Tangental surface from sphere onto arch

Hello. I am just having tough time creating this surface made of tangents from spherical surface onto an arch … problem is that i can do it with lines but i want to do it elegantly using some functions so that it is absolutely mathematicaly correct

Image explains everything. I have sphere and arch and i want to create surface of tangents (last tangent at the corner is actually only a point)

The techniques in this tutorial might prove useful:

Hi Juan - so the tangents would all be parallel in one view, correct? I mean, the tangents will determine the shape of the arch…?

This is rough and not a sphere but I think it is the same idea -

Is that what you are after, only making a surface through the lines and a more regular array of lines?
Like so:

Steps are:
Make vertical lines from the edge (Near Osnap, or Divide the edge to mark regular locations.
Match the lines to the edge (Match command, SurfaceEdge option) for Tangency, with Preserve other end as “none” and Perpendicular to edge not checked.

Extend or trim the resulting lines to the target surface or plane.
Loft the lines, or make a lower rail and Sweep2… depends what you want…
Did I guess about right?

@Juan_Gallo - here’s a more organized example.
TangentSrf.3dm (205.3 KB)