going back to basics, a rear mech has to have a parallelogram or a plunger to shift the chain from side to side, and at least a single pivot to accommodate chain tension variations. Leaving the added complication of the top pulley possibly not being centred on the lower pivot (and therefore moving when different chainrings are used) aside for the moment, you can see that the top pulley is constrained to move in a fixed path if the mech has a single pivot.
Pretty much every variation of parallelogram orientation and mounting has been tried. Some examples;
-chainstay mounted rear mechs with a horizontal parallelogram (Nivex etc, plus 80s/90s suntour, plus modern Dahon mechs
- Huret Svelto (and many others similar)
- Sun Tour slant parallelogram (first seen in a pressed steel mech in the early 60's)
- Campag Gran sport
Note that many of these mechs have a swivel at the hanger (to allow wheel removal), but this does not move when the mech is in use, so it is not a second pivot per se. All these mechs are single pivot (SP) mechs.
Of these some have a path that is only suitable for close ratios, or at least doesn't help to track the contours of a typical wide range freewheel. The Huret Svelto and campag gran sport are like this.
Others have a path that does help track a wide range freewheel; the Nivex does this, as does the Allvit, and so does the Sun Tour mech.
Amazingly the Nivex was the first ever parallelogram mech, yet it has a curved path for the pulley, that can be made to suit 'real world useful' wide range gear profiles, with fixed percentage intervals between the sprockets. The Huret Allvit was not a good mech in many ways, but is almost as clever as the Nivex, being in essence a Nivex style parallelogram, but mounted on a 'normal' gear hanger instead.
The Sun Tour slant SP parallelogram has a cruder (straight) path, but it is simple and the design can easily be varied (with slant angle) to suit different range freewheels. [IIRC the Gen 1 Campag chorus mech had a unique feature; an adjustable slant angle to the mech, so that the mech would work on a wide range freewheel or a close-spaced one.]
Sun Tour's slant parallelogram patent expired in about 1983, and Shimano (amongst others) immediately incorporated a slant parallelogram element to their mechs, starting with the seminal RD-7400. Prior to that, shimano had been using an idea originally pioneered by Simplex, in which there is a sprung 'B' pivot. Shimano called this dual pivot system 'servo pantograph' or somesuch.
in theory the Dual Pivot (DP) system can allow the mech to track any profile of gear cluster. All that is required is to have the correct balance of spring forces so that there is enough upwards force to overcome the weight of the chain lower run. After that the mech should sort itself out, adjusting itself as the angle of the chain run onto the sprocket varies. However the reality is that there are several things that stop this from happening in a perfect way
1) The weight of the slack chain run varies
2) there is friction in all the pivots
3) there will be free play in all the pivots
4) the springs in both pivots are constant rate, rather than constant force.
The last point is subtle but important. The springs in the pivots are fighting one another and must be made to fit inside small housings. This means they cannot be made constant force, and instead of the mech self adjusting to different sprocket sizes, it doesn't do this perfectly and indeed it cannot do this. This means that the DP method (in this form) only works over a small range. So adding a slant parallelogram to a DP mech simply means that the DP part has less work to do and can therefore work better. Campag, Shimano, Huret, Simplex, Sun Tour, (and others) all produced 'slant DP' mechs in the 1980s onwards. They worked OK too, but very commonly wear out (or seize up...) on the top pivot.
[BTW the flaws in basic mechs may instead be addressed by adding a second parallelogram; this gave rise to the Huret Duopar, the Sun Tour mountech, amongst others. These worked brilliantly when new, but were complex (expensive) and ultimately unreliable; all those extra pivots just wore out prematurely.]
In theory a DP mech will 'kick' on the upper pivot during some shifts, which may allow a smoother shift, even when the top pulley is running very close to the sprocket. In practice this isn't essential with modern chains/sprockets, plus the need to accommodate very different gear clusters with a single mech has been reduced; bikes are more specialised these days and fewer riders change the gearing appreciably; (they are more likely to buy another bike in fact....
). This means that the slant SP mech has come back into play; despite being suited to a small range of different gear clusters, and having an 'imperfect straight path' they are simpler (cheaper) to make and can wear better than slant DP mechs.
It is as well to note that much of the improvement in shift quality in recent years has come from subtle changes in chain and sprocket shaping. This can allow slick shifts even when the top pulley isn't that close to the sprocket.
A slant DP mech can allow fast shifts with a wide range of chain and sprocket combinations, by running a close pulley. However these shifts may not be slick and some may be baulky if you try to 'force' a bad combination. If you try to run a slant SP mech with the same (bad) chain and sprocket combination, you will also get poor results. You don't need much wear in any of this kit to reduce shift quality again, either.
My personal take on all this is that modern gearing can work beautifully, but it is getting less and less versatile and it mostly wears (to the point that the shift quality is lost) too quickly. The 'dream' of a truly versatile rear mech (that copes with a lot of different gear systems yet is simple, cheap and robust) has been abandoned by the manufacturers. I think this is a mistake, and that it is still attainable. I even have some ideas of how this could be done.
However, in the meantime, I'm also becoming less and less interested in changing gear at all. I quite often ride for hours and only use two or three gears. In hilly terrain, I am happier than I used to be with just a few widely spaced gears. When I ride with other folk, they seem to be endlessly prodding at their gear levers for no good reason, which is just annoying, even if you are only watching it!