TIling surface of a closed shape with a parallelogram

Hello,I have a shape of a capped cylinder. So it essentially consists of three parts: two sphere halves and a cylinder between them merged into one surface. I need to start tiling it with a parallelogram with different sides, say 9 and 10 mm. The key requirements are: (1) I would to start this tiling from a very specific position: the first position should sit at the center of that shape (intersection of two diameters), (2) the next level should start with the parallelogram offset from the previous one by half the length. I tired to sketch it on the surface. Any suggestions would be appreciated.