[Math] Distance between points on Earth

calculuscurvesgeodesygeometrytrigonometry

So the problem is this:
Assuming the surface of the Earth
is a sphere of circumference 40, 000 kilometers, estimate the distance between
Philadelphia and Paris.

I'm uncertain how to do this problem. I haven't done these kinds of question before, but besides doing some geometry of the earth how am I supposed to solve this? How do you go about using the latitude and longitude coordinates?

Best Answer

One approach using more familiar methods might be:

  • calculate the radius of the earth using the $2\pi r$ expression for the circumference
  • longitude $\lambda$ and latitude $\phi$ on a sphere radius $r$ correspond to the coordinates $\left(r \cos(\phi) \cos(\lambda), r \cos(\phi) \sin(\lambda), r \sin(\phi)\right)$ with the origin at the centre of the sphere (remember that W is negative $\lambda$)
  • you can find the angle between two points using the dot product, dividing by the product of their magnitudes (i.e. by $r^2$) and then taking the arccosine
  • you can find the corresponding great circle arc length distance by multiplying the angle (in radians) by the radius, i.e. $r \theta$

Apparently the real world (non-spherical) answer is just under $6000$km so if you are not close to that then there may be an error

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