NavList:

Hi Bill.

I think your device idea is great. If you can get it working, it should do just what you have said: minimize the sum of the square distances by settling to a minimum spring potential energy state. The sliders will have the be absolutely free to move: maybe little "cars" riding on the rods? If not cars, then maybe dip the whole thing in fine oil? The slightest friction would allow the thing to settle well out of the minimum energy equilibrium that you want.

The drawings you made, which you described as crude, are fine art, by the way, straight out of Leonardo's studio. Ha ha. OK, maybe not that good, but they certainly explain the design nicely.

It's a little hard for me to imagine anyone using something like this in a real-world navigation scenario. It's cumbersome, and probably easy to break... There are simpler methods in nearly every interesting case, and most apps do the "least squares" solution automatically. But that's OK because this device could have great teaching value. This is actually helpful for your design since then it could be built at any convenient scale. Even a virtual model of this analog computer could be helpful for some students to see where the least squares fix ends up. It's worth a try.

Of course after you have the whole thing built, and it's running friction-free, you'll discover a student at the first demo posing the big, fundamental question: "Er... why least squares?" By making the least squares energy so literal and physical with these springs, you bring that question to the forefront. It's an interesting problem... How would you answer that question in an intuitive fashion for a reasonably intelligent navigation student? Why least squares? Why not least distance (first power)? Or why not minimize distance raised to the power (3/2)? I have not come up with a perfect answer to this myself, so I'm interested to hear how others would respond...

Frank Reed