Path traced by a point on the periphery of the ball? Helix?

[AbeerJoshi]

TL;DR

Path traced by a point on the horizontal periphery of the ball (right or left), when spinned and launched.

So, If we imagine a ball, give it some initial angular velocity ω, and launch it straight up, with a translational initial velocity u.

What will be the path traced by a point on the periphery of the ball (the point on the either of the two extremes horizontally, left or right)?
(air resistance is neglected, weight is not).

I believe, It's going to end up sort of like a compressed helix (somewhat like in the image below...)
Here L(i) is the distance between 2 consecutive loops...
My question here is, what will be the ratio of L1:L2:L3 here? anything unique? an AP? a GP? Do we have to integrate something? (sorry for bad representation, but it was the best I could draw)
Also will the loops formed, be of equal radii? (Because, I believe them to be..?)
I know that ΣL(i)=u^2/2g (since, that is the maximum height achieved by the ball.)
Sorry for any rookie mistakes, I am a high schooler...

 

[Orodruin]

Simply write the position of the point on the perifery as the center of mass motion of the ball (parabolic motion) and the rotation of the point relative to the center of mass (uniform circular motion).

 

[Baluncore]

Welcome to PF.

The point will form a helix with the same radius as the ball.

As the ball slows and then falls back to Earth, the pitch of the helix will change, being proportional to the vertical velocity of the ball.

 

[AbeerJoshi]

Welcome to PF.

The point will form a helix with the same radius as the ball.

As the ball slows and then falls back to Earth, the pitch of the helix will change, being proportional to the vertical velocity of the ball.

Thank you!
But in what way can I find out the ratio of the pitches? What function do I assume to integrate and with regard to what?

 

[Baluncore]

But in what way can I find out the ratio of the pitches?

The x and y coordinates are simply due to the ball rotating against time, t.
x = r⋅Cos(w⋅t)
y = r⋅Sin(w⋅t)
The z coordinate is the height of the ball in its parabolic flight against time.
z = u⋅t+ ½⋅a⋅t²
where a = -9.8 m/s/s; the acceleration downwards due to gravity.

The pitch of the helix is the change in the z coordinate, after each full rotation of the ball.
If you work out the period of w; T = 2π/w, you can compute z for each full rotation.

 

[Orodruin]

If you work out the period of w, T = 1/w, you can compute z for each rotation.

Minor correction: The period is $T = 2\pi/\omega$.

 

[AbeerJoshi]

The x and y coordinates are simply due to the ball rotating against time, t.
x = r⋅Cos(w⋅t)
y = r⋅Sin(w⋅t)
The z coordinate is the height of the ball in its parabolic flight against time.
z = u⋅t+ ½⋅a⋅t²
where a = -9.8 m/s/s; the acceleration downwards due to gravity.

The pitch of the helix is the change in the z coordinate, after each full rotation of the ball.
If you work out the period of w; T = 2π/w, you can compute z for each full rotation.

It was indeed very helpful! Thanks a-lot!