in euclidean plane one can move polygons like rectangles, triangles etc. around by isometries, e.g. translations. For instance if we consider a rectangle with midpoint $0\in\mathbb{R}²$ then the image of this rectangle under the translation $T(y)=y+x$ is a rectangle around $x\in\mathbb{R}²$.
Is there an equivalent in hyperbolic geometry? Is there some polygon like rectangles etc. around a point $z$ which one can moove in hyperbolic space by isometris to any other point? Usual translations wouldn't work, since they are no isometries. Can anyone help me?
Best Regards.
Best Answer
In the Euclidean space, sides of polygons are lines. In the hyperbolic space, sides of polygons are hyperbolic lines; in the ball model these are the diameters of the ball and spherical arcs that meet the boundary orthogonally. The sides always bend in in the ball model, as compared to the Euclidean triangle with same corners.
Isometries conserve hyperbolic lines (since they are geodesics in the hyperbolic metric). Isometries in either of the mentioned models are precisely the Möbius transformations that fix the boundary. (In 2D we will also have to include complex conjugation as a Möbius mapping if we don't require preservation of orientation.) These isometries are the natural translations in hyperbolic geometry.
These two models are the Poincaré models of hyperbolic space. There are other models as well, and much more discussion, on that same Wikipedia page.