Drawing a box in TikZ - one edge closer to the observer than other edges - with perspective (Version 2)

Question

In the following code, a box is rendered with perspective. I would like the ratio of the dimensions to be 3:4:12. The sides going into the page should be the shortest.

How do I draw the same box with the side going across the page parallel to the top and bottom edges of the page?

\documentclass{amsart}

\usepackage{tikz}
\usetikzlibrary{calc,intersections}



\begin{document}


\begin{center}
\begin{tikzpicture}

%\clip (-3,-3) rectangle (3,3);
\coordinate (top_front) at (0,0);
\coordinate (bottom_front) at (0,-3);
\coordinate (top_right) at (15:2.5cm);
\coordinate (top_left) at (165:2.5cm);

%You can change the perspective by playing with the 5, 5, 15:
\coordinate (front_right) at ($(top_front)!3!(top_right)$);
\coordinate (front_left) at ($(top_front)!5!(top_left)$);
\coordinate (front_bottom) at ($(top_front)!6!(bottom_front)$);

\path[name path=bottom_right_path] (bottom_front) -- (front_right);   
\path[name path=right_back_path] (top_right) -- (front_bottom);
\path[name path=back_left_path] (bottom_front) -- (front_left);
\path[name path=left_back_path] (top_left) -- (front_bottom);
\path[name path=top_right_path] (top_left) -- (front_right);
\path[name path=top_left_path] (top_right) -- (front_left);

\coordinate[name intersections={of=bottom_right_path and right_back_path, by=back_right}];
\coordinate[name intersections={of=back_left_path and left_back_path, by=back_left}];
\coordinate[name intersections={of=top_right_path and top_left_path, by=top_back}];


%The bottom-back vertex of the cube is located.
\path[name path=bottom_left_path] (back_left) -- (front_right);
\path[name path=back_right_path] (back_right) -- (front_left);
\coordinate[name intersections={of=bottom_left_path and back_right_path, by=bottom_back}];
%
\draw[dashed] (back_left) -- (bottom_back);
\draw [dashed](back_right) -- (bottom_back);
\draw[dashed] (top_back) -- (bottom_back);



%A diagonal on a face of the box is labeled.
\path[draw,dashed] (top_left)--(bottom_front)node[midway, above, sloped, font=\scriptsize]{face diagonal};
\draw[dashed,blue] (back_right) -- (top_left)node[midway, sloped, above, font=\scriptsize]{box diagonal};



%\shade[right color=gray!10, left color=black!50, shading angle=105] (top_front) -- (bottom_front) -- (back_left) -- (top_left) -- cycle;
%\shade[left color=gray!10, right color=black!50, shading angle=75] (top_front) -- (bottom_front) -- (back_right) -- (top_right) -- cycle;
\begin{scope}
\clip (top_front) -- (top_right) -- (top_back) -- (top_left) -- cycle;
\shade[inner color = gray!5, outer color=black!50, shading=radial] (top_front) ellipse (3cm and 1.5cm);
\end{scope}

\draw (top_front) -- (bottom_front);
\draw (top_front) -- (top_right);
\draw (top_front) -- (top_left);
\draw (top_right) -- (back_right);
\draw (bottom_front) -- (back_right);
\draw (top_left) -- (back_left);
\draw (bottom_front) -- (back_left);
\draw (top_back) -- (top_right);
\draw (top_back) -- (top_left);



\end{tikzpicture}
\end{center}



\end{document}

Answer 1

This is what you can achieve with Max' great answer. I understand that we disagree, but as AndréC points out, "the confrontation of ideas in science is essential". So, even if you do not like Max' great macros, or find them unnecessary, this answer is meant for others having a similar question. IMHO Max' methods have superseded the older answer by Jan Hlavacek, which, at that time, was a great step in the right direction.

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{tikz-3dplot}
\usepgfmodule{nonlineartransformations}
% Max magic https://tex.stackexchange.com/a/447120/121799
\makeatletter 
% the first part is not in use here
\def\tikz@scan@transform@one@point#1{%
  \tikz@scan@one@point\pgf@process#1%
  \pgf@pos@transform{\pgf@x}{\pgf@y}}
\tikzset{%
  grid source opposite corners/.code args={#1and#2}{%
   \pgfextract@process\tikz@transform@source@southwest{%
     \tikz@scan@transform@one@point{#1}}%
   \pgfextract@process\tikz@transform@source@northeast{%
     \tikz@scan@transform@one@point{#2}}%
  },
  grid target corners/.code args={#1--#2--#3--#4}{%
   \pgfextract@process\tikz@transform@target@southwest{%
     \tikz@scan@transform@one@point{#1}}%
   \pgfextract@process\tikz@transform@target@southeast{%
     \tikz@scan@transform@one@point{#2}}%
   \pgfextract@process\tikz@transform@target@northeast{%
     \tikz@scan@transform@one@point{#3}}%
   \pgfextract@process\tikz@transform@target@northwest{%
     \tikz@scan@transform@one@point{#4}}%
  }
}

