Parabolic isometry of the hyperbolic plane
= Parabolic isometry of the hyperbolic plane
A nonidentity orientation-preserving isometry of the <hyperbolic plane> is parabolic when it has one fixed point on the ideal boundary and none in the plane. In the <upper half-plane model>, every parabolic isometry is conjugate to a nonzero horizontal translation.
This is one type of <isometry> of the <hyperbolic plane>, classified by its fixed points.