What earth radius should I use to calculate distances near the Poles?

Question

I'm monitoring a GPS unit which is on it's way from Cape Discovery in Canada, to the North Pole. I need to keep track of the distance travelled and distance remaining each day, so I'm using the Haversine Formula, which I'm told is very accurate for smaller distances.

I'm really bad a Math, but I have a sneaking suspicion that the accuracy depends greatly on the radius of the Earth, and since the universe decided to make Earth out of an oblate spheroid, I have a choice of approximations for Earths radius to choose from.

Since I'm monitoring coordinates very close to the north pole, I'm wondering wether anyone knows which radius is going to provide the most accuracy.

• Mean Equatorial: 6,378.1370km
• Mean Polar: 6,356.7523
• Authalic/Volumetric: 6,371km
• Meridional: 6367km

Or any other kind of Radius that anyone knows about?

I'm hoping some maths or cartography whizz might know the answer to this one.

Answer 1

You could approximate the actual radius at the point(s) where you're measuring the distance (provided that you calculate a sequence of relative small distances).

Assuming the earth being an ellipsoid with the main axis a being the mean equatorial radius and the second axis b being the mean polar radius, you can calculate the point on the ellipse represented by these two axes by using the current latidude. The calculation is shown and explained here.

(Note: This ellipse can be thought as a cross section of the earth through the poles and the point where you want to calculate the distance)

This gives you a point q=(qx,qy), the radius at this point being r=sqrt(qx^2+qy^2). That's what I'd use for calculating the Haversine formula.

Answer 2

It doesn't really matter - they are all going to be wrong if you just treat the earth as a sphere. I would probably use the polar since you are mostly going north.