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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 $r=1$, $c=14$:

3Dplots

Looking at $x(\theta)$ and $y(\theta)$ values:

xvals

enter image description here

¿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.

sayeg84
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    It would all depend on how you restrict a helix to "move" on the surface of the sphere, wouldn't it? The "mathexchange" spiral is designed to have the same change in $z$ coordinate for each complete turn, like a helix. That is, it is a cylindrical projection of a helix onto a sphere. The "Wikipedia" spiral has equal change in latitude for each complete turn. It then makes the very silly claim that a loxodrome is not a "spiral" since it doesn't have this property, after showing earlier how spirals can have very different properties from each other. – David K Apr 30 '20 at 03:06
  • @DavidK The Wikipedia entry for "spiral" looks poorly edited – sayeg84 Apr 30 '20 at 14:46
  • I agree. I think it is intrinsically difficult for Wikipedia to organize a page like that really well. There is plenty of useful information there but you have to think about how you can actually use it. – David K Apr 30 '20 at 16:58

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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.)

trisct
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    Wikipedia has an equal change in latitude for each complete turn. It's what you would get if you projected a helix onto the sphere by converting the $z$ coordinate to latitude above or below the equator. – David K Apr 30 '20 at 03:14
  • @trisct thanks for the long explanation. – sayeg84 Apr 30 '20 at 16:08