I have ask this half baked question many time in many places and I have received the lower tier answers all the time.
Driving on a multi-lane highways with no traffic (yes it can happen, eg upstate NY Adirondacks overnight ) you can straighten the road shifting lanes to always be on the inside of the curve. Calculate the shortest path.
I once vaguely phrased this what shortest line that can be drawn between two parallel curved line a constant distance apart. I think this is mathematically half baked and could be restates.
Perhaps a good starting point would be to look at a "chord"[1], which is a line segment between two points on a circle. Then you'd look at a chord that is tangent to the inner circle of a set of two concentric circles. You can find the length of the chord from the radii of the circles.
But what you're asking about is more general. You are looking for some sort of shortest path between two parallel curves [2]. Which seems like it would be a tricky problem to solve. Are you considering physics here? If you are, you are basically looking at a "racing line" [3]. And that's a complicated problem since it involves a lot of physics and engineering. You can't look at a single turn in a track and determine the best racing line. That's because the best racing line for a specific turn depends on the best racing lines of the nearby turns. You'd have to look at the track as a whole. There are entire theses written on this topic [4].
But a highway road is not the same as a racing track. In highway engineering they often use the Euler Spiral [5] to construct the turns, because of its nice properties. Funny enough, Euler Spirals play a role in optimizing your entry into turns on a track [6]!
I think those might be some good jump-off points for further inquiry.
One thing to note is that this shortest path won’t typically be much shorter. Consider a circular ring road. The circumference on the inside is pi * d and on the outside is pi * (d + w), where w is the width of the road, so the amount you save by driving on the inside is pi * w which is probably less than 100m.
In a more general simplified case where you always stick to the inside of every curve and jump from left to right for free, the amount you save compared to sticking to the middle lane is the line integral:
\frac14 w \int k \mathrm ds
where k is the curvature (strictly speaking k should be an adjusted curvature of 1/((1/K) - (w/2)) where K is the actual curvature).
Because roads are not very curvy, the integral of k will be small and so the distance saved is very low. And note this is an overestimate because it doesn’t cost in changing from side to side
The real gains come not from decreasing the distance travelled (sometimes the racing line does the opposite), they come from limiting the radius of the turn so you can carry more speed through. At least thats what I've learned from playing Grid 2 for far to many hours.
Simple. Put up walls all the way along the road, then pull a string tight from one end of where you're driving to the other. Then you just measure the string!
This looks a lot like the xyproblem[1] which I have fallen prey to many times. You start out by thinking about something - highways in your case - then you formulate a model, which might turn out to work or might not. But then you make the mistake of asking about your own formulation of the problem rather than the original problem. From my own experience, even if your formulation turns out to be right, asking about the original context will usually result in higher quality answers.
Ugh I loathe this concept, although it's real. But 9 times out of 10 when you ask about a complicated problem online, which you've reduced to its simplest components because it's impossible to explain the entire context to an outsider in a text field, you'll get some internet smartass screaming 'xy problem!' to dismiss your question (this is not directed at the parent comment, just a general observation)
Fun edge case: I first thought you should always drive either leftmost or rightmost lane in curves. But if a "curve" only turns a couple of degrees you might be better of ignoring its existence and just go straight through.
What's the question? Given a specific example you can do the calculation. Generally the ideal route is going to consist of hugging the insides of curves + straight lines that are tangent between two opposite-direction curves. But for arbitrary curves I don't think you can say much more that's interesting or useful.
Driving on a multi-lane highways with no traffic (yes it can happen, eg upstate NY Adirondacks overnight ) you can straighten the road shifting lanes to always be on the inside of the curve. Calculate the shortest path.
I once vaguely phrased this what shortest line that can be drawn between two parallel curved line a constant distance apart. I think this is mathematically half baked and could be restates.