How does this apply to surfing?
http://en.wikipedia.org/wiki/Brachistochrone_curve
(read the “History”…gotta love those guys)
It's an interesting notion.
Apparently though, I'm not intuitive enough to see the particular application you had in mind. So rather than assume what it was, I thought I'd wait and see if some further clarification was coming down the road. Was it a reference to wave forms? Or, was it a reference to the way surfer's surf those wave forms?
Anyway, that stated, let's go with '… the way surfer's surf those wave forms'.
You could say that MTB 'steady-state' treatment (and data) assumes a particular trajectory. But I'm not aware of any formal or informal treatment of the optimal way to move on the face of a wave, given the curvature of the wave face, and the changes in the flow along the face of the wave. Still, in the most general terms, it may be possible to develop some treatment using the kind of data that MTB has generated to argue for a particular trajectory.
My understanding of what's going on isn't sophisticated enough to treat such a general problem. I've tended to see the problem as the surfer continually trying to optimize the forces of planing with respect to some (changing) objective, under the constraint imposed by a given a curvature (of the flow.) This is likely to involve something similar to a brachistrochrone problem, which is basically a particular kind optimization problem, so in that respect the problems are similar.
Though perhaps you're only concerned with dropping through the gravitational field? Say a big bottom turn, or some other maneuver which utilizes a drop? For me however, I'd still argue that it would still involve the constraint of continual optimization of the forces of planing, but now with just an additional constraint. That additional constraint(s) being in both in the magnitude and direction of the net forces of planing.
If your argument was going to be something like, surfers always try to get from 'here' to 'there' in the least-time, fine. At this point, I'm still more inclined to see it more as 'in the right-time', which admittedly, may very well be the same as least-time, a lot of the time, but in either case, I'd still argue you still have the optimization of planing forces as a constraint.
Anyway, my comments are likely useless to you. But if you're inclined, it would be interesting to see where you were going with this.
kc