2 Questions on Geodesic Curves and the Earth's Surface.

Question

Maths PhD student Tom and his younger brother James decide to spend their holidays in Japan. They take the British Airways direct flight London-Tokyo.

a) What type of curve should the airplane fly in order to minimise flight time and why? [2 marks]

For this question, I answered that the airplane should fly in a geodesic curve because a geodesic is the shortest route between two points on the Earth's surface.

b) Tom tries to compute a parametrisation of the curve according to the airplane’s speed. What will he find? [8 marks]

I'm not exactly sure how to answer this question. We are given a picture of a map but that's about it. How would I go about answering this?

Thank you

Earth

Answer 1

a) The answer is correct provided the aircraft is flying high enough. To calculate the flying distance you can use the Heaver sine Formula. For greater accuracy, you can use Vincenty's Formula

b) Assuming that the aircraft is flying at high enough altitude, where it is not impacted by low earth winds, speed is irrelevant in choosing the path. Regardless of the speed, the path chosen is the shortest path over a 3-dimensional globe of the earth and not a 2-dimensional flat map of the earth.

Note: For low altitude flights, airlines chose the most fuel efficient route instead of the shortest flying route or geodesic. The most fuel efficient route will be along the direction of the winds/air currents such as trade winds, westerlies or the roaring forties so it is not uncommon to see airline deviate their path from a perfect geodesic.

Answer 2

Part a is correct. I am not sure how much content you are expected to give for part b; you may just need to show that you understand that the paramaterisation of a geodesic based on speed does not change the shape of the geodesic. If more is expected of you, you can discuss the fact that a shortest path on the given map (a straight line) is not a geodesic. A geodesic on the projection you have been given, the Mercator Projection looks sort of like a squished sine wave. In this image, the geodesic is in red, while the "shortest line" on the map is in blue.