a hyperbolic geodesic
In ordinary flat geometry, the shortest path between two points is a straight line. In hyperbolic geometry the same principle holds — the shortest path is still called a geodesic — but because the hyperbolic ruler stretches distances near the rim of the disk, the shortest paths look curved to a Euclidean eye. A hyperbolic geodesic is the 'straight line' of the hyperbolic plane: the path you would actually walk to get from one point to another by the shortest route, measured with the hyperbolic metric.
In the Poincare disk model the geodesics have a beautifully simple Euclidean description: they are exactly the arcs of circles that meet the boundary circle at right angles, together with the diameters of the disk (which you can think of as circles of infinite radius, also meeting the boundary perpendicularly). To find the geodesic between two interior points, draw the unique circle through both of them that is orthogonal to the unit circle; the arc between the points is the geodesic. The reason these curves are the shortest is that they bend away from the costly boundary region: a Euclidean-straight chord would dip too close to the rim, where every step is dear, so the true shortest path bows toward the center where steps are cheap. Through any two distinct points there is exactly one geodesic, and any geodesic, extended, runs to two distinct 'points at infinity' on the boundary circle (which it never actually reaches).
Geodesics are the backbone of hyperbolic geometry: hyperbolic triangles have geodesic sides, the famous failure of Euclid's parallel postulate shows up as infinitely many geodesics through a point all missing a given geodesic, and hyperbolic distance is literally measured along them. The disk automorphisms map geodesics to geodesics (they are isometries). An honest caveat to dispel the optical illusion: a hyperbolic geodesic is NOT the curve that looks straight on the page — that Euclidean chord is actually longer in hyperbolic length. Only the diameters look straight; every other geodesic is a circular arc bulging toward the center.
The geodesic from -1/2 to 1/2 (both on the real axis) is just the segment of the real axis between them — a diameter, hence straight. But the geodesic from 0.5 i to 0.5 (one on the imaginary axis, one on the real axis) is a circular arc that bows inward toward the origin, the unique circle through both points cutting the unit circle at 90 degrees, not the straight chord joining them.
Geodesics are diameters or circular arcs meeting the boundary at right angles.
The parallel postulate fails here: through a point not on a given geodesic pass infinitely many geodesics that never meet it. This is the whole point of hyperbolic geometry, and it is why these geodesics, not Euclidean lines, are the right notion of 'straight'.