II.off [美 ɔf, ɑf, 英 ɒf]副off often appears as the second element of certain verb structures in English (break off, pay off, take off, etc). For translations, see the relevant verb entry (break, pay, take, etc).
I.on [美 ɑn, ɔn, 英 ɒn]介词on often appears as the second element of certain verb structures in English (count on, lay on, sign on, etc). For translations, see the relevant verb entry (count, lay, sign, etc).
Note that on the "affine" plane, one might push off to a parallel line, so (thinking geometrically) the number of intersection points depends on the choice of push-off.