The meridian is that of a hyperbolic pseudo-sphere.
One set of lines is of Geodesics.
The other set is orthogonal to Geodesics, that I can call Involutes. Involutes have an interesting property of constant minimum distance between each member of the Involute set. They are also called geodesic parallels.
I cannot recall the reference, it was first called as Involutes by Leibniz. If in 2D around a circle are marked say 10 equal parts on its perimeter and Involutes drawn by taut string, you can see 10 parallel curved lines separated by one tenth of perimeter.
This comes out in the treatment by geodesic polar co-ordinates on surfaces of revolution in differential geometry.
To verify that there in fact would be no gaps even in 3D, between neighbouring pipes, I undertook 3D printing.... If interesting to anyone, I shall upload one more picture.