Curved Longitude Lines - Raytracing Brain Teaser

Here's a brain teaser,

So when you stretch an image of the sky coordinate grid onto a sky dome,
then look at it in a hemispherical view, does anybody know what the
mathematical equations (sin, cos, etc. functions) for the "former
longitudes" (curved "vertical" lines) are on the projected 2D circle? Quite
an odd question, I'm thinking.

Here's an image to get you started, showing half the sky, latitude lines,
"former longitude" lines, ground, and a sun, projected onto a 2D circle: .