Reference system with all grid lines being great circles

I was thinking of a problem where it might be easier if instead of latitude (and parallels) we would have a system with an additional east/west pole and great circles through them would provide a second coordinate to go along with longitude.

Was thinking surely this has been done before, but I'm missing a name for such a reference system.

Is this known/used and by what name so I can read more?

edit: a sketch as requested:

• Can you provide a sketch of what you imagine?
– Erik
Nov 25, 2020 at 15:44
• Did you read this? gis.stackexchange.com/questions/366890/… It does not answer your question, but for the reasons explained there, I doubt that anyone has seriously conceptualized such a coordiante system. Nov 26, 2020 at 13:53
• @babel I'm thinking in the opposite direction of that question, not make lines of longitude parallel but the other wide around replace lines of equal latitude with great circles. Though both aim to increase symmetry between the two coordinates. Nov 26, 2020 at 15:42
• It's just that I simply have difficulties to image what benefit such a system would have, what would be the advantage of it. For that I'm not sure if such a system was ever conceived. But curious to learn... Nov 26, 2020 at 15:50
• You would run into a number of issues with such a system. For instance, consider the great circle arc from your fictitious East Pole to the North Pole. Any point on that arc would be ambiguous, because their longitude would be the same, and their 'pseudo-latitude' would also be the same. There would be no way to define a point on that arc unless you resort to normal latitudes. Nov 26, 2020 at 17:25