Uniform $\lambda $ spaced circles of unit radius through origin are given by
$$ (x-\lambda)^2 +(y-\lambda)^2 =1$$
include a net of constant curved differential length rhombuses.
What is the equal area mapping of such unit circles onto a lune of unit sphere spherical coordinates latitude/longitude
$$ 0<\theta<\pi/2,\; -\pi/2<\phi<\pi/2 ?\;$$
They both have an area equal to $ \pi 1^2.$
Thanks for any references / answer.