Ellipsis is not a term much used in modern syntax or grammar. It's not nearly specific enough, since it refers only to something being "left out" of the sentence, without specifying what it was, where it was, how big it was, or why it got left out. Linguists don't use it because it's handwaving, using a general term for a lot of different phenomena.
There are dozens of rules that could be (and often are) called "ellipses", but modern syntax requires specifics of each deletion type and the conditions it obtains under. For instance, the absence of you in
can be called "ellipsis", but it's an ellipsis that (a) only occurs in imperatives and (b) only occurs with second-person subjects. It's not just any old ellipsis; it has constraints. They all do.
There's Conversational Deletion, which produces
- Ever been to Miami?
- Not exactly what I was hoping for.
under certain circumstances. Under other circumstances, Whiz- Deletion omits a Wh- word (or that) subject, and its following be auxiliary from a relative clause, converting, for instance,
- the man who was standing there to the man standing there, and
- the book that is on the landing to the book on the landing.
Or there is rule-based deletion of individual function words like to-deletion that is obligatory with modal auxiliaries and sense verbs
- *They may/should/can/must to arrive at 5, but They may/should/can/must arrive at 5.
- *I saw/heard/felt him to slam the door, but I saw/heard/felt him slam the door.
or the for that often occurs along with to in infinitives
- For him to say such a thing is surprising.
- They didn't expect (for) him to say such a thing
... and so on. You can find lots more in this list of English transformational rules.