Forget Einstein, let's talk about Reality! All of this is experimental, not theoretical. I've been doing this for 16 years as the technology has changed.
It's easy to calculate the distance between any 2 points accurately. Stand at A, write down numbers. Quickly go to B, the error will be the same, write down numbers. Assuming a straight line the math isn't so bad. You might be off by 1/2 meter.
I've got a great algorithm to calculate the radius given 3 data points nearly in a line. With this, centripetal acceleration, force, lean, friction etc. No one else has published this according to Google.
Interesting problem for you: How to calculate lean angle for a motorcycle for less than $50k. This is important for racing. I've got 2 solutions, 1 untested. More than 50k?
Another: Record the exact line, within a lane, while racing. 1ft acc would be nice. You can drive around the course slowly right before the race begins to calibrate. Impossible! All you can do is compare yourself to other racers at the same time, assuming you have the same satellites locked. WAAS is not even close.
There's some weird math going on with GPS data. It doesn't matter how fast or slow you go. It doesn't matter how often you sample, even 20x a second. Only relative accuracy matters for this case, even if the error is changing slowly over time, that's OK, a separate issue. It doesn't matter how far apart the sample points are in feet or inches.
No matter what you do, it's difficult to calculate the distance traveled accurately. A mechanical odometer from 1950 is much more accurate. The rope with knots Columbus threw into the water in 1400AD. As accurate as GPS. Why?
I mean within 1-2%. Ignoring things like change in altitude and the weird shape of the planet. Nobody even calculates that. There's no point in trying. Seems like you need AI or a human and additional satellite image data. You could do better than simply adding up the distance between points by instead measuring the distance between accurate turns, in the city blocks for example. Or pick a few points on a curved road radius. Straight is easy using a map and a ruler. Adding up the data points is not the way to go. It never works out.
You have to make additional assumptions. How well are the corners defined? Perfectly square like in theory? With a finite radius? How much zig zag within your lane or while hiking a straight trail? The GPS measurements don't match the reality.
Speed is another problem. Even at a constant speed, you cannot measure this correctly using GPS. You can only measure the speed between any 2 points further apart. By looking at the GPS time and calculating the distance of the path. Which brings you back to the original problem above.
What matters anyway? The actual speed, which would be correct if you averaged 100 points. Or the speed it WOULD have been if you went perfectly straight? Or assuming a certain amount in % of wasted zigging? What matters for distance? Every little step to the side of the trail? Or how far it could have been in theory if you walked perfectly? There is no right answer. And no way to measure it today.
Einstein was wrong... ;) Thoughts?
Using realtime data from the internet, would a more accurate odometer be helpful to anyone you know?