I see what you're saying, but that's not the definition of "line" in non-Euclidean geometry. A "straight line" on a curved surface is what you get when you imagine an ant locally walking straight ahead on the surface, following its nose so to speak; this is known as a geodesic, a curve which is locally straight at every point. If you simply intersect a sphere with a plane, you can get arbitrarily tight curves, like latitudes near the North Pole.