I really like how the code + images enhance the post, but I think this model is a little simplistic for how analytical minds actually cross roads. The post assumes that my discomfort level is a constant, but in reality I become more comfortable the further across the road I've walked (as I become more confident that I'm not going to be flattened by a car). This leads to a smoothed curve on the latter half of the street.
Does Fermat's principle allow for variable refractions?
Does Fermat's principle allow for variable refractions?
Yes, and a common example is how light from the sky can bend to run through a layer of hot air right over hot sand, causing a mirage.
However Fermat's principle is a local rule. That is, there should be no way to improve the path by adjusting it a little bit, but it might not be a global minimum. The classic example demonstrating this is a mirror. The light bouncing off the mirror often had a shorter path available (just go directly there), but there was no local variant of the path which was better than the one that it took.
Fermat's principle holds for different things for different reasons. For instance light follows the principle because of how the wave front expands. Ants follow the principle because ants that followed a faster path tend to lay a fresher scent trail. But my other comments remain true regardless of why it holds for any particular type of thing.
"Does Fermat's principle allow for variable refractions?"
Yes, and that's where is gets interesting! Light will take a curved path through a tank of salt water, because the salinity, and through that the index of refraction, will not be constant but will be a continuous function of the depth.
I believe in the curve: as fur as there is no car in the road, I curve my trajectory to walk less (and stay a little longer in the road). The curve should have something like: given "St" the securityTime, at any time t, v(t) * St is located on the other side of the road. With v(t) the vector velocity witch is not constant (the angus changes, not the speed).
At this point you may very well not have analytical solutions anymore. But that's fine, you could do a numeric simulation and thereby obtain your maximum-comfort curved trajectory. :)
Does Fermat's principle allow for variable refractions?