\def\tikzgridtransform{%
  \pgfextract@process\tikz@current@point{}%
  \pgf@process{%
    \pgfpointdiff{\tikz@transform@source@southwest}%
      {\tikz@transform@source@northeast}%
  }%
  \pgf@xc=\pgf@x\pgf@yc=\pgf@y%
  \pgf@process{%
    \pgfpointdiff{\tikz@transform@source@southwest}{\tikz@current@point}%
  }%
  \pgfmathparse{\pgf@x/\pgf@xc}\let\tikz@tx=\pgfmathresult%
  \pgfmathparse{\pgf@y/\pgf@yc}\let\tikz@ty=\pgfmathresult%
  %
  \pgfpointlineattime{\tikz@ty}{%
    \pgfpointlineattime{\tikz@tx}{\tikz@transform@target@southwest}%
      {\tikz@transform@target@southeast}}{%
    \pgfpointlineattime{\tikz@tx}{\tikz@transform@target@northwest}%
      {\tikz@transform@target@northeast}}%
}

% Initialize H matrix for perspective view
\pgfmathsetmacro\H@tpp@aa{1}\pgfmathsetmacro\H@tpp@ab{0}\pgfmathsetmacro\H@tpp@ac{0}%\pgfmathsetmacro\H@tpp@ad{0}
\pgfmathsetmacro\H@tpp@ba{0}\pgfmathsetmacro\H@tpp@bb{1}\pgfmathsetmacro\H@tpp@bc{0}%\pgfmathsetmacro\H@tpp@bd{0}
\pgfmathsetmacro\H@tpp@ca{0}\pgfmathsetmacro\H@tpp@cb{0}\pgfmathsetmacro\H@tpp@cc{1}%\pgfmathsetmacro\H@tpp@cd{0}
\pgfmathsetmacro\H@tpp@da{0}\pgfmathsetmacro\H@tpp@db{0}\pgfmathsetmacro\H@tpp@dc{0}%\pgfmathsetmacro\H@tpp@dd{1}

%Initialize H matrix for main rotation
\pgfmathsetmacro\H@rot@aa{1}\pgfmathsetmacro\H@rot@ab{0}\pgfmathsetmacro\H@rot@ac{0}%\pgfmathsetmacro\H@rot@ad{0}
\pgfmathsetmacro\H@rot@ba{0}\pgfmathsetmacro\H@rot@bb{1}\pgfmathsetmacro\H@rot@bc{0}%\pgfmathsetmacro\H@rot@bd{0}
\pgfmathsetmacro\H@rot@ca{0}\pgfmathsetmacro\H@rot@cb{0}\pgfmathsetmacro\H@rot@cc{1}%\pgfmathsetmacro\H@rot@cd{0}
%\pgfmathsetmacro\H@rot@da{0}\pgfmathsetmacro\H@rot@db{0}\pgfmathsetmacro\H@rot@dc{0}\pgfmathsetmacro\H@rot@dd{1}

\pgfkeys{
    /three point perspective/.cd,
        p/.code args={(#1,#2,#3)}{
            \pgfmathparse{int(round(#1))}
            \ifnum\pgfmathresult=0\else
                \pgfmathsetmacro\H@tpp@ba{#2/#1}
                \pgfmathsetmacro\H@tpp@ca{#3/#1}
                \pgfmathsetmacro\H@tpp@da{ 1/#1}
                \coordinate (vp-p) at (#1,#2,#3);
            \fi
        },
        q/.code args={(#1,#2,#3)}{
            \pgfmathparse{int(round(#2))}
            \ifnum\pgfmathresult=0\else
                \pgfmathsetmacro\H@tpp@ab{#1/#2}
                \pgfmathsetmacro\H@tpp@cb{#3/#2}
                \pgfmathsetmacro\H@tpp@db{ 1/#2}
                \coordinate (vp-q) at (#1,#2,#3);
            \fi
        },
        r/.code args={(#1,#2,#3)}{
            \pgfmathparse{int(round(#3))}
            \ifnum\pgfmathresult=0\else
                \pgfmathsetmacro\H@tpp@ac{#1/#3}
                \pgfmathsetmacro\H@tpp@bc{#2/#3}
                \pgfmathsetmacro\H@tpp@dc{ 1/#3}
                \coordinate (vp-r) at (#1,#2,#3);
            \fi
        },
        coordinate/.code args={#1,#2,#3}{
           \pgfmathsetmacro\tpp@x{#1} %<- Max' fix
            \pgfmathsetmacro\tpp@y{#2}
            \pgfmathsetmacro\tpp@z{#3}
        },
}

