5

Given that a regular pyramid T.ABCD in which

  • P is on TC such that TP:PC = 1:3
  • R is on the extension of BC such that BR:BC = 1:3
  • Q is arbitrarily on the plane TAD but neither outside nor on the boundary

Draw the cross section when the plane PQR cuts the pyramid.

MWE

My own solution is as follows but I am looking for other simpler ways (if any).

enter image description here

\documentclass[pstricks,border=0pt,12pt]{standalone}
\usepackage{pst-eucl}
\newpsstyle{aux}
{
    linestyle=dashed,
    linecolor=lightgray,
}
\begin{document}
\begin{pspicture}[showgrid=false](-5,-2)(11,11)
    \pstGeonode[PosAngle={-45,0,45,180}](0,0){A}(8,0){B}(4,2){C}(-4,2){D}
    \pstMiddleAB[PointName=none,PointSymbol=none]{A}{C}{O}
    \pstGeonode(O|0,9){T}
    \psline(D)(A)(B)
    \psline[linestyle=dashed](B)(C)(D)
    \psline[linestyle=dashed](T)(C)
    % additional
    \pstHomO[HomCoef=\pscalculate{1/4}]{T}{C}[P]
    \pstHomO[HomCoef=\pscalculate{-1/3}]{B}{C}[R]
    \pstGeonode[PosAngle=180](.65,5){Q}
    %
    \pstMiddleAB{A}{B}{E}
    \pstMiddleAB{C}{D}{F}
    \pstTranslation[PosAngle={0,-135}]{O}{T}{E,F}
    %
    \pstInterLL{T}{B}{P}{R}{S}
    \pcline[style=aux,nodesep=-5mm](T)(B)
    \psline(T)(B)
    %
    \pstInterLL[PosAngle=45]{E'}{F'}{P}{R}{U}
    \pcline[style=aux,nodesep=-5mm](U)(R)
    \pcline[style=aux,nodesep=-5mm](E')(F')
    %
    \pstInterLL{T}{D}{U}{Q}{V}
    \pcline[style=aux,nodesep=-5mm](T)(D)
    \psline(T)(D)
    %
    \pstInterLL[PosAngle=180]{T}{A}{U}{Q}{W}
    \pcline[style=aux,nodesep=-5mm](U)(W)
    \pcline[style=aux,nodesep=-5mm](T)(A)
    \psline(T)(A)
    \pspolygon[fillstyle=solid,fillcolor=blue,opacity=.1,linestyle=none](F)(F')(E')(E)
    \pspolygon[fillstyle=solid,fillcolor=yellow,opacity=.3,linestyle=none](W)(S)(P)(V)
\end{pspicture}
\end{document}

Note: This question is more mathematical but it is still much more educational than asking to draw pigs, ducks, and their families in the zoo.

9
7

Thanks for the comment. Yes, I think that this can be done. We need "only" to figure out the slope of the line of P-S line. This translates on a slope of the projected plane on the ATD plane, which has to run through Q, of course. To keep the discussion general I do not use the information of how P and S are constructed, all I use is that they sit on the edges BT and CT, respectively, and Q is on the ATD plane. The reason why I got stuck is that I cannot run the animation over more than 6 iterations. I have no idea why, so this comes with a short animation. (I also see that the animation jumps because the bounding box grows, but this is a minor complaint compared to the "Generic error" I get when running larger animations. I have produced tons of animations and never gotten such errors. )

