On driving a car on a sphere

Question

Suppose I was driving a car on a solid sphere of radius $R$ at a constant speed. I constantly take a left turn always at the same angle. If the sphere was flat — flat Earth? :) — I would drive in circles of radius $T$.

1. How does my trajectory on the sphere look like?

2. When will my path be periodic?

3. If the path is not periodic, is it space-filling or will there be regions I will never get close to?

4. Will I visit certain places more often?

5. How many self-intersections will my trajectory make and where will they be?

6. Will my self-intersections be at all kinds of angles or a only at a few?

7. Is the trajectory chaotic when it is not periodic?

8. Is there a closed form?

Answer 1

If you mean to say that the path the car takes has the same curvature at every point, then of course the car travels in a (small) circle. I don’t see any other way of interpreting your word “always”.

Answer 2

I take this image from my MO post, Curves of constant curvature on an ellipsoid:

There I say,

the curves of constant geodesic curvature on a geometric sphere are all circles: simple, closed curves that are geometric circles lying in a plane.