# Calculate turn required to follow a line of latitude

Lines of latitude are not geodesic except at the equator. So to follow perfect eastern course in the northern hemisphere in a vehicle a left turn is constantly required. We can express a turn as the radius of a circle that the turn would produce on the flat. So, at the pole the turning radius would be zero and at the equator the turning radius would be infinite. Near the pole the turning radius would be almost equal to the distance from the pole.

Does anyone know of an equation that would provide the turning radius at a given latitude required to follow that line of latitude?

I will try to clarify with some examples.

If I were to drive a car on an eastern course at a location of 100m south of the north pole, I could not drive in a straight line otherwise I would eventually be driving south. So, to maintain an eastern course, I must turn the car and circle the pole with an approximate turning radius of 100m.

If I were 200m south of the north pole the turning radius would be approximately 200m. However, the further south I am from the north pole the less turn is required and the more it differs from my distance to the pole.

At the equator no turn is required because I will be travelling a geodesic, the turning radius at this point is infinite (meaning no turn).

Is there a mathematical function (f) that given a latitude provides the turning radius required to preserve that latitude and maintain an eastern course?

r = f(l)

where: l - latitude, r – turning radius

I have performed extensive searches and cannot find an answer.

• There are scores of duplicate posts on this topic. I suggest you review some of them. – Vince Aug 20 '17 at 2:22
• I find your question very confusing. If I drive due East, I don't need to correct my course at all to follow the latitude line that I'm on. Do you instead mean how to follow a geodesic line? – mkennedy Aug 20 '17 at 3:18
• I cannot find any posts that provide me with a method of determining the turning radius. – Gareth H Aug 20 '17 at 10:29
• To keep a constant Eastern course you need to apply a turn correction. For example if I am in a car 100m south of the north pole, to travel east I would need to apply a turn radius of approximately 100m to circle the pole. – Gareth H Aug 20 '17 at 10:31
• @gareth-h I believe that my answer provides you with the information you asked for. If you agree, please mark the answer accordingly. Thanks. – cffk Sep 4 '17 at 21:51