Floyd-
I agree about looking first at the leading edge, although I believe that the choice of l.e. radius is only part of it. That design above was made to optimize the transition from the extreme l.e. to the high point, an area I suspect does much to define handling characteristics. The maximum thickness probably is more of a variable depending on this than a goal in itself for determining max lift and stall characteristics. I don't think the aft part is quite as important, but I also don't think we always have to fudge it for structural reasons (see my fourth figure above). I would change the t.e. in our typical thicknesses just to use the exit strategy of my choice. You can hinge the airfoiled trailing edges (e.g. Keith Trostle's FW), but the research I've seen , from NACA to Al Rabe's, indicates that the thin flap is superior anyway.
I'm not sure how far you meant to go with that; so here's what I think. First, while the theoretical sections do usually end in sharp edges, there are other reasons besides delicacy (or dope warpage,...whatever) that we don't always build them that way. In fact, sometimes the desired flap hinge location is at a point of too much thickness. For me a reason not to build the theoretical t.e. contours is that credible sources say that a sharp point is not the best termination anyway; the recommended technique (e.g. by Barnaby Wainfan) is to end the wing (flapless or the hinged section's t.e.) with a sharp, vertical cut ('squared off'). The air (Kamm effect?) then breaks cleanly and continues to rejoin better than if there had been a "sharp" edge, without the skin friction past the new t.e. Apparently the wing acts as though it has that extra area.
I did find in an old NACA report as well, as in recommendations from John Miller, the belief that flat stationary flaps with sharply cut vertical edges make better lift than sections continuing the airfoil contour to a sharp edge. The unfortunately least examined figure in my post above shows XFOIL's "opinion" that the recommended "flapless" airfoil with that stationary flap not only has a higher maximum lift than without the flap, but that with or without, it still exceeds the maximum lift of the regular NACA section and has a significantly better stall characteristic. That stationary flap is flat and squared off.
When I built my last flapless - the one with the recommended wing - I have to admit that my Tom Morris "New Millenium" type structure made a very nice sharp airfoiled t.e., and I delayed adding my planned stationary flap to try out the sharp t.e. It seemed to fly well, but I never got a chance to really test high aoa's before I let it fall in from overhead (incidentally, the t.e. was not the weak link). I will build another similar plane with a simpler structure, and I'll use the stationary flap. You'll see on the last figure a vertical line chopping off the sharp t.e. The chord of that wing was computed to allow for the proper thickness there for the flap to attach - hinged or not.
That's how I use Profili. I choose a chord that leaves the desired thickness at the chord point where I want the t.e. This is easy with the ordinates it automatically computes for anything in their database, whether or not it comes from you, or you can simply play with the graphical editing until it comes out how you want it - a minute or so. Otherwise, I calculate the defining function to end at a chosen height, as I did for the flapped section (third figure) I concocted with the elliptical front and NACA 00xx aft parts. Mike Pratt's "P-Force", by the way, has an NACA 0018 or 0017 section with sharp t.e. and is perfectly strong enough due to the 1/16" sheet that meets at the t.e. It can be built rounded, sharp, or squared off.
Larry-
There is literature out there on that, but I have forgotten what it says. What I think I remember is that the actual value is closer to 24% for symmetrical airfoils. It does move a little with varying maximum thickness, and I would expect it to move forward for forward points of maximum thickness, but I think it moves only a tiny bit, if at all. I know I built the last wing with all 1/4-chord points lined up along a straight spanwise line and placed the "high point" up at 24% to get my longest tail arm. However, I did not do so thinking that I'd get much change in chord-wise position that way; it was just to keep the quarter-chord points from sweeping aft of the root point and moving the MAC rearward.
I'm pretty sure I have some research data around here that says something about this, but I'm about at the end of my allotted time her. I have to assemble the club newsletters and make sure that I don't have to get stronger through humiliation at tomorrow's orchestra rehearsal. While my lit here is probably sufficient, you can probably find that answer on NASA's NACA report server site. I too will look - later.
SK