grammar - Reducing the adverbial particle and the NP1 and "to be"

Preparing the rats for the dissection session, they came across a queer phenomenon.

I think it was this before the ellipsis:

While they were preparing the rats for the dissection session, they came across a queer phenomenon.

Here the adverbial particle while and the NP1 which is exactly the NP1 of the main sentence and a form of the verb be has been omitted. Could you give me a link which explains this kind of ellipsis and the like?

Another instance:

Obviously sad, she entered the room.

which has been:

While she was obviously sad, she entered the room.

Answer

This isn't an ellipsis. An ellipsis is when there is omission that is implied from context (usually it's a repeated part that is omitted), e.g.

Should I call you, or you me? = Should I call you, or [should] you [call] me?

Your examples on the other hand are phrases that describe the subject.

e.g.

Obviously sad, she entered the room.

Here, "obviously sad" is an adjective phrase that describes her. You can rephrase to something longer, but that's not the same as an ellipsis.

Here's your other example:

Preparing the rats for the dissection session, they came across a queer phenomenon.

Again, "preparing the rats for the dissection session" is a phrase describing them. In this case it's a participle phrase rather than an adjective phrase because it contains a verb (preparing), but it functions the same way: describing the subject of the main clause.