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{backgrounds}
\newcounter{coordi}
\begin{document}
\tdplotsetmaincoords{70}{-30} 
\foreach \BB in {1.5,1.6,...,2} %,1.9,1.8,...,1.6}
{\begin{tikzpicture}[tdplot_main_coords,declare function={B=3;H=B*sqrt(2);}]
 \foreach \Y in {-B,B} {\foreach \X in {-B,B}
 {\stepcounter{coordi} \path ({-1*sign(\Y)*\X},\Y,0) coordinate[label=right:$\Alph{coordi}$] (\Alph{coordi});}}
 \path (0,0,H) coordinate[label=right:$T$] (T);
 \draw[thick] (B) -- (T);
 \draw[thick,dashed] (B) -- (C) -- (D)  (C) -- (T);
 \draw[thick] (A) -- (T) -- (D) -- (A) -- (B);
 % this implements BR:BC = 1:3
 \path (C) -- (B) coordinate[pos=4/3,label=right:$R$] (R)
  (T) -- (C) coordinate[pos=1/4,label=right:$P$] (P)
  (intersection cs:first line={(B)--(T)}, second line={(R)--(P)}) 
 coordinate[label=right:$S$] (S);
 \foreach \X/\Y [count=\Z] in {B/S,C/P}
  {\path let \p1=($(\X)-(T)$),\p2=($(\Y)-(\X)$),
  \n1={veclen(\x2,\y2)/veclen(\x1,\y1)} in \pgfextra{\pgfmathsetmacro{\myratio}{\n1}
  \pgfmathsetmacro{\myz}{\myratio*H}
  \ifnum\Z=1
  \xdef\LstZ{\myz}
  \else
  \xdef\LstZ{\LstZ,\myz}
  \fi}; }
 \typeout{\LstZ} %<- contains the z-values of S and P
 % this brings you in the A-T-D plane 
 \begin{scope}[shift={(D)},x={($(A)-(D)$)},y={($(T)-(D)$)}]  
   \path (0.25,{\BB/3}) coordinate[label=right:$Q$] (Q);
 \end{scope}
 % the B-T and C-T lines have the equation p=(B*(1-t),\mp B*(1-t),H*t), t\in[0,1]
 % so for a given z value y=\mp B*(1-z/H)
 % or dy/dz=\pm B/H, dx/dz=\pm B/H
 % this gives us the slope of the plane in the ATD plane
 \path ($(Q)+({(B/H)*({\LstZ}[1]-{\LstZ}[0])},
 {B*(1-{\LstZ}[1]/H)+B*(1-{\LstZ}[0]/H)},
 {{\LstZ}[1]-{\LstZ}[0]})$) coordinate(aux);
 \path (intersection cs:first line={(Q)--(aux)}, second line={(A)--(T)}) 
  coordinate[label=right:$W$] (W)
  (intersection cs:first line={(Q)--(aux)}, second line={(D)--(T)}) 
  coordinate[label=right:$V$] (V);
 \begin{scope}[on background layer]
  \fill[yellow,fill opacity=0.4] (W) -- (S) -- (P) -- (V) -- cycle;
 \end{scope}
\end{tikzpicture}}
\end{document}

enter image description here

OLD ANSWER: It think the overall way you construct this is fine, +1 for that. However, there are three points which I thought could be useful to add.

  1. Orthographic projections. This is why I am using tikz-3dplot here and I assume that could be done with pstricks, too.
  2. An arguably simpler way of fixing Q on the desired plane: switch to plane with origin, say, at D and axes A-D and T-D.
  3. 3D ordering. This is the main point of this answer: compute the line where the yellow and blue planes intersect and draw the hidden part of the yellow plane on the background layer.

It tried to mark the relevant pieces in the code. And I think these things could be redone in pstricks, but not by me, I've not been using it actively now for a while.

