Several months ago I helped somebody with formulae for Lunettes to be used to construct an approximately spherical object out of 20-30 strips of thin veneer.

Consider a series of equilateral polygons inscribed inside circles of constant latitude. Also consider equilateral polygons circumscribed around circles of constant latitude. If you imagine the shape generated by the polygons, it should approximate a sphere if the polygons have 20-30 (or more) sides.

I came up with the following formulae for polygons with N equal sides (not difficult to figure out).

InnerSide = 2*Radius*Sin(Pi / N)*Cos(Latitude)
OuterSide = 2*Radius*Tan(Pi / N)*Cos(Latitude)
Ark = 2*Radius*(Pi / N)*Cos(Latitude)

Ark is the arc of a circle of constant latitude. Any of the above formulae could be used to calculate widths for the desired Lunettes. It is my guess that the Ark formula is likely to be the best approximation of the three, but the other two formulae are guaranteed to result in developable surfaces. For those not familiar with the term, cylinders & cones are developable surfaces because local geometry on these surfaces is exactly equivalent to plane geometry. Such a surface can be straightened out and made into a piece of a plane without tearing or stretching it.

Does anybody know the formula for the arc of a great circle which connects two points at the same latitude?

For Latitude = 0 (the Equator), the above formula is correct, but what is the formula for other latitudes? I think that using such a formula might be more likely to result in a developable surface. In practice, I do not think it matters, but in theory, a developable surface is more correct.