The ārūḍha-pada system is one of Jaimini's most-distinctive contributions and one of the most under-utilized in modern practice. The idea: for each bhāva, compute a 'projected-image' rāśi using a specific rule. The rule for any bhāva-N: count from N to the rāśi occupied by N's lord, then count the same distance from that lord-rāśi forward. Two exceptions: if the result is the bhāva itself OR the 7th from itself, take the 10th from the result instead.
The most-important ārūḍha is the ārūḍha-lagna (AL) — the ārūḍha of the 1st bhāva. AL represents the projected-self, the public image, how the world sees you (vs. the lagna which represents who you actually are). When AL and lagna are the same rāśi (or close), self-perception and public-perception align; when they're far apart, they diverge.
The upapada-lagna (UL) is the ārūḍha of the 12th bhāva and is the primary marriage-indicator in Jaimini's system (Adhy. 2 · Pāda 1). UL combined with the 7th-from-UL gives spouse-quality indications independent of the 7th-from-lagna analysis. This is Jaimini's parallel-track to Pārāśara's 7th-house treatment.