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{backgrounds}
\newcounter{coordi}
\begin{document}
\tdplotsetmaincoords{70}{-30} 
\begin{tikzpicture}[tdplot_main_coords,declare function={B=3;H=B*sqrt(2);}]
 \foreach \Y in {-B,B} {\foreach \X in {-B,B}
 {\stepcounter{coordi} \path ({-1*sign(\Y)*\X},\Y,0) coordinate[label=right:$\Alph{coordi}$] (\Alph{coordi});}}
 \foreach \Y in {-B,B}
 {\stepcounter{coordi} 
 \path (0,\Y,0) coordinate[label=right:$\Alph{coordi}$] (\Alph{coordi})
 (0,\Y,H) coordinate[label=right:$\Alph{coordi}'$] (\Alph{coordi}');}
 \path (0,0,H) coordinate[label=right:$T$] (T);
 \draw[thick] (B) -- (T);
 \draw[thick,dashed] (B) -- (C) -- (D)  (C) -- (T);
 \fill[blue!50,fill opacity=0.4] (E) -- (F) -- (F') -- (E') -- cycle;
 \draw[thick] (A) -- (T) -- (D) -- (A) -- (B);
 % this implements BR:BC = 1:3
 \path (C) -- (B) coordinate[pos=4/3,label=right:$R$] (R);
 \path (intersection cs:first line={(R)--(F')}, second line={(C)--(T)}) 
 coordinate[label=right:$P$] (P)
 (intersection cs:first line={(R)--(F')}, second line={(B)--(T)}) 
 coordinate[label=right:$S$] (S);
 \draw[gray,thin,dashed] (R) -- (F');
 % this brings you in the A-T-D plane 
 \begin{scope}[shift={(D)},x={($(A)-(D)$)},y={($(T)-(D)$)}]  
   \path (0.25,2/3) coordinate[label=right:$Q$] (Q);
 \end{scope}
 % compute intersection points V and W
 \path (intersection cs:first line={(Q)--(F')}, second line={(A)--(T)}) 
 coordinate[label=right:$W$] (W)
 (intersection cs:first line={(Q)--(F')}, second line={(D)--(T)}) 
 coordinate[label=right:$V$] (V);
 % compute intersections of W-S and V-P with the blue plane
 \foreach \X/\Y [count=\Z] in {A/W,B/S,D/V,C/P}
  {\path let \p1=($(\X)-(T)$),\p2=($(\Y)-(T)$),
  \n1={veclen(\x2,\y2)/veclen(\x1,\y1)} in \pgfextra{\pgfmathsetmacro{\myratio}{\n1}
  \ifnum\Z=1
  \xdef\LstRatios{\myratio}
  \else
  \xdef\LstRatios{\LstRatios,\myratio}
  \fi}; }
 %\typeout{\LstRatios}
 \path (W) -- (S) coordinate[pos={{\LstRatios}[0]/({\LstRatios}[0]+{\LstRatios}[1])}] (WS)
  (V) -- (P) coordinate[pos={{\LstRatios}[2]/({\LstRatios}[2]+{\LstRatios}[3])}] (VP);
  \draw (WS) circle (1pt);
 \begin{scope}[on background layer]
  \fill[yellow,fill opacity=0.4] (WS) -- (S) -- (P) -- (VP) -- cycle;
 \end{scope}
 \fill[yellow,fill opacity=0.4] (WS) -- (W) -- (V) -- (VP) -- cycle;
\end{tikzpicture}
\end{document}

enter image description here

2
  • @ArtificialOdorlessArmpit I believe so. How much do you assume to be known about Q? Do we assume to know its 3d coordinates? – user121799 May 5 '19 at 19:30
  • @ArtificialOdorlessArmpit OK, I tried to keep it as general as possible. – user121799 May 5 '19 at 21:18
5

Perhaps you get a big surprise when you see my code. I calculated all coordinates of points of intersection.

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


 \tikzset{ hidden/.style = {thick, dashed}}
   \tikzset{%
     add/.style args={#1 and #2}{
         to path={%
  ($(\tikztostart)!-#1!(\tikztotarget)$)--($(\tikztotarget)!-#2!(\tikztostart)$)%
   \tikztonodes}}}

 \begin{document}

 \tdplotsetmaincoords{70}{20}
 \begin{tikzpicture}[tdplot_main_coords,scale=1.3]
 \pgfmathsetmacro\a{4}
 \pgfmathsetmacro\h{5}

 % definitions
 \path
coordinate(A) at (0,0,0)
coordinate (B) at (\a,0,0)
coordinate (C) at(\a,\a,0)
coordinate (D) at (0,\a,0)
coordinate (T) at (1/2*\a, 1/2*\a, \h) 
coordinate (Q) at (1/6*\a, 1/2*\a, 1/3*\h)               
coordinate (P) at  (2/3*\a, 2/3*\a, 2/3*\h)
coordinate (R) at  (\a, -1/3*\a, 0)
coordinate (Y) at  (5/18*\a, 13/18*\a, 5/9*\h)
coordinate (E) at (5/6*\a, 1/6*\a, 1/3*\h)
coordinate (V) at (1/3*\a, 0, 0)
coordinate (Z) at (0, 1/6*\a, 0)
coordinate (X) at  (1/2*\a, 7/6*\a, \h);

 \draw[hidden,thick]
        (A) -- (C)  (B) -- (D)  (A)--(D)   (T)-- (C)  (T)-- (D) (C) -- (D) (X) -- (Z) (Y) -- (P) (V) -- (Z);
\draw [thick] (T)-- (C) (A) -- (B) -- (C) (T)-- (A) (T)-- (R) -- (X) -- cycle (T)-- (B) (E) -- (V) -- (R)

;

