Coriolis Drift of Discrete Objects

[Sooty]

I am writing a program aimed at 'gun nuts' designed to display Coriolis Drift of bullets (ie, 'discrete objects in free-fall', and not large fluid masses). Using the 2 equations below, I am able to calculate and display the values of Coriolis Drift (in terms of X & Y (vertical and horizontal)) as seen from a Horizontal Plane on the Earth's surface for all cases of Latitude and Direction, (the second 2 essential input terms).
These are my 2 equations:-

X:= T *Range * Sin(Lat) * Omega

Y:= Sin(Dir) * T * Range * Cos(Lat) * Omega;

Where:-
X = Horizontal Drift
Y = Vertical Drift
T = Time to Target
Range = Distance from gun to Target
Dir = Direction/Azimuth of fire in 'compass degrees' - 0 to 360
Lat = Latitude in degrees - from North (+90degrees) to South(-90degrees)
Omega = a Constant for Earth's Rotational Velocity

Now, I am attempting to extend my Coriolis Drift equations to address the same question for all values of Latitude and Direction beyond a Horizontal Plane (as above), to include Elevation above and below the Horizontal (ie, shooting uphill and downhill, +90deg/-90deg).

In short, what I am looking for is the necessary modifications/additions to the above equations (in a form that can easily be implemented in a Pascal program, hopefully using no more exotic math than basic Trig functions!) necessary to take the 7 terms above, plus:-

Elevation of fire from Vertically Up(+90degs) to Vertically Down(-90degs)

and return X and Y (Horizontal and Vertical relative to the Earth at the point in question) values for Drift, as seen by a viewer looking through a scope mounted on a rifle delivering the shot as described.

Thank you!

 

[FactChecker]

I have always tracked trajectories by simulating the motion in very small steps. But those simulations had so much going on during the flight that it was completely different from a ballistic trajectory. I do not know if there are usable closed-form solution for your problem.

 

[Sooty]

I have always tracked trajectories by simulating the motion in very small steps. But those simulations had so much going on during the flight that it was completely different from a ballistic trajectory. I do not know if there are usable closed-form solution for your problem.

'Very small steps'... yes, an 'iterative approach'... that is how I have done it in terms of getting figures for complete ballistic trajectories... literally 'finite element' calculations, a metre/meter at a time, and it gives excellent results. My current problem is really 'nit-picking on top of what is already close to being nit-picking'... so I may just have to accept that although there must be a solution possible, (hey, it's 'only geometry'!) I'm not going to find it!

Tnx for your input.

 

[Sooty]

Reviewing my previous question, I believe I can 're-model' it in a way that is 'mathematically equivalent' and will so will allow me to continue with my project.
It addresses the question in terms of Great Circle/Spherical geometry, which although I can find relevant articles on the internet, they present their maths in ways that are waaay beyond my understanding.
So, instead of referring to anything about Coriolis, consider this:

So. Knowing point A on a sphere,
where the position of A is defined solely by Lat, where Lat = Latitude in degrees (as before)

And point B, whose distance from point A is defined (not by actual distance travelled) by the angle Asector, Where Asector = Angle of the sector of a Great Circle path that runs between them.

Now, the direction of point B relative to point A is Dir, where Dir = Direction/Azimuth in 'compass degrees' - 0 to 360 (as measured at point A) along the Great Circle path.

Questions:-
what is the value of Lat of point B?
what is the value of Dir at point A relative to point B on the great Circle path?
- - -
Hopefully, to clarify, if it's not already clear...
2 points on a Great Circle... Lat and Dir for each, obviously different... for my scenario, the distance between the 2 points is defined by the 'included angle' of the 'earth sector' covered, and not by the distance between them on the Earth's surface...