How would a straight, flat road appear visually?

Suppose there is a really long road. This road is absolutely straight, and as a result, does not follow the curvature of the Earth. This straight road begins tangent to the curvature of the Earth, and extends a very long way into space.

Now if you were to stand the point where the road meets the Earth, and walk along the Earth in the direction of the road, what would the shape of the road look like? Since, in your frame of reference, you are moving in a straight line, would the road appear to you to be curved?

I imagine it might. As you walk further round the Earth, the road would get higher and higher. When you have walked 90 degrees from your original position, the road would appear to be vertical. Then, as you continued moving round the Earth, the road would start to point in the direction opposite to your own!

Or perhaps, this wouldn’t happen at all. Maybe the road would always appear straight, but having an absolute line of reference of this sort would allow you to perceive your true circular motion!