[Questions] Modeling a Baseball Pitch Trajectory in 3D Space

[lavendersmell]

TL;DR Summary

I am attempting to model the trajectory of a baseball pitch with spin in 3D. I am looking to find the location at a specific point in time during the pitch, and build the model knowing the pre-determined start/end points of the pitch.

I am currently taking some time off of college (I am a sophomore), and I'm trying to continue coding and experimenting with Calculus-y math as I'll be going into Calculus II and then III when I go back. I am currently trying to develop a 3D baseball pitch visualizer for my own purposes. I am not sure if this is better suited in the homework help forum, if so, I will gladly post there (though this is not homework).

I have taken introductory physics, but my knowledge doesn't run too deep, but it is the branch of science I am most passionate about, so I'm trying to understand this better (as a comp-sci major who wants to go into Aerospace).

I have primarily been referencing this webpage for the physics/math of pitching: https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/curveball-trajectory/. Here's a quick summary:

The ball move along a flight path than can be described as an arc segment. To get the radius of that path:

Magnitude of force acting on ball: $$F = Cl \cdot 4 / 3 \cdot (4 \pi^2 \cdot r \cdot s \cdot V \cdot b^3)$$
where: $\text{ Cl = lift coefficient, r = density of air, s = ball spin, V = ball velocity, b = ball radius}$

Radius of curvature of flight path: $$R = V^2 \cdot a$$
where: $\text{a = acceleration, a = F/m by Newton's second law}$

Combined equations: $$R = (3mV) / (16 \cdot Cl \cdot r \cdot s \cdot b^3 \cdot \pi^2)$$
where: $\text{F = force on acting on ball, m = mass of baseball}$

And for the total ball deflection (Y$_d$) along the flight path:

$$R^2~ – ~D^2 = (R -Y_d)^2$$
$$\sqrt{R^2~ – ~D^2} = R - Y_d$$
$$Y_d = R ~–~ \sqrt{R^2~ – ~D^2}$$

Here are my questions:

1. Does the methodology on this page account for 3D space, or just 2D? If so, how can you plot these 2D arc segments in 3D? To plot in 3D, would I simply apply the force of gravity on the vertical axis? Or does the force acting on the ball also affect the trajectory in that direction? Some pitches do drop faster than others (I believe with pitches backspin do), but I'm unsure if this is just a result of the slower pitch velocities in breaking balls.
2. With that information, how does one get a specific point in time along the arc segment? In other words, I want to obtain the X, Y, and Z position of the ball at a specific point in time. Would I use arc length to accomplish this, and then parameterize it? If so, how would I begin finding the arc length of the segment?
3. Lastly, how (generally) can I work backwards if I already know the start AND end positions of the arc? In other words, how can I build the arc/determine the radius if I already know the exact starting and end points? I was thinking this would be simple, but I cannot wrap my head around it.

I am sorry if any of my questions are not clear. I am NOT asking anyone to solve this for me -- I truly want to understand the underlying math here, though any tips, hints, and/or links to resources would be appreciated.

Thank you!

 

[Baluncore]

Start with Magnus effect.
https://en.wikipedia.org/wiki/Magnus_effect
The lift of a spinning sphere is a fundamental of aerodynamics.

For a 3D simulation you must have a 3D position, a 3D velocity, with a 3D oriented spin axis and rate of rotation. Will there also be a wind velocity in the x-y plane?

Modelling a trajectory from the start point A to a specified point B, a straight line distance r, will be difficult. I suggest you launch, at an elevation angle, from the origin into the x-z plane. Once the ball has travelled a distance of r, and you have computed the change in y, due to spin or wind etc, you can rotate the trajectory into your 3D space. If needed, change the launch elevation to get a better gravity dependent z value at the destination.

I expect you will end up with a 3D matrix solution.