\fill[yellow,fill opacity=0.4] (P) -- (Y) -- (Z) -- (V) -- (E) -- cycle;
 \foreach \point/\position in {A/below,B/below,C/below,D/right,T/above,P/right,R/below,E/right,Q/left,V/below,X/right,Y/left,Z/right}
 {
   \fill (\point) circle (1.2pt);
   \node[\position=.3pt] at (\point) {$\point$};
 }
\end{tikzpicture}
 \end{document} 

enter image description here

With \tdplotsetmaincoords{70}{290}

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


 \tikzset{ hidden/.style = {thick, dashed}}
    \begin{document}

 \tdplotsetmaincoords{70}{290}
 \begin{tikzpicture}[tdplot_main_coords,scale=1.3]
 \pgfmathsetmacro\a{4}
 \pgfmathsetmacro\h{5}

 % definitions
 \path
coordinate(A) at (0,0,0)
coordinate (B) at (\a,0,0)
coordinate (C) at(\a,\a,0)
coordinate (D) at (0,\a,0)
coordinate (T) at (1/2*\a, 1/2*\a, \h) 
coordinate (Q) at (1/6*\a, 1/2*\a, 1/3*\h)               
coordinate (P) at  (2/3*\a, 2/3*\a, 2/3*\h)
coordinate (R) at  (\a, -1/3*\a, 0)
coordinate (Y) at  (5/18*\a, 13/18*\a, 5/9*\h)
coordinate (E) at (5/6*\a, 1/6*\a, 1/3*\h)
coordinate (V) at (1/3*\a, 0, 0)
coordinate (Z) at (0, 1/6*\a, 0)
coordinate (X) at  (1/2*\a, 7/6*\a, \h);

 \draw[hidden,thick]
        (A) -- (C)  (B) -- (D)  (B) -- (C)   (T)-- (C)  (T)-- (C) (C) -- (D)  (Y) -- (P) (V) -- (Z)  (B) -- (R);
\draw [thick] (T)-- (D)  (T)-- (A) (T)-- (R) -- (X) -- cycle (T)-- (B) (E) -- (V) -- (R) (A)--(D) (A)--  (B) (X) -- (Z)

;

\fill[yellow,fill opacity=0.4] (P) -- (Y) -- (Z) -- (V) -- (E) -- cycle;
 \foreach \point/\position in {A/below,B/below,C/below,D/below,T/above,P/right,R/below,E/above,Q/left,V/below,X/above,Y/left,Z/below}
 {
   \fill (\point) circle (1.2pt);
   \node[\position=.3pt] at (\point) {$\point$};
 }
\end{tikzpicture}
 \end{document} 

enter image description here

This code can use with Q inside the triangle TAD (see at https://en.wikipedia.org/wiki/Convex_combination). You can view at many angles by changing the value of u and v.

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


 \tikzset{ hidden/.style = {thick, dashed}}
    \begin{document}

 \tdplotsetmaincoords{70}{290}
 \begin{tikzpicture}[tdplot_main_coords,scale=1.3,line join = round, line cap = round]
\pgfmathsetmacro\a{6}
\pgfmathsetmacro\h{5}
\pgfmathsetmacro\u{2/5}
\pgfmathsetmacro\v{1/6}

% definitions
\path
coordinate(A) at (0,0,0)
coordinate (B) at (\a,0,0)
coordinate (C) at(\a,\a,0)
coordinate (D) at (0,\a,0)
coordinate (T) at (1/2*\a, 1/2*\a, \h) 
coordinate (Q) at ({1/2*\v*\a}, {1/2*\v*\a+(1-\v-\u)*\a}, {\v*\h})              
coordinate (P) at  (2/3*\a, 2/3*\a, 2/3*\h)
coordinate (R) at  (\a, -1/3*\a, 0)
coordinate (Y) at ({\a*(3*\u+2*\v)/(2*(2+3*\u))}, {(3*\u-2*\v+4)*\a/(2*(2+3*\u))}, {(3*\u+2*\v)*\h/(2+3*\u)})
coordinate (E) at (5/6*\a, 1/6*\a, 1/3*\h)
coordinate (V) at ({\a*(3*\u+5*\v-3)/(3*\u+6*\v-4)},0,0)
coordinate (Z) at (0,{\a*(3*\u+5*\v-3)/(3*(-1+\v))},0)
coordinate (X) at  (1/2*\a, 7/6*\a, \h);