\tikzset{
    view/.code 2 args={
        \pgfmathsetmacro\rot@main@theta{#1}
        \pgfmathsetmacro\rot@main@phi{#2}
        % Row 1
        \pgfmathsetmacro\H@rot@aa{cos(\rot@main@phi)}
        \pgfmathsetmacro\H@rot@ab{sin(\rot@main@phi)}
        \pgfmathsetmacro\H@rot@ac{0}
        % Row 2
        \pgfmathsetmacro\H@rot@ba{-cos(\rot@main@theta)*sin(\rot@main@phi)}
        \pgfmathsetmacro\H@rot@bb{cos(\rot@main@phi)*cos(\rot@main@theta)}
        \pgfmathsetmacro\H@rot@bc{sin(\rot@main@theta)}
        % Row 3
        \pgfmathsetmacro\H@m@ca{sin(\rot@main@phi)*sin(\rot@main@theta)}
        \pgfmathsetmacro\H@m@cb{-cos(\rot@main@phi)*sin(\rot@main@theta)}
        \pgfmathsetmacro\H@m@cc{cos(\rot@main@theta)}
        % Set vector values
        \pgfmathsetmacro\vec@x@x{\H@rot@aa}
        \pgfmathsetmacro\vec@y@x{\H@rot@ab}
        \pgfmathsetmacro\vec@z@x{\H@rot@ac}
        \pgfmathsetmacro\vec@x@y{\H@rot@ba}
        \pgfmathsetmacro\vec@y@y{\H@rot@bb}
        \pgfmathsetmacro\vec@z@y{\H@rot@bc}
        % Set pgf vectors
        \pgfsetxvec{\pgfpoint{\vec@x@x cm}{\vec@x@y cm}}
        \pgfsetyvec{\pgfpoint{\vec@y@x cm}{\vec@y@y cm}}
        \pgfsetzvec{\pgfpoint{\vec@z@x cm}{\vec@z@y cm}}
    },
}

\tikzset{
    perspective/.code={\pgfkeys{/three point perspective/.cd,#1}},
    perspective/.default={p={(15,0,0)},q={(0,15,0)},r={(0,0,50)}},
}

\tikzdeclarecoordinatesystem{three point perspective}{
    \pgfkeys{/three point perspective/.cd,coordinate={#1}}
    \pgfmathsetmacro\temp@p@w{\H@tpp@da*\tpp@x + \H@tpp@db*\tpp@y + \H@tpp@dc*\tpp@z + 1}
    \pgfmathsetmacro\temp@p@x{(\H@tpp@aa*\tpp@x + \H@tpp@ab*\tpp@y + \H@tpp@ac*\tpp@z)/\temp@p@w}
    \pgfmathsetmacro\temp@p@y{(\H@tpp@ba*\tpp@x + \H@tpp@bb*\tpp@y + \H@tpp@bc*\tpp@z)/\temp@p@w}
    \pgfmathsetmacro\temp@p@z{(\H@tpp@ca*\tpp@x + \H@tpp@cb*\tpp@y + \H@tpp@cc*\tpp@z)/\temp@p@w}
    \pgfpointxyz{\temp@p@x}{\temp@p@y}{\temp@p@z}
}
\tikzaliascoordinatesystem{tpp}{three point perspective}

