Let's go to to the normal infinite plane for a moment.

You can use a map that is inside a circle with r=1. The objects get deformed, but points have a 1 to 1 correspondence. Lines that pass though 0 look straight, but other lines are curved.

Measuring a distance is hard, you have to use some weird rules.

If you draw a segment of length 0.001 segment in the circular map, it has almost the same length in the real infinite map.

If you draw a segment of length 0.001 segment near the border of the circular map, it's a huge thing in the infinite map.

Moreover, a line that pass thorough 0 has apparent length 2 in the map, but represent an infinite length in the plane

Note that the border of the circle is outside the plane.

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The reverse happen if you have a map of the Earth. You can draw on the map with a pencil a long segment near the pole, but it represents a small curved segment in the Earth.

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Back to your question ,,,

It's on the hyperbolic plane, not in the usual euclidean plane. So the map is only the top half, and the horizontal line = axis x is outside, it's the border.

Length are weird, and a 0.001 segment draw with a pencil on the map far away from the x axis is small in the actual hyperbolic plane, but a 0.001 segment draw with a pencil on the map near the x axis is very long in the actual hyperbolic plane.

The circles "touch" the x axis. In spite they look short when you draw them with a pencil, they part that is close to the x axis has a huge length in the hyperbolic plane.