Preferred developing language

I say you should stick to Processing.js. It is quite capable of handling 2-D games, and makes it fairly easy to add to websites. Who wants to download and install a 2-D game, when it could run instantly in their browser? And browser game websites like Kongregate would be happy to host it, and you would get some ad revenue.

Of course, Unity games can run in a browser as well.

Instead of being set in stone, ships and other vehicles will be haphazard combinations of skeletons and modules.

pretty simple, but with enough complexity to force me to work hard at it.

Whatever you say… 😛

If you need a few tips on the Physics engine, I have already created one with Processing.js on Khan Academy.

EOC flight demo

The code is outdated compared to what I have now, but the basic OOP and trigonometry is correct.

The link still doesn’t work for me. Perhaps something in your account settings does not let other people view your programs? I’m not sure. Perhaps if you give me the link to your profile, rather than a specific program, that will work.

I know what your problem is. Here is the code starting at line 172. I added the rotation variable and the two println() statements for debugging purposes.
if (char.inship === true && control.occupied === true) {
char.a = control.a;
var rotation = atan2(mouseY-width/2,mouseX-height/2)-90;
println(rotation);
println(control.a);
if (keys[SHIFT]) {
if (control.a < atan2(mouseY-width/2,mouseX-height/2)-90) { control.a += control.agility; } if (control.a > atan2(mouseY-width/2,mouseX-height/2)-90) {
control.a -= control.agility;
}

}
Plug this code into your program and you will see the problem. At the left hand side, the rotation variable switches from 270 to -90, which obviously affects whether control.a is greater than or less than the mouse angle.

To solve this, we need to designate an optimal situation. We can do this in the form of an IPO (input, processing, output) chart. The inputs are the ship rotation, and the mouse rotation to the center of the screen. I think the desired output should be the angle between these two angles. When the ship is facing the mouse, the output should be 0. As the mouse moves clockwise around the ship, the angle should increase, reaching 180 behind the ship. As the mouse moves counter-clockwise around the ship, the angle should decrease, reaching -180 behind the ship. The switching point between positive and negative angles should always be behind the ship.

The final stage is the processing stage. How do we get from input to output? First, we define a variable for each input. We already have control.a defined, so lets add var rotation = atan2(mouseY-height/2,mouseX-width/2); for the mouse angle.

Now, let’s say the ship is rotated 100 degrees, and is facing the pointer. Both input variables will be the same. If we subtract control.a from rotation, the angle will be 0 whenever the ship is facing the pointer.
var rotation = atan2(mouseY-height/2,mouseX-width/2)-control.a;
Now, not only is rotation 0 when the ship is facing the mouse, but the angle increases as the mouse moves clockwise around the ship, and decreases when the mouse moves counterclockwise around the ship, reaching -180 behind the ship. However, the number continues to decrease beyond that point, until it switches to a positive number at the left.

To fix this, all we have to do is add 360 to var rotation when rotation is less than -180. At the same time, we should subtract 360 when rotation is greater than 180.
if(rotation >= 180) {
rotation -= 360;
}else if(rotation < -180) { rotation += 360; }
I originally used a while loop instead of an if statement, because the manual rotation of the ship allows control.a to reach values much higher than 180, 360, or even 1000. However, I then discovered this section of code in your program:
if (control.a > 360) {
control.a = 0;
}
This solves the problem if control.a reaches high values, but not if it reaches low values. So I would suggest changing it to this:
if (control.a >= 360) {
control.a = 0;
}
if (control.a < 0) { control.a = 360; }

Edit: I forgot to mention, you need this code to utilize the output.
if (keys[SHIFT]) {
if(rotation > 0) {
control.a += control.agility;
}
if(rotation < 0) { control.a -= control.agility; } }

Now, everything should be working properly.

For clarity, here is the entire fixed program: program.
I hope this helps!

P.S. Because var rotation is larger the further your mouse angle is from the angle of the ship, you could tell the ship to rotate faster or slower as well.

P.P.S. The ship will vibrate around its focal point. This is because var rotation is not being updated when control.a is being updated during the execution stage (the code shown in my edit). The code below fixes the problem, albeit not very efficiently.
if (keys[SHIFT]) {
if(rotation > 0) {
control.a += control.agility;
}
var rotation = atan2(mouseY-height/2,mouseX-width/2)-control.a;
if(rotation >= 180) {
rotation -= 360;
}
if(rotation < -180) { rotation += 360; } if(rotation < 0) { control.a -= control.agility; } }