\makeatother

\begin{document}
\foreach \X in {-87,-84,...,-3}
{  \tdplotsetmaincoords{70}{\X}
  \begin{tikzpicture}
  \path[use as bounding box] (-6,-2) rectangle (6,10);
  \begin{scope}[scale=6,font=\sffamily,
  view={\tdplotmaintheta}{\tdplotmainphi},
            perspective={
                p = {(4,0,1.5)},
                q = {(0,4,1.5)},
            }
  ]
  \begin{scope} % top face
   \draw[clip]   (tpp cs:0,0,1) coordinate (top_front)
  -- (tpp cs:1,0,1) coordinate (top_right)
   -- (tpp cs:1,1,1) coordinate (top_back)
   -- (tpp cs:0,1,1) coordinate (top_left) --
    cycle;
    \shade[inner color = gray!5, outer color=black!50, shading=radial] (top_front) ellipse (3cm and 1.5cm);
  \end{scope}
  % right face  
  \draw   (top_front) -- (top_right) -- (tpp cs:1,0,0) coordinate (bottom_right)
  -- (tpp cs:0,0,0)  coordinate (bottom_front) --   cycle;
  % top face  
  \draw   (top_front) -- (top_left) -- (tpp cs:0,1,0) coordinate (bottom_left)
  -- (bottom_front) --  cycle;
  % your midway coordinates with Max perspective coordinate system
  \coordinate (front_right) at (tpp cs:0.5,0,1);
  \coordinate (front_left) at (tpp cs:0,0.5,1);
  \coordinate (front_bottom) at (tpp cs:0,0,0.5);
  \end{scope}
  \end{tikzpicture}}
\end{document}

As you requested. The dimensions of the boxes are stored in the macros \boxX, \boxY and \boxZ. Of course getting the desired output here requires slightly more effort here since, in addition to the two angles defining the view (or rotation) you also have the perspective parameters that you need to adjust.

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{tikz-3dplot}
\usepgfmodule{nonlineartransformations}
% Max magic https://tex.stackexchange.com/a/447120/121799
\makeatletter 
% the first part is not in use here
\def\tikz@scan@transform@one@point#1{%
  \tikz@scan@one@point\pgf@process#1%
  \pgf@pos@transform{\pgf@x}{\pgf@y}}
\tikzset{%
  grid source opposite corners/.code args={#1and#2}{%
   \pgfextract@process\tikz@transform@source@southwest{%
     \tikz@scan@transform@one@point{#1}}%
   \pgfextract@process\tikz@transform@source@northeast{%
     \tikz@scan@transform@one@point{#2}}%
  },
  grid target corners/.code args={#1--#2--#3--#4}{%
   \pgfextract@process\tikz@transform@target@southwest{%
     \tikz@scan@transform@one@point{#1}}%
   \pgfextract@process\tikz@transform@target@southeast{%
     \tikz@scan@transform@one@point{#2}}%
   \pgfextract@process\tikz@transform@target@northeast{%
     \tikz@scan@transform@one@point{#3}}%
   \pgfextract@process\tikz@transform@target@northwest{%
     \tikz@scan@transform@one@point{#4}}%
  }
}

\def\tikzgridtransform{%
  \pgfextract@process\tikz@current@point{}%
  \pgf@process{%
    \pgfpointdiff{\tikz@transform@source@southwest}%
      {\tikz@transform@source@northeast}%
  }%
  \pgf@xc=\pgf@x\pgf@yc=\pgf@y%
  \pgf@process{%
    \pgfpointdiff{\tikz@transform@source@southwest}{\tikz@current@point}%
  }%
  \pgfmathparse{\pgf@x/\pgf@xc}\let\tikz@tx=\pgfmathresult%
  \pgfmathparse{\pgf@y/\pgf@yc}\let\tikz@ty=\pgfmathresult%
  %
  \pgfpointlineattime{\tikz@ty}{%
    \pgfpointlineattime{\tikz@tx}{\tikz@transform@target@southwest}%
      {\tikz@transform@target@southeast}}{%
    \pgfpointlineattime{\tikz@tx}{\tikz@transform@target@northwest}%
      {\tikz@transform@target@northeast}}%
}

% Initialize H matrix for perspective view
\pgfmathsetmacro\H@tpp@aa{1}\pgfmathsetmacro\H@tpp@ab{0}\pgfmathsetmacro\H@tpp@ac{0}%\pgfmathsetmacro\H@tpp@ad{0}
\pgfmathsetmacro\H@tpp@ba{0}\pgfmathsetmacro\H@tpp@bb{1}\pgfmathsetmacro\H@tpp@bc{0}%\pgfmathsetmacro\H@tpp@bd{0}
\pgfmathsetmacro\H@tpp@ca{0}\pgfmathsetmacro\H@tpp@cb{0}\pgfmathsetmacro\H@tpp@cc{1}%\pgfmathsetmacro\H@tpp@cd{0}
\pgfmathsetmacro\H@tpp@da{0}\pgfmathsetmacro\H@tpp@db{0}\pgfmathsetmacro\H@tpp@dc{0}%\pgfmathsetmacro\H@tpp@dd{1}

