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¿Cómo dibujar un cubo que pueda inscribirse dentro de un cono circular recto?

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¿Cómo dibujar un cubo que pueda inscribirse dentro de un cono circular recto?

Estoy tratando de dibujar un cubo que pueda inscribirse dentro de un cono circular recto (cono de radio r y altura perpendicular h). veo aquíhttps://www.geeksforgeeks.org/largest-cube-that-can-be-inscribed-within-a-right-circular-cone/Puedo encontrar la longitud del lado del cubo. Pero no puedo dibujarlo. Lo intenté

    \documentclass[border=3.14mm,12pt,tikz]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{calc,backgrounds}
\begin{document}
%polar coordinates of visibility
\pgfmathsetmacro\th{65}
\pgfmathsetmacro\az{110}
\tdplotsetmaincoords{\th}{\az}
%parameters of the cone
\pgfmathsetmacro\R{4} %radius of base
\pgfmathsetmacro\v{5} %hight of cone
\begin{tikzpicture} [scale=1, tdplot_main_coords, axis/.style={blue,thick}]
\path
coordinate (O) at (0,0,0)
coordinate (A) at (\R,0,0)
coordinate (B) at (0,\R,0)
coordinate (C) at ($ 2*(O) - (A) $)
coordinate (D) at ($ 2*(O) - (B) $)
coordinate (S) at (0,0,\v)
;
\fill (S) circle[radius=1pt] node[above] {$S$};
\fill (A) circle[radius=1pt] node[below] {$A$};
\fill (B) circle[radius=1pt] node[below] {$B$};
\fill (C) circle[radius=1pt] node[above] {$C$};
\fill (D) circle[radius=1pt] node[above] {$D$};
\fill (O) circle[radius=1pt] node[below] {$O$};

 \draw[thick] (A) -- (S) (S) -- (B);
\draw[dashed] (A) -- (B) -- (C) -- (D) -- cycle (S) -- (C) (A) -- (C) (B) -- (D) (S) -- (O);

\pgfmathsetmacro\cott{{cot(\th)}}
\pgfmathsetmacro\fraction{\R*\cott/\v}
\pgfmathsetmacro\fraction{\fraction<1 ? \fraction : 1}
\pgfmathsetmacro\angle{{acos(\fraction)}}

% % angles for transformed lines
\pgfmathsetmacro\PhiOne{180+(\az-90)+\angle}
\pgfmathsetmacro\PhiTwo{180+(\az-90)-\angle}

% % coordinates for transformed surface lines
\pgfmathsetmacro\sinPhiOne{{sin(\PhiOne)}}
\pgfmathsetmacro\cosPhiOne{{cos(\PhiOne)}}
\pgfmathsetmacro\sinPhiTwo{{sin(\PhiTwo)}}
\pgfmathsetmacro\cosPhiTwo{{cos(\PhiTwo)}}

% % angles for original surface lines
\pgfmathsetmacro\sinazp{{sin(\az-90)}}
\pgfmathsetmacro\cosazp{{cos(\az-90)}}
\pgfmathsetmacro\sinazm{{sin(90-\az)}}
\pgfmathsetmacro\cosazm{{cos(90-\az)}}

% % draw basis circle
\tdplotdrawarc[tdplot_main_coords,thick]{(O)}{\R}{\PhiOne}{360+\PhiTwo}{anchor=north}{}
\tdplotdrawarc[tdplot_main_coords,dashed]{(O)}{\R}{\PhiTwo}{\PhiOne}{anchor=north}{}

% % displaying tranformed surface of the cone (rotated)
\draw[thick] (0,0,\v) -- (\R*\cosPhiOne,\R*\sinPhiOne,0);
\draw[thick] (0,0,\v) -- (\R*\cosPhiTwo,\R*\sinPhiTwo,0);
 \end{tikzpicture}
  \end{document}

ingrese la descripción de la imagen aquí

Respuesta1

Suponiendo que la fórmula de su enlace sea correcta, este es un posible cubo. La cifra parece confirmar la fórmula.

