[Pisces]

Your document with formulas seem entirely correct. I just found a spot where you are cutting corners a bit in explaining why the numbers are as you state. Bottom of page 4, just after the ATI calculation in the example. Otherwise, nice job.

And since our initial course is not directly to the target, we find the course to the target as 21◦

It's true, but you do not show how that is found based on the data given in the example. Likely you will be losing a lot of people here. As the earlier used drawing of the intercept triangle does not match this setup. The drawing assumes the submarine is pointed to the target directly. Only when the situation drawing is recreated based on the given information can the reader determine that it is not in this example. Which I would suggest to readers to do anyway anyhow!!!

... You can find this by using the protractor on the map (if playing Silent Hunter) or by adding the bearing to target to your current course and subtracting 360◦ if the result is greater than 360◦.

Again true, but only on the assumption that you actually mean the relative bearing to the target. As given by the periscope dial or crew reports. (I suppose it can be considered obvious, as the true bearing is the one that we are trying to figure out here) But that one (relative bearing) isn't given in the initial data. Also, as intercepts are usually starting from beyond visual range, the AOB also cannot be determined visually. So using AOB and relative bearings is bit of an inappropriate example to explain the most frequent beyond-visual-range cases.

To solve this anyway, work your way back from the target course, via the AOB, from the target point of view, to the submarine point of view:

The relevant data given is your current course, targets current course and AOB. To calculate the course(true bearing) to target the starboard AOB needs to be added to the target course (port-side deducted). That gives the course/bearing to you from the target's point of view. To get the course from you to the target you add or subtract 180 degrees. (whichever one keeps you between 0 and 360)

target course + stb. AOB, or target course - port AOB; then +/- 180 degrees:

90+111=201 degrees (you are at 201 degrees from the target's position)

201-180= 21 degrees (course to target from your position)

I rechecked the calculations and the calculations for the time to intercept was wrong. It should all work now.

It is valid for where you can intercept (if you can't, the math *may* still work out) and for when you are in a position that the angle on the bow is less than 90 degrees. Again, if you are in a position, relative to the target that this isn't the case, the math on a calculator will work out, but, it will be in error.

For those of you interested in why, it has to due with the domain on the arcsin function. It is restricted to [-pi/2, pi/2]. I will spend more time trying to define this and explain it better in the documentation, but for now, in most cases it works as a good rough estimate for an intercept course, especially once you've closed to visual range and you need an exact course to travel to intercept the target.

I wouldn't worry about the sin function doing funky business if angles out side of 0-90 degrees are entered. Angles up to 180 degrees would still give positive values. ATI can only be negative if the AOB is given as a negative angle between 0 and -180. Which theoretically can indicate port vs starboard, but is never done in the context of the game(s). Nor have I ever seen AOB be reported as negative.

The intercept cannot happen when the submarine speed is less than target speed*sin(AOB). You might come closer if he is approaching (AOB < 90), but never enough to 'ram'. In order to gain in on a target that is moving away (AOB=90 to 180) the submarine must be going faster than the target speed. No other way around that. That is important to know to even consider the intercept.