Trampoline physics: Why do rectangular trampolines provide more "bounce" The guys over at reddit.com/r/trampoline say that a rectangular trampoline has a larger sweet spot AND provides more air.  Typical dimensions are 9x15 10x16 to 10x18
I can see the following cases:

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*Consider a line running down the middle of the long dimension.  Suppose the trampoline is 2 W wide by W long.  Once you get within W of the end, you start having edge effects, maybe a bit further from the end.  This gives you a long narrow strip of equal response.  Since many bounces are either flat or sitting, this gives all parts of your body even treatment.

By contrast a round trampoline  will be 'softer' in the centre than around it.

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*A large round trampoline requires that more air be moved with the mat motions.  This energy loss is parasitic.

The argument made on one forum is the rectangular matt has more springs.  This seems specious.  If it were that simple, they could use stronger springs, or space them closer together.

Comment asks what would equivalent sizes be?  Hard to tell.  In effect the outer foot to 1.5 feet are unusable.  In round sizes the range is from 10-16 feet, with 13-15 being most common, although there are a lot of 11's.  12 isn't common.
In rectangular the minimum I've seen is 4x9 which gives only a 2x7 usable area. In squares 11x11 and 13x13 are common. But the guys who are doing athletics say that a 10x16 or a 9x14 works better than a square one.
Both square and round ones tend to have an radial component to the rebound inward, the round one more so than the square. This is touted as a safety feature, as you have to be more "off target" on your landing to land on the frame on your next bounce.
Comment asks what an athlete consdiers important: A: bounce.  How high can he get with good control.  I've seen some matts that are actually nets with 1/8" cord and about 1/2" spaces.  I think this is to reduce the air movement.  In principal this depends mostly on how closely the springs are to ideal.  One company, "Springfree" uses fiberglass rods isntead of springs.  This allows them to put the rods under the frame, reducing the amount of frame you can land on, and eliminting putting your ankle between two springs from 10 feet up.
B: Sweet spot.  Rectangular ones have a longer, but narrower sweet spot I think.  This makes for a more predictable bounce landing on face or back.
A: Edit:  Realized that this doesn't answer the question.  Note that this part of the answer is very subjective, and may not meet the standards for answers here.
A round trampoline has radial symmetry.  When you aren't in the centre, there is a sideways component to your rebound vector because you are closer to one edge.
A rectangular trampoline has a larger sweet spot.  More significantly, almost all the tricks gymnasts use are in a single plane that contains the long axis of the mat.  Jumping errors are much more likely to be along this axis.  Having an extended sweet spot makes recovery from small errors easier.
This means that the gymnast feels more control over his trajectory.  And I, at least, have difficulty adding to my energy while correcting for horizontal movement.
Watching youtube videos I don't see rectangular trampoline users going much higher unless they are generally more skilled, and I've certainly seen bounces over the top of the net on round trampolines.
##Factors with large effects on trampoline bounce:A:  Material of the mat.
Cheap garden trampolines have a tightly woven polypropylene mat.  Top end competition tramponlines have mats composed of pairs of strings with significant spacing.  Roughly half to 2/3 of the mat is air holes.  This seriously increases the elasticity of the system as a whole.  This is the single biggest factor.  My gym has identical frames,but one has an open weave poly mat, (about twice as much air holes as a cheap tramp mat) The difference in response is instantly noticeable when you step onto the mat.  Air pumping is the single largest energy loss in a trampoline
B:  Stretchiness of the mat fabric.
Much of the stretch of a mat depends on the material and the weave.  The ideal mat is not stretchy.  Elastic (used in the common meaning, not the physics meaning) fabrics aren't very elastic (high hysteresis)  The advantage of an stretchy fabric is that multiple jumpers interact less.  You have a conical deformation around your feet, that quickly decreases to 'background noise'  So that if you are 3-4 feet from another jumper, you aren't nearly as affected when you are on a competition mat.
C:  Length of the spring
Medium grade springs can extend by 50% of their length before their elasticity departs from Hooke's law.  A longer spring of similar alloy and turning will expand more.  Cheap tramps have 5 inch springs -- total length -- with the active coils only about 3.5 inches.  Mid range tramps will have 10 inche springs, about 8.5" active.  Competition tramps will have up to 15" springs.
D: Spring alloy.
I'm not sure what the standard alloy is.  Piano wire springs can be stretched 200% of base length (So a 10" spring can be stretched to 20") without running into spring failure modes.  However these alloys are not corrosion resistant, and generally are only used on indoor trampolines.  Literature mentions either chrome plating or powder coating them, but coating damage can lead to early failure.
Longer springs give a trampoline a longer throw:  The time spent decellarting and accelerating is a longer distance.  This is a win, as some simple algebra shows that average g forces correspond to the rise height / the rebound depth.  So if an athlete is jumping 5 meters, and doing his rebound in a meter, he's pulling an average of 5 g's.  In practice this is concentrated at the end, as first order approximation, force goes up as the square of the displacement.
