Creating a version of conical helix (parametric) about the border of a paraboloid.

Question

Im just getting to know a little more about helix's and i thought it was cool to visualize these helix's in 3D as regular helix's that extend uniformly across the z axis but what was far more interesting was the visualizations of conical as well as spherical spirals. They usually operate by making the radius of the spiral a function of Phi but naturally after examining this i began to wonder what a spiral about the edge of a paraboloid would look like and how you would represent this parametrically. I should start by asking if these equations exist somewhere where i could see them and if not what would they be.

In other-words for a paraboloid with the equations

x(θ,φ) = cos(φ) * θ
y(θ,φ) = sin(φ) * θ
z(θ,φ) = θ^2

what are the equations for a spiral whos radius is about the border of the paraboloid

Answer 1

You can take a whole family of spirals $S_k$ like this, for a given fixed $n$, (on the figure below, there are $n=15$ spirals):

$$\begin{cases}x&=&\cos(t+2\pi k/n) t\\y&=&\sin(t+2 \pi k/n)t\\z&=&t^2\end{cases}$$

(check that $x^2+y^2=z$, proving that the different spirals are on the paraboloid).

The spirals have different colors, from $k=0$ (pure red) to $k=n-1$ (pure blue).

or, in a lateral view:

Matlab code for the figure:

clear all;close all;hold on;axis([-5,5,-5,5,0,25])
[X,Y]=meshgrid(-4:0.01:4);
Z=min(X.^2+Y.^2,26);
surf(X,Y,Z,'edgecolor','none');alpha(0.4);view([14,70])
n=15;t=0:0.005:4.;
for k=0:n-1
    T=t+2*pi*k/n;
    plot3(cos(T).*t,sin(T).*t,t.^2,'k','color',[1-k/n,0,k/n]);
end;

Answer 2

Okay i ended up finding a paper exploring spirals on surfaces of revolutions which describes the derivation for these equations

Equations