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If you were a pilot flying from Tokyo, Japan, to Vancouver, B.C., what would you do to minimize the time required to complete the flight?

The Earth is approximately a sphere, and the way to minimize distance (and thus, travel time, all other things equal) is to travel along a great circle, a circle that covers the full circumference of the sphere. Any other path along the surface of the sphere that travels between those two points will necessarily be longer.

Where this becomes counter-intuitive is that you will, in general, not stay at the same latitude, nor simply interpolate between the two latitudes. (If you are traveling between two points on the equator, such as from Quito, Ecuador to Kampala, Uganda, you do stay along the equator; but that's the only time you would.) The great circle that contains Tokyo and Vancouver actually passes through the southern peninsula of Alaska, despite the fact that Alaska is at a higher latitude than either Tokyo or Vancouver. This also has the advantage that if necessary the flight could ditch in Alaska, whereas flying over the Pacific Ocean would leave them with few options.

To help intuit this a little better, it can help to actually look at a globe, rather than a map (since maps inherently distort the shape of the Earth); also, if you think about traveling between two points that are far north on opposite sides of the globe, you wouldn't try to stay at the same latitude, you'd go over the North Pole. The trip from Tokyo to Vancouver doesn't require traversing the North Pole, but it's the same principle; the shortest route curves up to a higher latitude and then back down.

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