I was searching in online resources for the correct parametrization of an Spherical Spiral over a sphere or radius $r$. The definition of Spherical Spiral that I have is an Helix restricted to move over the surface of a sphere. (Note that I am not looking for the parametrization of the Loxodrome, the line that goes from North Pole to South Pole having the same angle with respect to all meridians)
According to Wikipedia for a number $c > 1 \in \mathbb{R}$ that counts the half circle turns of the spiral, the parametrization is the following:
$$
\begin{split}
x(\theta) &= r \sin{\theta} \cos{c\theta} \\
y(\theta) &= r \sin{\theta} \sin{c\theta} \\
z(\theta) &= r \cos{\theta}\\
\\
\theta &\in [0,\pi]
\end{split}
$$
However, according to Emre's Answer of this question, the parametrization is the following
$$
\begin{split}
x(t) &= r \sqrt{1-t^2} \cos{a \pi t} \\
y(t) &= r \sqrt{1-t^2} \sin{a \pi t} \\
z(t) &= r t\\
\\
t &\in [-1,1]
\end{split}
$$
If I make the transformation $t = \cos{\theta}$, this equations become:
$$
\begin{split}
x(\theta) &= r \sin{\theta} \cos{\left( a \pi \cos{\theta} \right)} \\
y(\theta) &= r \sin{\theta} \sin{\left( a \pi \cos{\theta} \right)} \\
z(\theta) &= r \cos{\theta} \\
\\
\theta &\in [0,\pi]
\end{split}
$$
Which implies the parameters $a$ and $c$ are related as $c = a \pi$. However, this two parametrizations are not the same curve. We can see the difference by plotting them for $r=1$, $c=14$:
Looking at $x(\theta)$ and $y(\theta)$ values:
¿Which parametrization is the accurate one? I trust more the Wikipedia parametrization as it goes accordingly to my definition, but the other parametrization makes me wonder.
I tried to review the source of Wikipedia's parametrization but it is a German textbook with no translation.
Best Answer
Wikipedia's spiral: $\phi$ is parametrized proportional to $\theta$.
Emre's spiral: $\phi$ is parametrized proportional to the height $t=\cos\theta$ of the sphere.
Which one is the one you want depends on how you define "restricted". For Emre's answer, you may think of it as restricting a helix horizontally to the sphere. For example, if $p$ is a point on a normal helix, and does not lie on the sphere, then you move it horizontally towards the vertical axis of the sphere so that it lands on the sphere.
The case in the Wikipedia page is hard to describe intuitively, but apprently different from Emre's answer. (Probably along the radial direction, but I am not sure.)