\documentclass[border=3.14mm,12pt,tikz]{standalone}
\usepackage{tikz-3dplot}
\begin{document}
%polar coordinates of visibility
\pgfmathsetmacro\th{65}
\pgfmathsetmacro\az{110}
\tdplotsetmaincoords{\th}{\az}
%parameters of the cone
\begin{tikzpicture}[scale=1, tdplot_main_coords, axis/.style={blue,thick}]
  \pgfmathsetmacro\R{4} %radius of base
  \pgfmathsetmacro\v{5} %hight of cone
  \path (0,0,0) coordinate (O)  
      (\R,0,0) coordinate (A)  
      (0,\R,0) coordinate (B)  
      (-\R,0,0) coordinate (C)  
      (0,-\R,0) coordinate (D)  
      (0,0,\v) coordinate (S);
  \foreach \point/\pos in {S/above,A/below,B/below,C/above,D/above,O/below} 
  {\fill (\point) circle[radius=1pt] node[\pos] {$\point$};}
  \draw[thick] (A) -- (S) (S) -- (B);
  \draw[dashed] (A) -- (B) -- (C) -- (D) -- cycle (S) -- (C) (A) -- (C) (B) -- (D) (S) -- (O);

  \pgfmathsetmacro\cott{{cot(\th)}}
  \pgfmathsetmacro\fraction{\R*\cott/\v}
  \pgfmathsetmacro\fraction{\fraction<1 ? \fraction : 1}
  \pgfmathsetmacro\angle{{acos(\fraction)}}

  % % angles for transformed lines
  \pgfmathsetmacro\PhiOne{180+(\az-90)+\angle}
  \pgfmathsetmacro\PhiTwo{180+(\az-90)-\angle}

  % % coordinates for transformed surface lines
  \pgfmathsetmacro\sinPhiOne{{sin(\PhiOne)}}
  \pgfmathsetmacro\cosPhiOne{{cos(\PhiOne)}}
  \pgfmathsetmacro\sinPhiTwo{{sin(\PhiTwo)}}
  \pgfmathsetmacro\cosPhiTwo{{cos(\PhiTwo)}}

  \tdplotdrawarc[tdplot_main_coords,dashed]{(O)}{\R}{\PhiTwo}{\PhiOne}{anchor=north}{}

  \pgfmathsetmacro{\a}{\v*\R*sqrt(2)/(\v+sqrt(2)*\R)} % edge
  \pgfmathsetmacro{\mya}{sqrt(1/2)*\a} % half diagonal

  \pgfmathsetmacro{\myalpha}{90}
  \begin{scope}[canvas is xy plane at z=0] 
   \path foreach \X in {1,...,4}{ (\myalpha+90*\X:\mya) coordinate (P\X)};
  \end{scope}
  \begin{scope}[canvas is xy plane at z=\a] 
   \path foreach \X in {1,...,4}{ (\myalpha+90*\X:\mya) coordinate (Q\X)};
  \end{scope}
  \draw[dashed] (P4) -- (P1) -- (P2)  (P1) -- (Q1);
  \draw (P4) -- (P3) -- (P2) -- (Q2) -- (Q3) -- (Q4) -- (Q1) -- (Q2) 
  (Q4) -- (P4) (P4) (P3) -- (Q3);

  % % draw basis circle
  \tdplotdrawarc[tdplot_main_coords,thick]{(O)}{\R}{\PhiOne}{360+\PhiTwo}{anchor=north}{}

  % % displaying transformed surface of the cone (rotated)
  \draw[thick] (0,0,\v) -- (\R*\cosPhiOne,\R*\sinPhiOne,0);
  \draw[thick] (0,0,\v) -- (\R*\cosPhiTwo,\R*\sinPhiTwo,0);


\end{tikzpicture}
\end{document}

ingrese la descripción de la imagen aquí

Si giras el cubo, todavía cabe.

