As a carpenter/framer/house builder we encountered a similar problem back before personal computers, (but after color tv).
To craft the hand rail out of a steel rod, for a spiral stair case with a diameter of 5’, we wanted to roll a length of 5/8" round steel rod into a circle to establish the handle’s circular shape. Then we could pull up one end to match the height of the spiral stair case, but of course the diameter of the circle would shrink, a lot.
After a long weekend end of ‘special oracle’ consultation someone figured out that it was a comparison in right angled triangles; The height of the stairs and diameter of 5’ were the altitude and base of one right triangle; The height of the stairs and the circumference of the spiral were the altitude and base of the 2nd right triangle. The hypotenuse of the 2nd triangle was the railing length and our new circle’s circumference.
I think what ‘Chris’ is describing at the end of his ramp, is a constantly increasing ratio of what is described above, or has introduced yet another dimension of each upward iteration being also smaller in shape. Wow, NASA stuff.
I’ll bet that water going down a drain is the algebra he’s looking for.