# The places my satnav doesn't want me to see

If I take a spare day and some sandwiches, program my gps/satnav for home and take the opposite of whatever it tells me, where will I go? On a grid system it looks like I will just move away from home in the same direction as I was to start with, but the roads round here are not in a grid. Most roads join at T junctions so there is only a choice of right or left. If travelling north when my home is directly east, the gps will say turn right so I will turn left. The next T junction will take me either north or south again and possibly for some miles (it isn't a grid) so I wonder if I could end up circling around my destination rather than just moving away from it.
Is it possible to have a road system where doing the opposite of the shortest route to home leaves you circling round it?
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Commented:
If you come to a T-junction along one of the arms instead of up the stem and you are directed to "Turn right", would going straight be the same as doing "the opposite"? If so, it's easy to imagine a road system that would keep you circling your destination. It's possible that that's what would actually happen, considering that actual road systems are generally eventually bounded by oceans.

Tom

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It depends entirely on you specific road system, in particular on the length of the stem of the T
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Yes, it is possible
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Assume that it is possible.

Since the problem is symmetrical you could travel in a clockwise circle around your home.

Any right turn will bring you closer to your home and any left turn will take you farther away but you will need to continue making right turns to maintain the clockwise circle. Therefore, when you come to any intersection that offers both, the only way to maintain the circling would be to turn right, but that takes you closer and thus you would be forced to move farther away and out of your circle.

You could certainly spiral out if the roads were set up correctly, but could not travel in a cycle unless there were no intersections (at all) that offered both left and right turns.

You could set this up if you had a big circular piece of road that had only T intersections.

Another odd point is that if a particular stretch of road moved away and then curved back on itself, you could have the situation where the GPS would suddenly say to continue on your path in which case you would naturally be forced to make a u-turn (turn around in case that's a U.S. only term).
So you could oscillate if not circle.
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Commented:
This is assuming that if you pass a T intersection and the GPS tells you to turn, the 'opposite' is to just keep going which doesn't make sense because if that were true, then when you passed the other T intersections where the GPS did not tell you to turn, the 'opposite' should be to turn. So even T intersections don't allow you to keep circling if you set the problem up fairly.

The "opposite of the shortest route" would surely send you farther and farther with each choice. The only way to make it possible is to have a convoluted road system and to cheat on the T intersections.

Of course, you could also cheat by making a circular road with no left turns available (assuming clockwise travel still).
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Given a perfectly spherical Earth covered in roads you'd eventually go around in a circle of points equidistant from your home and its antipode.
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Given a normal shaped earth, you'd probably end up driving down some road that dead ends on a beach and get stuck there.
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Author Commented:
>when you passed the other T intersections where the GPS did not tell you to turn, the 'opposite' should be to turn

I'd only do the opposite of what it tells me, if I pass a turning that it doesn't mention I just carry on following the road.

I can now imagine a circular road with 'spokes' heading in toward home, there can be roads leading out from this circle as long as they never form a crossroads with one of the spokes. Once on there for instance travelling clockwise, the gps would continue to say to turn right so of course I take the other option which is to go straight on (straight on round the corner, you know what I mean). It would never tell me to take one of the left turns leading outside the circle so I can eat my sandwich while going round and round, but never quite arriving.

I suppose that by chance there may be a road system somewhere like this, hopefully less obviously circular, but one that obeys these fundamental rules.

Thanks for the suggestions, sometimes just asking the question helps to focus on it long enough to find an answer.
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