How should we define a motion of the Poincaré universe? Intuitively, in the Euclidean plane, a motion is a transformation that "moves things around", without changing their size or shape, or the relationships among them. We'd like motions in the Poincaré universe to do the same. A "transformation"? We need a bijection (one-to-one correspondence) from the set of points on the unit disk onto itself. "Size"? The transformation should at least preserve distances. "Shape"? It should preserve angles as well. "Relationships among objects"? Well, it should at least transform all points on one line into points on another.

Now, a line is part of a Möbius circle, so perhaps we should be looking at transformations which preserve these circles. We already have a large collection of these: the Möbius transformations. If we choose those which map the Poincaré universe onto itself (we don't care what they do to points outside the edge), we have bijective transformations. [All Möbius transformations which map the edge onto itself either leave the inside in (like reflection in the real axis, for example), or switch inside and outside (like inversion in the edge). We want only the former type.] These Möbius transformations also map Poincaré lines onto lines, since they map Möbius circles perpendicular to the edge into circles perpendicular to the edge. Furthermore, it seems likely that these transformations will also preserve distances and angles, since both are calculated in terms of cross ratios: this part, we'll have to check.

So we make the following definition: a Poincaré motion is the restriction to the disk of any Möbius transformation mapping the circle onto itself and the interior of this circle onto the interior. The motion will be called direct or opposite whenever the corresponding Möbius transformation is direct or opposite.