Thank you Yma16 for your assistance. If in my original drawing Rm = 22, L =30, x = 11, and gamma = 50 degrees, would you demonstrate your suggested method for finding the length of the geodesic from point A to point B? Thanks so much for your help.
Here's how I would do this.
Your points A and B, as I understand it, are on the cylindrical surface where the two vertical lines intersect the circle in the top view. If we assign coordinates to the horizontal diameter shown (with 0 at the center), the two points on it are at coordinates x (positive as shown) and x-L (negative as shown because L>x).
You have not stated the vertical distance between A and B in the side view, but it can be calculated as z = L tan(gamma).
"Unwrapping" the cylindrical surface to make a rectangle, the horizontal distance between A and B is the arc length on the circle. The angles to A and B are, respectively, arccos(x/r) and arccos((x-L)/R), so the arc length between them is y = R[arccos((x-L)/R) - arccos(x/r)].
Therefore, by the Pythagorean theorem, the distance from A to B on the surface is sqrt(z^2 + y^2), that is,
sqrt(L^2 tan^2(gamma) + R^2 [arccos((x-L)/R) - arccos(x/r)]^2)
For your numbers, I get z = 35.75, y = 34.45, so the distance is 49.65.