglobalexpandafterletcsname smuggle@arabic{smuggle}endcsname#1%
      
    aftergroupletaftergroup#1expandafteraftergroupcsname smuggle@arabic{smuggle}endcsname
      
}
      
DeclareRobustCommandsmuggle[2][1]{%
      
    smuggleone{#2}%
      
    ifnum#1>1
      
    aftergroupsmuggleaftergroup[expandafteraftergroupthenumexpr#1-1aftergroup]aftergroup#2%
      
    fi
      
}
      

      
defparsecoord(#1,#2,#3)>(#4,#5,#6){%
      
    def#4{#1}%
      
    def#5{#2}%
      
    def#6{#3}%
      
    smuggle{#4}%
      
    smuggle{#5}%
      
    smuggle{#6}%
      
}
      
defSPTD(#1,#2,#3).(#4,#5,#6){#1*#4+#2*#5+#3*#6}
      
defVPTD(#1,#2,#3)x(#4,#5,#6){(#2*#6-#3*#5,#3*#4-#1*#6,#1*#5-#2*#4)}
      
defVecMinus(#1,#2,#3)-(#4,#5,#6){(#1-#4,#2-#5,#3-#6)}
      
defVecAdd(#1,#2,#3)+(#4,#5,#6){(#1+#4,#2+#5,#3+#6)}
      
tikzset{intersection of line trough/.style args={#1 and #2 with plane
      
        containing #3 and normal #4}{%
      
        /utils/exec={pgfmathsetmacro{ltest}{abs(SPTD#2.#4-SPTD#1.#4)}
      
            parsecoord#1>(myAx,myAy,myAz)
      
            parsecoord#2>(myBx,myBy,myBz)
      
            ifdimltest pt<0.01pt
      
            typeout{Planespace andspace linespace arespace parallel!ltest}
      
            pgfmathsetmacro{myd}{0}
      
            else
      
            pgfmathsetmacro{myd}{(SPTD#3.#4-SPTD#1.#4)/(SPTD#2.#4-SPTD#1.#4)}
      
            fi
      
            %typeout{({myAx+myd*(myBx-myAx)},{myAy+myd*(myBy-myAy)},{myAz+myd*(myBz-myAz)})}
      
            defmyP{({myAx+myd*(myBx-myAx)},{myAy+myd*(myBy-myAy)},{myAz+myd*(myBz-myAz)})}
      
            smugglemyP},
      
        insert path={%
      
            myP}
      
}}
      

      
begin{document}
      
    tdplotsetmaincoords{75}{110}
      
    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 (a,b,0) 
      
    coordinate (D) at (0,b,0)                            
      
    coordinate (S) at (0,0,h);                
      
    defmynormal{VecMinus(a,b,0)-(0,0,h)}
      
    typeout{mynormal:(a,b,-h)}
      
    path[intersection of line trough={(0,0,h) and (a,0,0) with plane containing (0,0,0) and normal (a,b,-h)}]  coordinate (E)
      
    [intersection of line trough={(0,0,h) and  (0,b,0) with plane containing (0,0,0) and normal (a,b,-h)}]  coordinate (F)
      
    [intersection of line trough={(0,0,h) and  (a,b,0) with plane containing (0,0,0) and normal (a,b,-h)}]  coordinate (K);
      

      

      
    draw[dashed, thick]
      
    (A) -- (B)   (D)--(A) (S)--(A);
      
    draw[dashed, thick]
      
    (E) --(A)  -- (F);
      
    draw[ultra thick]
      
    (S) -- (B) -- (C) -- (D)--cycle (S)--(C);
      

      
    draw [thick, dashed] (A) -- (K) (A) -- (C) (B) -- (D) (E) -- (F) ;
      
    draw [thick] (E) -- (K) -- (F) ;
      

      
    foreach point/position in {A/left,B/below,C/below,S/above,D/right,E/left,D/right,F/right,K/above right}
      
    {
      
        fill (point) circle (1.5pt);
      
        node[position=3pt] at (point) {$point$};
      
    }
      

      
    end{tikzpicture}
      
        end{document}


      I got
      enter image description here






      tikz-pgf tikz-3dplot




      tikz-pgf tikz-3dplot






      share|improve this question













      share|improve this question











      share|improve this question




      share|improve this question










      asked 11 mins ago









      minhthien_2016minhthien_2016

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