Hello Barb, this is an excellent question! Let me see if I can muster a quick and simple explanation and then I will revisit this question at a later date because I feel like this is one of those questions that can really benefit from pictures and I need time to prepare those.
Section 31 of the International Residential Code governs the plumbing vent systems and it does outline distance limitations that need to be kept in mind designing a floor layout with trapped plumbing fixtures (i.e. those having a P-trap on their drain). However, I think you've got the design objective in reverse here: since all trapped plumbing fixtures have to be vented and the maximum distance to the vent opening is limited by the amount of slope allowed for a particular diameter pipe, you actually want to locate your fixtures as close to the vent (be that the stack vent or a branch vent) as possible. In other words, you want to be able to install that connection to the existing vent rather than having the fixture so far from the vent that it requires you to install another vent. It's the installation of a new vent and the additional work, space, roof penetration etc. that you want to prevent, not eliminate a connection to an existing vent.
But the actual physical limitation of the fixture arm (drain pipe from individual fixture) length is the requirement to have its bottom at the P-trap end not be any higher than its top at the opposite end, the one connected to the vent. Otherwise, if enough water is rushing through that pipe, it will suck in the water that is supposed to stay in the P-trap when you turn the faucet off. If new air cannot enter from the vent, the water will keep going and will create negative pressure that will drain the P-trap water and the trap will essentially become useless.Â
This may not be the simplest explanation of the subject matter but I hope pictures will help, I'll try to post them here in a couple of days.