\draw[hidden,thick]
(A) -- (C)  (B) -- (D)  (B) -- (C)   (T)-- (C)  (T)-- (C) (C) -- (D)  (Y) -- (P) (V) -- (Z)  (B) -- (R) ;
\draw [thick] (T)-- (D)  (T)-- (A) (T)-- (R) -- (X) -- cycle (T)-- (B) (E) -- (V) -- (R) (A)--(D) (A)--  (B) (X) -- (Z)
;

\fill[yellow,fill opacity=0.4] (P) -- (Y) -- (Z) -- (V) -- (E) -- cycle;
\foreach \point/\position in {A/below,B/below,C/below,D/below,T/above,P/right,R/below,E/above,Q/left,V/below,X/above,Y/left,Z/below}
{
    \fill (\point) circle (1.2pt);
    \node[\position=.3pt] at (\point) {$\point$};
}
\end{tikzpicture}

\end{center}

\begin{center}
    \begin{tikzpicture}[tdplot_main_coords,scale=1.3,line join = round, line cap = round]
    \pgfmathsetmacro\a{6}
    \pgfmathsetmacro\h{5}
    \pgfmathsetmacro\u{3/5}
    \pgfmathsetmacro\v{1/3}

    % definitions
    \path
    coordinate(A) at (0,0,0)
    coordinate (B) at (\a,0,0)
    coordinate (C) at(\a,\a,0)
    coordinate (D) at (0,\a,0)
    coordinate (T) at (1/2*\a, 1/2*\a, \h) 
    coordinate (Q) at ({1/2*\v*\a}, {1/2*\v*\a+(1-\v-\u)*\a}, {\v*\h})              
    coordinate (P) at  (2/3*\a, 2/3*\a, 2/3*\h)
    coordinate (R) at  (\a, -1/3*\a, 0)
    coordinate (Y) at ({\a*(3*\u+2*\v)/(2*(2+3*\u))}, {(3*\u-2*\v+4)*\a/(2*(2+3*\u))}, {(3*\u+2*\v)*\h/(2+3*\u)})
    coordinate (E) at (5/6*\a, 1/6*\a, 1/3*\h)
    coordinate (V) at ({\a*(3*\u+5*\v-3)/(3*\u+6*\v-4)},0,0)
    coordinate (Z) at (0,{\a*(3*\u+5*\v-3)/(3*(-1+\v))},0)
    coordinate (X) at  (1/2*\a, 7/6*\a, \h);

    \draw[hidden,thick]
    (A) -- (C)  (B) -- (D)  (B) -- (C)   (T)-- (C)  (T)-- (C) (C) -- (D)  (Y) -- (P) (V) -- (Z)  (B) -- (R) ;
    \draw [thick] (T)-- (D)  (T)-- (A) (T)-- (R) -- (X) -- cycle (T)-- (B) (E) -- (V) -- (R) (A)--(D) (A)--  (B) (X) -- (Z)
    ;

    \fill[yellow,fill opacity=0.4] (P) -- (Y) -- (Z) -- (V) -- (E) -- cycle;
    \foreach \point/\position in {A/below,B/below,C/below,D/below,T/above,P/right,R/below,E/above,Q/left,V/below,X/above,Y/left,Z/below}
    {
        \fill (\point) circle (1.2pt);
        \node[\position=.3pt] at (\point) {$\point$};
    }
    \end{tikzpicture}

with \pgfmathsetmacro\u{3/5} \pgfmathsetmacro\v{1/3}, we get enter image description here

6
  • You still used the same algorithm with the plane FF'E'E. – Let Me Be Your Lost Disciple May 6 '19 at 15:36
  • @ArtificialOdorlessArmpit I think, the most important is when the section is a pentagon, and quadrilateral? Can you answer this question? – minhthien_2016 May 6 '19 at 15:39
  • The main problem is "Is there any other method to draw but without using the plane given above?". Drawing pentagon, quadrilateral or other shapes does not matter. – Let Me Be Your Lost Disciple May 6 '19 at 15:40
  • @ArtificialOdorlessArmpit I think, you should ask this question at math.stackexchange.com – minhthien_2016 May 6 '19 at 15:48
  • Have you read this? :-) – Let Me Be Your Lost Disciple May 6 '19 at 15:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.