How can I mark the preimage of covering map (spiral on circle)?

Question

I'm currently trying to create the following image with TikZ:

I currently have:

But the orange part is missing. How can I get this?

(By the way: This looks much better as PDF ... there seems to be a problem in my toolchain for PNG-creation. But that's not part of this question.)

That's the source:

\documentclass[varwidth=true, border=2pt]{standalone}
\usepackage{amsmath,amssymb}% math symbols / fonts
\usepackage{pgfplots}
\usetikzlibrary{decorations.markings}
\pgfplotsset{compat=newest}

\begin{document}
\begin{tikzpicture}
\draw[->, thick] (7,5) -- (7, 2) node [midway, right] {$p$};
\node at (1,5.5) {$\mathbb{R}$} ;
\node at (6.4,0.7) {$\color{blue} S^1$};
\node at (4.9,1.1){$x$};
\begin{axis}[
 view={-30}{-45},
 axis lines=middle,
 zmax=60,
 height=8cm,
 xtick=\empty,
 ytick=\empty,
 ztick=\empty,
 enlarge y limits=true,
 enlarge x limits=true,
]

\addplot3+[->,ytick=\empty,yticklabel=\empty,
  mark=none,
  thick,
  black,
  domain=0:14.8*pi,
  samples=400,
  samples y=0,
]
({sin(deg(x))},{cos(deg(x)},{x+15});
\addplot3+[ytick=\empty,yticklabel=\empty,
  mark=none,
  thick,
  dotted,
  black,
  domain=-1:0,
  samples=100,
  samples y=0,
]
({sin(deg(x))},{cos(deg(x)},{x+15});

\addplot3+[,ytick=\empty,yticklabel=\empty,
  mark=none,
  thick,
  blue,
  domain=0:14.7*pi,
  samples=400,
  samples y=0,
]
({sin(deg(x))},{cos(deg(x)},{0});

%%%%%%%%%%%%% Point
\addplot3+[
  mark options={color=black},
  mark=*
] 
coordinates {({sin(deg(45)},{cos(deg(45))},0)};
%%%%%%%%%%%%%

\end{axis}
\end{tikzpicture}
\end{document}

Answer 1

This is another case where Asymptote is so much easier:

Here's the code:

\documentclass{standalone}
\usepackage{asymptote}

\begin{document}

\begin{asy}[width=10cm,height=10cm]
import graph3;

usepackage("amsfonts");

size3(200);

currentprojection=orthographic(4,6,3);

// parametrization
real x(real t) {return cos(2pi*t);}
real y(real t) {return sin(2pi*t);}
real z(real t) {return 0.5*t;}
real z0(real t) {return 0;}

scale(true);

// some parameters
real delta = 0.01;
real phix = 0.1;
real phim = 6.7;

// spiral
path3 spiral1 = graph(x,y,z,0.9,1,operator ..);
draw(spiral1,dotted);
path3 spiral2 = graph(x,y,z,1,phim,operator ..);
draw(spiral2,Arrow3);

// blue circle
draw(unitcircle3, blue);

// orange segments
pen sp = orange+1;

draw(graph(x,y,z0,phix-delta,phix+delta,operator ..),sp);
for(real i=1; i<phim; ++i) {
  draw(graph(x,y,z,i+phix-delta,i+phix+delta,operator ..),sp);
}

// the dot x
triple px = (x(phix),y(phix),0);
dot(px);
label("$x$",px,S);

// axes and labels
xaxis3("",red,Arrow3);
yaxis3("",red,Arrow3);
zaxis3("",red,Arrow3);

label("$\mathbb{R}$",(1,-1,4));

draw((-1,1,4)--(-1,1,1),Arrow3);
label("$p$",(-1,1,2.5),E);

label("$S^1$",(-1,1,0),W,blue);
\end{asy}

\end{document}

Answer 2

Well easiness I disagree :P Two minutes of tweaking with QTikZ (by no means a generic answer)

\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.9}
\usetikzlibrary{decorations.markings}
\begin{document}
\begin{tikzpicture}[scale=2,mydeco/.style={decoration={
    markings,
    mark=between positions 0.115 and 1 step 0.2 with { 
        \node at (-2.8mm,0) {(};
        \node at (2.8mm,0) {)};
        }
    }
}]
\begin{axis}[hide axis,height=7cm,width=4cm,view={50}{7},samples y=0,no marks]
\addplot3+[-latex,samples=200,domain=0:10*pi,
    postaction={-,dashed,draw,orange,thick,dash pattern=on 6mm off 3.275cm,dash phase=19.5mm,
    postaction={mydeco,decorate}}
    ] ({sin(deg(-x))},{cos(deg(x))},x);
\addplot3+[blue,dashed,samples y=0,domain=-pi:0] ({sin(deg(-x))},{cos(deg(x))},x);
\end{axis}
\end{tikzpicture}
\end{document}