I have a paper to write before class.
I can say this, if I'm not mistaken, this way:
I have a paper which is to be written before class.
What has happened here? What is the name of this?
Another example:
The inherent desire to develop that technique caused him to spend 11 years on it.
Here this can be said in this way:
The inherent desire (which was to/in order to) develop that technique caused him to spend 11 years on it.
Could you give me a link for further study?
... a paper to write / a paper to be written
... a desire to develop that technique
These are two very different NOUN TO VERB constructions.
The first "derives" from a clause in which NOUN is the Patient of VERB (its Direct Object in the first, active version, its Subject in the second, passive version). This, as you correctly observe, may be paraphrased with a relative clause which is to write / which is to be written.
The second, however, derives from a construction in which NOUN is originally a verb taking an infinitive clause as its complement:
I desire apples. I desire a fulfilling life. I desire to develop that technique.
Your proposed relative clauses don’t work there, because they disrupt this particular syntactic relationship: now the noun desire lacks its necessary complement.
Unless you want to sound extremely formal and educated, just use the infinitive.
I have a paper which is to be written before class.
Would be said to a boss that you've never met before when you're trying to get a day off to write the paper, possibly over email, for example.
I have a paper to write before class.
Would be used pretty much everywhere else.
I’m not sure of all the details, but I believe that the infinitive is not modifying the noun. Rather, the noun is the (direct) object of the verb (infinitive).
I have a paper to write before class.
=
I have to write a paper before class.
It seems that “have” is being used to mean something like “need”. Note that this construct can work without the noun object:
I have to eat.
You can see this mentioned in this EL&U question: Omitting the last “to” in “All {I need to / have to / must} do is (to?) do something”. The answer to When should a verb be followed by a gerund instead of an infinitive? addresses the large issue, but, curiously, doesn’t say much about “have”.
