Hello ,

It depends on your defining metric.

All certain circles in hyperbolic geometry are deemed to be straight due to a certain defining or modeling metric function.

In a simpler example if you define

∫((r^2+(rdθ)^2)^0.5)/ (r^2 )

as your arc length , then any circle through the origin is quite straight.

As , It can be seen from the above equation that the curvature of a circle is the inverse of its radius. Since the radius of a particular circle is constant, its curvature is same at its every point.

The less is the radius, the sharp is the curvature.

Hence , If the curvature of a line is zero and constant, it means that the rate of change of its direction at its ANY point is zero, means it does not change its direction at any point, means it is a straight line.

Hope it Helps !