On which 2D surfaces in 3D can a Kagome lattice pattern be drawn using three sets of parallel lines"?

Question

In the background/introduction to my Space SE question Have Kagome lattice patterns been used as structural reinforcement in spacecraft in non-Iranian spacecraft? Can we help Scott Manley "unsee" this one? I wanted to highlight that the Kagome-like cross-hatched pattern seen there can be uniform and repeating on the cylindrical surface of a rocket body.

So to illustrate that I said:

On a flat or cylindrical surface the pattern has three sets of parallel lines, call them A, B, and C, but instead of all three intersecting at the same points, they are offset so that AB, BC, and CA intersections are at different points, dividing the surface into twice as many triangles as hexagons.

Question(s):

1. Can a Kagome lattice or trihexagonal tiling be made from three sets of parallel lines on a flat and/or cylindrical surface?
2. Are there other 2D surfaces in 3D space on which it can be made from three sets of parallel lines?

notes:

1. "can be made" doesn't mean will always be, just that it is possible.
2. corrections of vocabulary are just as welcome as those of mathematics. For example I may be conflating lattice with tiling

Answer 1

1. Can a Kagome lattice or trihexagonal tiling be made from three sets of parallel lines on a flat and/or cylindrical surface?

I think we already have the flat case. In the case of the cylinder, the wrapping preserves parallel lines, but we would be restricted to cylinders where the lattice's dimensions are an integer multiple of the circumference, since the lines would have to wrap around and meet again.

2. Are there other 2D surfaces in 3D space on which it can be made from three sets of parallel lines?

Any surface where parallel lines in more than one direction do not meet each other would be possible. That includes flat planes, cylinders, as well as other surfaces which are curved in only one dimension, such as $z=x^2$. You could not make this lattice on a sphere for example.

The technical term for a surface like this is a developable surface, a surface which has zero Gaussian curvature.

However, like the cylinder, in each case the boundary conditions may be difficult, such as for a cone.