Spiral on a truncated cone

Question

I need to plot a spiral around an inverted truncated cone, starting and ending at the centre of both bases. Something like this:

I've searched and found a similar solution, but for cylinders only: Helix on a cylinder

I couldn't adapt it to my geometric shape in question, so any help would be appreciated.

The code for the cone only:

\documentclass[border=0.125cm]{standalone}
\usepackage{tikz}
\usetikzlibrary{3d,calc}

\begin{document}

\begin{tikzpicture}

    \draw (-1,0) arc (180:370:2cm and 1cm);
    \draw (-1,0) arc (180:10:2cm and 1cm);
    \draw (0,-3) arc (180:360:1cm and 0.5cm);
    \draw[dashed] (0,-3) arc (180:0:1cm and 0.5cm);
    \draw(0,-3)  -- (-0.97,-0.18);
    \draw(2,-3)  -- (2.97,-0.18);

    \shade[left color=blue!5!white,right color=blue!60!white,opacity=0.3] (-0.97,-0.18) arc (190:350:2 and 1) -- (2,-3) arc (360:180:1cm and 0.5cm) -- cycle;
    \shade[left color=blue!5!white,right color=blue!60!white,opacity=0.3] (1,0) ellipse (2cm and 1cm);

    \draw[dashed] (0.7,-3) arc (180:0:0.3cm and 0.15cm);
    \draw[dashed] (0.7,-3) arc (180:360:0.3cm and 0.15cm);

\end{tikzpicture}

\end{document}

Thanks in advance,

Dinis Nunes

EDIT:

Following Ignasi suggestion, I've added a spiral function, but my goal was making the line follow closely the wall of the cone.

Code:

\node[xshift=1cm,yshift=-3cm] (dummy) {};
\draw[densely dotted,scale=0.09,variable=\x,shift=(dummy)] plot[domain = 0:10*pi, samples = 400] ({\x*0.6*sin(0.3*pi*deg(\x))},{\x},{\x*0.6*cos(0.3*pi*deg(\x)});

Answer 1

Here's one possibility using pgfplots; the spiral is drawn first and used to draw the truncated code:

The code:

\documentclass[dvipsnames,border=5pt]{standalone}
\usepackage{pgfplots}
\usetikzlibrary{calc}
\pgfplotsset{compat=newest}

\colorlet{myblue}{blue!30}

\def\Pointi{42.67}
\def\Pointii{12.6}

\begin{document}

\begin{tikzpicture}
\begin{axis}[
 view={-10}{-10},
 axis lines=none,
 zmax=60,
 clip=false,
 height=16cm,
 width=15cm,
 xtick=\empty,
 ytick=\empty,
 ztick=\empty
]
\addplot3+[
  draw=none,
  no marks,
  domain=10.3:14.405*pi,
  samples=400,
  samples y=0,
]
  ({5*x*sin(0.20*pi*deg(x))},{5*x*cos(0.20*pi*deg(x)},{x});

\coordinate (br)
  at ({5*\Pointii*sin(0.20*pi*deg(\Pointii))},{5*\Pointii*cos(0.20*pi*deg(\Pointii)},{\Pointii});
\coordinate (ur)
  at ({5*\Pointi*sin(0.20*pi*deg(\Pointi))},{5*\Pointi*cos(0.20*pi*deg(\Pointi)},{\Pointi});

\coordinate (BR) at ( $(ur)!1.113!(br) $ );
\coordinate (UR) at ( $(br)!1.24!(ur) $ );

\path
  (BR) 
  arc [start angle=0,end angle=360,x radius=38pt,y radius=10pt]
  coordinate[midway] (BL);
\filldraw[ball color=MidnightBlue!20,fill opacity=0.7,draw=black]
  (UR) 
  arc [start angle=0,end angle=360,x radius=204pt,y radius=20pt]
  coordinate[midway] (UL);
\filldraw[ball color=MidnightBlue!20]
  (UR) --
  (BR) arc [start angle=360,end angle=180,x radius=38pt,y radius=10pt] --
  (BL) -- 
  (UL)
  arc [start angle=180,end angle=360,x radius=204pt,y radius=20pt];
\draw[dashed]
  (BR) 
  arc [start angle=0,end angle=180,x radius=38pt,y radius=10pt];
\draw[dashed]
  ([xshift=-19pt]BR) 
  arc [start angle=0,end angle=360,x radius=19pt,y radius=4pt];
\addplot3+[
  mark=none,
  ultra thick,
  BrickRed,
  domain=10.3:14.405*pi,
  samples=400,
  samples y=0,
]
  ({5*x*sin(0.20*pi*deg(x))},{5*x*cos(0.20*pi*deg(x)},{x});
\end{axis}
\end{tikzpicture}

\end{document}