Of course you wouldn't mind which is the point of origin, given there is no true longitudinal zero for the fake sphere.

That's completely irrelevant. Longitudinal datum is indeed entirely arbitrary - a historic debate won by the British. But that's equally true of the monopole FE map, which is, I'm assuming, your preferred model? You and Tom have both agreed that lat/long coordinates work just fine on FE, so where's the problem there?

Origins are generally arbitrary things. Your formula, using cartesian coordinates, would return the same distance between two places regardless of where the origin was placed. It could be one of the two places, or somewhere entirely different - it would all cancel out.

Your problem isn't the origin. Your problem is you've triumphantly presented a formula for cartesian coordinates but you've been given lat/long, measured in degrees. So plugging degrees into a formula designed for distances isn't going to work. You need some way of converting the two, especially since you and Tom keep saying how simple it is and how, for example, mariners have been navigating that way for ages. Well, ok...but they've been using lat/long for centuries, and very happily calculating distance between points. So you need to explain how that could be possible on a FE.

If you are going with the monopole model, what you effectively have in a lat/long is a set of polar coordinates. These are normally a distance and an angle, of course, but on the monopole FE that's kind of what you have. Longitude makes sense on either RE or FE. Latitude though can't be an angle on the monopole FE. So you need some way of converting degrees latitude to a distance from north pole - the origin of your polar coordinate system. I don't know what you want to use for this conversion because you guys never really say - I guess 1nm per minute of latitude, as per conventional navigational thinking? That would give a radial distance for the FE of 180 * 60 = 10,800 nautical miles, which is pretty close to the Wiki's estimate for the radius.

Assuming you're happy with that conversion, you should just be able to plug in any lat/long to a polar/cartesian coordinate converter and, hey presto, out will come the x/y pairings you need to plug into your formula. Then we can get some distances out of you.

With me so far?