That's where the Jones paper comes in! It turns out that minimum drag for given heeling moment occurs when the induced velocity distribution varies linearly over the span. You can get this by shaping the planform, or twist, or both.
If there's no constraint on heeling moment, here's what the optimum span loads look like:
If there's absolutely no gap between the sail and the surface, you get the half-ellipse that people expect. But if there's even a small gap, then the optimum loading looks a whole lot like a full ellipse that gets a bit egg-shaped as you close off the gap. There's a big hit to the induced drag due to the gap, too.
You've got a good point about having to balance the drag vs the heeling moments. Here's the design tradeoff for the minimum drag loadings shown above:
The circled point at the bottom corresponds to the sealed half-ellipse case that's the best you can possibly do for a given span. As you open up the gap, you see the drag shoot up to half again greater than the theoretical best. For very large gaps, the drag would approach twice the minimum and the center of effort would be very high because you've lifted the rig way up into the air.
If you try to recover the drag by making the rig taller, the center of effort goes up. Each of the points in this carpet plot correspond to a different design, optimized for that combination of span and gap. The lines running up and to the left correspond to anchoring the foot and stretching the mast. The drag goes down, but the center of effort goes up.
So let's say you adopted the Jones approach, and tapered the loading so the induced velocity distribution was a maximum at the foot and tapered to zero at the head. This is what the spanloads would look like:
If you use a linear twist, you can get each spanwise station to have the same lift coefficient and the planform shape will look just like the span loading. These sure look a lot like sailboard rigs to me! Which isn't surprisng, because the sailboard were able to evolve without any artificial constraints due to rating rules, etc.
Here's what the design tradeoffs look like for these rigs:
At comparable spans and gaps at the foot, these planforms have more drag than the previous ones. But here are the design trades for the two laid over top of each other:
For any given heeling moment, the more tapered planform allows you to increase the span enough to reduce the drag without exceeding the design heeling moment. Or you can reduce the height of the center of effort without taking a drag penalty by making the rig both more tapered and a little higher.
In principle, there's no end to how much you can apply this principle. If you taper the induced velocities more and more, you start to produce negative lift at the head of the sail. And you can drive the center of effort right down to zero. The negative lift at the head balances the greater lift in the body of the rig, leading to a net gain with no heeling moment at all! The performance is terrible for a given size of mast, but if you make the rig really tall...
Obviously you're not going to get this with typical a soft sail - it can only be done with a highly twisted rigid rig or perhaps a gaff vanged to leeward. Personally, I think the planforms with the induced velocity driven to zero at the head are the practical limit.
All this is based on simple theory and a uniform wind. But the trends and conclusions are still good when you consider the wind gradient. And the optimum span loadings are good for both cat rigs and for other rigs when you add together the contributions for all the sails.
