How to find point on arch to draw vertical line to?

I am lost, or more truthfull! ( an idot ) can some one help me to find the location on a moving arch. So i can draw the vertical line to stop at the arch!
I want to eventualy draw multipal spaced vertical lines within the arch, and to adjust the arch, befor sending to Rhino!

Here is one example. There are many ways. (6.8 KB)

Also here with multiple lines. (9.4 KB)

Thank you so much, You have no idea how many days i have spent with a numb head over this.
I have been all over the lawn, ( grasshopper ) ! trying to fathom this out.
thank you