I’m terrible at math, but I had this thought while playing around in Blender. The car near the pole needs to turn the wheel to maintain distance, but the car on the equator drives in a “straight” line. At what distance from the pole is the transition? I hope this makes the question clear, I ran into a character limit in the title.
Anything not on the equator needs to turn the steering wheel.
You would have to specify which car is being driven to determine how far the wheel would need to be turned due to differences in steering control mechanisms. Some cars use rack & pinion gears for steering while others are street by wire. The latter have no fixed relationship between steering wheel being rotation angle and the amount the tires are being turned (that ratio changes based on speed)
I think we could probably rephrase the question to specify what angle the front car tires have to be turned to make the circle. The answer would then be simply whatever angle is needed to have the car tires be tangent to the curve.
Gotta narrow it further. Which tire, both? When turning the inner tire turns more than the outer (see Ackerman geometry).
Nah, you right I forgot that inner turns slightly more aggressively. Can we instead argue that the midline of the car has to be tangent on the line of longitude that we’d be traveling on to maintain radial position?
Side note: it’s not radial position on the surface skin of a sphere but I couldn’t think of a better term. Any ideas for a better wording?
This seems obvious now that I read it.
Image, because that’s how I understand things:
I would also note that turning ratios aren’t the only consideration, but distance traveled and therefore speeds.
Also not at all a mathologist.
Note that the lines you drew are straight, no turning needed, but they are all equivalent to driving on the equator if you’re on a sphere.
You don’t get the north pole route by turning the equator route, but by translating it northwards.