\documentclass[border=3.14mm,12pt,tikz]{standalone}
\usepackage{tikz-3dplot}
\begin{document}
%polar coordinates of visibility
\pgfmathsetmacro\th{65}
\pgfmathsetmacro\az{110}
\tdplotsetmaincoords{\th}{\az}
\foreach \Angle in {0,2,...,88}
{\begin{tikzpicture}[scale=1, tdplot_main_coords, axis/.style={blue,thick}]
  \pgfmathsetmacro\R{4} %radius of base
  \pgfmathsetmacro\v{5} %hight of cone
  \path (0,0,0) coordinate (O)  
      (\R,0,0) coordinate (A)  
      (0,\R,0) coordinate (B)  
      (-\R,0,0) coordinate (C)  
      (0,-\R,0) coordinate (D)  
      (0,0,\v) coordinate (S);
  \foreach \point/\pos in {S/above,A/below,B/below,C/above,D/above,O/below} 
  {\fill (\point) circle[radius=1pt] node[\pos] {$\point$};}
  \draw[thick] (A) -- (S) (S) -- (B);
  \draw[dashed] (A) -- (B) -- (C) -- (D) -- cycle (S) -- (C) (A) -- (C) (B) -- (D) (S) -- (O);

  \pgfmathsetmacro\cott{{cot(\th)}}
  \pgfmathsetmacro\fraction{\R*\cott/\v}
  \pgfmathsetmacro\fraction{\fraction<1 ? \fraction : 1}
  \pgfmathsetmacro\angle{{acos(\fraction)}}

  % % angles for transformed lines
  \pgfmathsetmacro\PhiOne{180+(\az-90)+\angle}
  \pgfmathsetmacro\PhiTwo{180+(\az-90)-\angle}

  % % coordinates for transformed surface lines
  \pgfmathsetmacro\sinPhiOne{{sin(\PhiOne)}}
  \pgfmathsetmacro\cosPhiOne{{cos(\PhiOne)}}
  \pgfmathsetmacro\sinPhiTwo{{sin(\PhiTwo)}}
  \pgfmathsetmacro\cosPhiTwo{{cos(\PhiTwo)}}

  \tdplotdrawarc[tdplot_main_coords,dashed]{(O)}{\R}{\PhiTwo}{\PhiOne}{anchor=north}{}

  \pgfmathsetmacro{\a}{\v*\R*sqrt(2)/(\v+sqrt(2)*\R)} % edge
  \pgfmathsetmacro{\mya}{sqrt(1/2)*\a} % half diagonal

  \pgfmathsetmacro{\myalpha}{\Angle}
  \begin{scope}[canvas is xy plane at z=0] 
   \path foreach \X in {1,...,4}{ (\myalpha+90*\X:\mya) coordinate (P\X)};
  \end{scope}
  \begin{scope}[canvas is xy plane at z=\a] 
   \path foreach \X in {1,...,4}{ (\myalpha+90*\X:\mya) coordinate (Q\X)};
  \end{scope}
  \draw (P4) -- (P3) -- (P2) -- (Q2) -- (Q3) -- (Q4) -- (Q1) -- (Q2) 
  (Q4) -- (P4) (P4) (P3) -- (Q3) (P4) -- (P1) -- (P2)  (P1) -- (Q1);

  % % draw basis circle
  \tdplotdrawarc[tdplot_main_coords,thick]{(O)}{\R}{\PhiOne}{360+\PhiTwo}{anchor=north}{}

  % % displaying transformed surface of the cone (rotated)
  \draw[thick] (0,0,\v) -- (\R*\cosPhiOne,\R*\sinPhiOne,0);
  \draw[thick] (0,0,\v) -- (\R*\cosPhiTwo,\R*\sinPhiTwo,0);
\end{tikzpicture}}
\end{document}

ingrese la descripción de la imagen aquí

Respuesta2

También uso la fórmula de tu enlace.

