In both Euclidean and hyperbolic geometries the reflection in a line m is the transformation of the plane, which leaves each point of m fixed and transforms a point A not on m as follows: Let M be the projection from A to m. Then, the image of A under this reflection is the unique point A' lying on the other side of m with respect to A, and M is equidistant from A and A'.

Figure 21

Let m be a u-line in the upper half-plane and let r be the circle containing m (see Figure 21). Suppose A is a point not on m, let A' be the inverse of A in r. By the theorem mentioned in the section ‘perpendicular from point’, the u-line n through A and A' is orthogonal to m, and let M be the projection of A on m (in non-Euclidean sense). We can also see that the end point P of n is the inverse of another end point P' in r and points on r are fixed under inversion in r.

Since inversion preserves the cross-ratio used to define hyperbolic length (will be introduced later), we have:

In conclusion, reflection across the u-line m is represented under the upper half-plane model by inversion in the circle r, which contains the u line m.

By the above discussions, we use the original tool ‘Inverse’ in Cabri Geometry II to produce our reflection macros. We omit the details.

Download This Macro reflect_point.mac