In Euclidean plane, a set of points that have the same distance from a given line is a parallel line (or two parallel lines, one on each side, depending on how exactly you define it). However, as we've seen, two lines cannot be parallel in the usual sense of word in the hyperbolic geometry. Instead, a set of points with the same distance from a given line is an equidistant curve. In Poincaré projection, equidistant curves look like circle arcs that intersect the horizon in two points. In fact, straight lines can be considered a special case of equidistant curves where the distance from the guiding straight line is zero.