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

Question

I control a GPS device that is located this way from Cape Discovery in Canada to the North Pole. I need to track the distance traveled and the distance left every day, so I use the Haversin Formula , which I am told very accurately for smaller distances.

I have very poor mathematics, but I have a secret suspicion that accuracy depends largely on the radius of the Earth, and since the universe decided to pull the Earth out of a flattened spheroid, I have a choice of approximations for the radius of the Earth to choose.

Since I control the coordinates very close to the north pole, I wonder if anyone knows which radius will provide the maximum accuracy.

• Average equatorial: 6.378.1370 km
• Medium Polar: 6,356.7523
• Authalic / Volumetric: 6,371km
• Meridional: 6367km

Or any other radius that everyone knows about?

I hope that some mathematicians or cartographic whistles may know the answer to this question.

Answer 1

You can get closer to the actual radius at the point (s) where you are measuring the distance (assuming you are calculating a sequence of relative small distances).

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

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

This gives you the point q = (qx, qy), the radius at this point is r = sqrt (qx ^ 2 + qy ^ 2). This is what I would use to calculate the Haversin formula.

Answer 2

It doesn’t matter - all of them will be wrong if you simply treat the earth as a sphere. I would probably use polar explorers, since you mostly travel north.