%Initialize H matrix for main rotation
\pgfmathsetmacro\H@rot@aa{1}\pgfmathsetmacro\H@rot@ab{0}\pgfmathsetmacro\H@rot@ac{0}%\pgfmathsetmacro\H@rot@ad{0}
\pgfmathsetmacro\H@rot@ba{0}\pgfmathsetmacro\H@rot@bb{1}\pgfmathsetmacro\H@rot@bc{0}%\pgfmathsetmacro\H@rot@bd{0}
\pgfmathsetmacro\H@rot@ca{0}\pgfmathsetmacro\H@rot@cb{0}\pgfmathsetmacro\H@rot@cc{1}%\pgfmathsetmacro\H@rot@cd{0}
%\pgfmathsetmacro\H@rot@da{0}\pgfmathsetmacro\H@rot@db{0}\pgfmathsetmacro\H@rot@dc{0}\pgfmathsetmacro\H@rot@dd{1}

\pgfkeys{
    /three point perspective/.cd,
        p/.code args={(#1,#2,#3)}{
            \pgfmathparse{int(round(#1))}
            \ifnum\pgfmathresult=0\else
                \pgfmathsetmacro\H@tpp@ba{#2/#1}
                \pgfmathsetmacro\H@tpp@ca{#3/#1}
                \pgfmathsetmacro\H@tpp@da{ 1/#1}
                \coordinate (vp-p) at (#1,#2,#3);
            \fi
        },
        q/.code args={(#1,#2,#3)}{
            \pgfmathparse{int(round(#2))}
            \ifnum\pgfmathresult=0\else
                \pgfmathsetmacro\H@tpp@ab{#1/#2}
                \pgfmathsetmacro\H@tpp@cb{#3/#2}
                \pgfmathsetmacro\H@tpp@db{ 1/#2}
                \coordinate (vp-q) at (#1,#2,#3);
            \fi
        },
        r/.code args={(#1,#2,#3)}{
            \pgfmathparse{int(round(#3))}
            \ifnum\pgfmathresult=0\else
                \pgfmathsetmacro\H@tpp@ac{#1/#3}
                \pgfmathsetmacro\H@tpp@bc{#2/#3}
                \pgfmathsetmacro\H@tpp@dc{ 1/#3}
                \coordinate (vp-r) at (#1,#2,#3);
            \fi
        },
        coordinate/.code args={#1,#2,#3}{
           \pgfmathsetmacro\tpp@x{#1} %<- Max' fix
            \pgfmathsetmacro\tpp@y{#2}
            \pgfmathsetmacro\tpp@z{#3}
        },
}

\tikzset{
    view/.code 2 args={
        \pgfmathsetmacro\rot@main@theta{#1}
        \pgfmathsetmacro\rot@main@phi{#2}
        % Row 1
        \pgfmathsetmacro\H@rot@aa{cos(\rot@main@phi)}
        \pgfmathsetmacro\H@rot@ab{sin(\rot@main@phi)}
        \pgfmathsetmacro\H@rot@ac{0}
        % Row 2
        \pgfmathsetmacro\H@rot@ba{-cos(\rot@main@theta)*sin(\rot@main@phi)}
        \pgfmathsetmacro\H@rot@bb{cos(\rot@main@phi)*cos(\rot@main@theta)}
        \pgfmathsetmacro\H@rot@bc{sin(\rot@main@theta)}
        % Row 3
        \pgfmathsetmacro\H@m@ca{sin(\rot@main@phi)*sin(\rot@main@theta)}
        \pgfmathsetmacro\H@m@cb{-cos(\rot@main@phi)*sin(\rot@main@theta)}
        \pgfmathsetmacro\H@m@cc{cos(\rot@main@theta)}
        % Set vector values
        \pgfmathsetmacro\vec@x@x{\H@rot@aa}
        \pgfmathsetmacro\vec@y@x{\H@rot@ab}
        \pgfmathsetmacro\vec@z@x{\H@rot@ac}
        \pgfmathsetmacro\vec@x@y{\H@rot@ba}
        \pgfmathsetmacro\vec@y@y{\H@rot@bb}
        \pgfmathsetmacro\vec@z@y{\H@rot@bc}
        % Set pgf vectors
        \pgfsetxvec{\pgfpoint{\vec@x@x cm}{\vec@x@y cm}}
        \pgfsetyvec{\pgfpoint{\vec@y@x cm}{\vec@y@y cm}}
        \pgfsetzvec{\pgfpoint{\vec@z@x cm}{\vec@z@y cm}}
    },
}