\documentclass[border=2mm,12pt,tikz]{standalone}
\usepackage{fouriernc}
\usepackage{tikz-3dplot}
\usetikzlibrary{calc,backgrounds}
\begin{document}
%polar coordinates of visibility
\pgfmathsetmacro\th{65}
\pgfmathsetmacro\az{110}
\tdplotsetmaincoords{\th}{\az}
%parameters of the cone
\pgfmathsetmacro\R{4} %radius of base
\pgfmathsetmacro\v{5} %hight of cone
\pgfmathsetmacro\a{{(\v * \R * sqrt(2)) / (\v + sqrt(2) * \R)}}
\pgfmathsetmacro\b{{1/2*\a*sqrt(2)}}
\begin{tikzpicture} [scale=1, tdplot_main_coords, axis/.style={blue,thick}]
\begin{scope}[canvas is xy plane at z=0]
\path
coordinate (O) at (0,0)
coordinate (A) at (\R,0)
coordinate (B) at (0,\R)
coordinate (C) at (-\R,0)
coordinate (D) at (0,-\R,0)
coordinate (M) at (\b,0)
coordinate (N) at (0,\b)
coordinate (P) at (-\b,0)
coordinate (Q) at (0,-\b);
\end{scope}
\begin{scope}[canvas is xy plane at z=\a] 
\path coordinate (M') at (\b,0)
coordinate (N') at (0,\b)
coordinate (P') at (-\b,0)
coordinate (Q') at (0,-\b)
;
\end{scope}
\coordinate (S) at (0,0,\v);

\foreach \v/\position in {O/below,A/below, B/right,  C/right, D/left, S/above} {\draw[draw =black, fill=black] (\v) circle (1.5pt) node [\position=0.2mm] {$\v$};
}
\foreach \X in {A,B} \draw[thick] (\X) -- (S);
\foreach \X in {M,N,P,Q} \draw[dashed] (\X) -- (\X');
\draw[dashed] (A) -- (B) -- (C) -- (D) -- cycle 
(M) -- (N) -- (P) -- (Q) -- cycle
(M') -- (N') -- (P') -- (Q') -- cycle
(A) -- (C) (B) -- (D)
(S)--(O)  (S)-- (C);

\foreach \Y in {M, N, P, Q, M', N', P', Q'} 
\fill (\Y) circle[radius=1.2pt];

\begin{scope}[canvas is xy plane at z=0]
\draw[dashed, blue]  (O) circle[radius=\b];
\end{scope}


\pgfmathsetmacro\cott{{cot(\th)}}
\pgfmathsetmacro\fraction{\R*\cott/\v}
\pgfmathsetmacro\fraction{\fraction<1 ? \fraction : 1}
\pgfmathsetmacro\angle{{acos(\fraction)}}

% % angles for transformed lines
\pgfmathsetmacro\PhiOne{180+(\az-90)+\angle}
\pgfmathsetmacro\PhiTwo{180+(\az-90)-\angle}

% % coordinates for transformed surface lines
\pgfmathsetmacro\sinPhiOne{{sin(\PhiOne)}}
\pgfmathsetmacro\cosPhiOne{{cos(\PhiOne)}}
\pgfmathsetmacro\sinPhiTwo{{sin(\PhiTwo)}}
\pgfmathsetmacro\cosPhiTwo{{cos(\PhiTwo)}}

% % angles for original surface lines
\pgfmathsetmacro\sinazp{{sin(\az-90)}}
\pgfmathsetmacro\cosazp{{cos(\az-90)}}
\pgfmathsetmacro\sinazm{{sin(90-\az)}}
\pgfmathsetmacro\cosazm{{cos(90-\az)}}

% % draw basis circle
\tdplotdrawarc[tdplot_main_coords,thick]{(O)}{\R}{\PhiOne}{360+\PhiTwo}{anchor=north}{}
\tdplotdrawarc[tdplot_main_coords,dashed]{(O)}{\R}{\PhiTwo}{\PhiOne}{anchor=north}{}

% % displaying tranformed surface of the cone (rotated)
\draw[thick] (0,0,\v) -- (\R*\cosPhiOne,\R*\sinPhiOne,0);
\draw[thick] (0,0,\v) -- (\R*\cosPhiTwo,\R*\sinPhiTwo,0);
 \end{tikzpicture}
  \end{document}

ingrese la descripción de la imagen aquí

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