It tries to boil the problem down to a 'discomfort' level, where what really matters is how long I think I can comfortably stay on the road, how fast I can cross the road, and how long I expect to wait to cross the road.
Let's assume all sections of the road are equally good for crossing (no pedestrian crossings etc). The road will then have a 'maximum angle of attack' which is a function of how fast I am travelling and how long I can stay on the road. I will walk in roughly the same pattern as in the link HOWEVER I will never cross at an angle greater than the maximum and I will potentially cross earlier or later depending on gaps in the traffic; it's faster to keep walking then to wait for a gap in traffic.
The main shortcoming in the model is that there is no upper limit on the angle of attack, and so it is easy to find example situations that are unrealistic.
I frequently cross the road near a highway. There is a light in the distance that works in busy times and it tends to be a busy highway around the time I usually cross. When the cars stop for the light I just cross in between them, giving me a straight across path. Otherwise there is too much traffic to attempt to cross while it's moving. I just walk along the road until the cars stop.
Also I never cross in a straight line. Most of the time I cross in an s-curve shape, meaning the road itself has a variable index of refraction for me.
When I lived in West Philly, I had to walk 10 blocks east and a few blocks north to reach school. Just before campus I had to cross a busy N/S street with a light, but most of the other intersections were just stop signs. I thought all the time about whether I should go straight east to the busy street or take the prettier route by cutting north a couple blocks first.
Because you're thinking "physics like" not "math like"
In physics, yes, you have the refraction factor, etc, and this calculation works.
But it also works (from the math point of view) saying that light will take the path that takes the least amount of time for it to cross between two points.
Light will travel in the fastest way possible under a certain set of well known constraints. I think trying to apply this same model to how 'mathematicians' cross the road is flawed, although it is a good approximation when the start and end point are not displaced too far along the road.
As a simple counter example, think of a laser pointing along the length of a very long piece of glass. If you angle the laser slightly down, so that it hits the glass at a very small angle of attack, the laser will travel for a long distance inside the glass before exiting. It will not travel straight through the glass, but neither will it 'cross' to the other side very quickly. Regardless of the difference in the refractive index of the air and glass, you can always point the laser at an angle that causes the laser beam to travel for an arbitrary length of time in the glass.
Compare this to a mathematician walking along a very long highway. The time they will take to cross the road is NOT dependent on JUST how far they are walking. If it has heavy traffic then they will cross just as soon as their is a suitable gap. Extending the length of their journey (equivalent to decreasing the angle of attack for the laser) does not continue to increase the time they take crossing the road unboundedly.
The model is flawed for this obvious counter example, but it is flawed in simpler situations as well, mostly because the reality is that every section of road and every moment in time are not as ideal as each other for crossing the road.
hmm, i don't think you can model this without a possible random distribution of cars and their and my respective speeds. i actually cross the street in a curved manner, sometimes dangerously close to passing cars. it also depends on my estimate on when the nearest light turns green.
So yeah, I agree that it's oversimplified. Yes, it doesn't consider pedestrian crossings, but what about the cars, their acceleration, and the lights at those crossings?
It tries to boil the problem down to a 'discomfort' level, where what really matters is how long I think I can comfortably stay on the road, how fast I can cross the road, and how long I expect to wait to cross the road.
Let's assume all sections of the road are equally good for crossing (no pedestrian crossings etc). The road will then have a 'maximum angle of attack' which is a function of how fast I am travelling and how long I can stay on the road. I will walk in roughly the same pattern as in the link HOWEVER I will never cross at an angle greater than the maximum and I will potentially cross earlier or later depending on gaps in the traffic; it's faster to keep walking then to wait for a gap in traffic.
The main shortcoming in the model is that there is no upper limit on the angle of attack, and so it is easy to find example situations that are unrealistic.