\tikzset{
    perspective/.code={\pgfkeys{/three point perspective/.cd,#1}},
    perspective/.default={p={(15,0,0)},q={(0,15,0)},r={(0,0,50)}},
}

\tikzdeclarecoordinatesystem{three point perspective}{
    \pgfkeys{/three point perspective/.cd,coordinate={#1}}
    \pgfmathsetmacro\temp@p@w{\H@tpp@da*\tpp@x + \H@tpp@db*\tpp@y + \H@tpp@dc*\tpp@z + 1}
    \pgfmathsetmacro\temp@p@x{(\H@tpp@aa*\tpp@x + \H@tpp@ab*\tpp@y + \H@tpp@ac*\tpp@z)/\temp@p@w}
    \pgfmathsetmacro\temp@p@y{(\H@tpp@ba*\tpp@x + \H@tpp@bb*\tpp@y + \H@tpp@bc*\tpp@z)/\temp@p@w}
    \pgfmathsetmacro\temp@p@z{(\H@tpp@ca*\tpp@x + \H@tpp@cb*\tpp@y + \H@tpp@cc*\tpp@z)/\temp@p@w}
    \pgfpointxyz{\temp@p@x}{\temp@p@y}{\temp@p@z}
}
\tikzaliascoordinatesystem{tpp}{three point perspective}

\makeatother

\begin{document}
\tdplotsetmaincoords{70}{-70}
  \begin{tikzpicture}
  \pgfmathsetmacro{\boxX}{0.3}
  \pgfmathsetmacro{\boxY}{0.4}
  \pgfmathsetmacro{\boxZ}{1.2}
  \begin{scope}[scale=6,font=\sffamily,
  view={\tdplotmaintheta}{\tdplotmainphi},
            perspective={
                p = {(4,0,1.5)},
                q = {(0,4,1.5)},
            }
  ]
  \begin{scope} % top face
   \draw[clip]   (tpp cs:0,0,\boxZ) coordinate (top_front)
  -- (tpp cs:\boxX,0,\boxZ) coordinate (top_right)
   -- (tpp cs:\boxX,\boxY,\boxZ) coordinate (top_back)
   -- (tpp cs:0,\boxY,\boxZ) coordinate (top_left) --
    cycle;
    \shade[inner color = gray!5, outer color=black!50, shading=radial] 
    (top_front) ellipse (3cm and 1.5cm);
  \end{scope}
  % right face  
  \draw   (top_front) -- (top_right) -- (tpp cs:\boxX,0,0) coordinate (bottom_right)
  -- (tpp cs:0,0,0)  coordinate (bottom_front) --   cycle;
  % top face  
  \draw   (top_front) -- (top_left) -- (tpp cs:0,\boxY,0) coordinate (bottom_left)
  -- (bottom_front) --  cycle;
  \end{scope}
\tdplotsetmaincoords{70}{-90}
  \begin{scope}[xshift=6cm,scale=6,font=\sffamily,
  view={\tdplotmaintheta}{\tdplotmainphi},
            perspective={
                p = {(4,0,1.5)},
                q = {(0,4,1.5)},
            }
  ]
  \begin{scope} % top face
   \draw[clip]   (tpp cs:0,0,\boxZ) coordinate (top_front)
  -- (tpp cs:\boxX,0,\boxZ) coordinate (top_right)
   -- (tpp cs:\boxX,\boxY,\boxZ) coordinate (top_back)
   -- (tpp cs:0,\boxY,\boxZ) coordinate (top_left) --
    cycle;
    \shade[inner color = gray!5, outer color=black!50, shading=radial] 
    (top_front) ellipse (3cm and 1.5cm);
  \end{scope}
  % right face  
  \draw   (top_front) -- (top_right) -- (tpp cs:\boxX,0,0) coordinate (bottom_right)
  -- (tpp cs:0,0,0)  coordinate (bottom_front) --   cycle;
  % top face  
  \draw   (top_front) -- (top_left) -- (tpp cs:0,\boxY,0) coordinate (bottom_left)
  -- (bottom_front) --  cycle;
  \end{scope}
  \end{tikzpicture}
\end{document}