In this chapter, we use inversive geometry to construct the model of a hyperbolic plane — a neutral plane that is not Euclidean. Namely, we construct the so-called conformal disc model of the hyperbol...In this chapter, we use inversive geometry to construct the model of a hyperbolic plane — a neutral plane that is not Euclidean. Namely, we construct the so-called conformal disc model of the hyperbolic plane. This model was discovered by Beltrami and is often called the Poincaré disk model.
The Poincaré disk model for hyperbolic geometry is the pair (D,H) where D consists of all points z in C such that |z|<1, and H consists of all Möbius transformations T for which T(D)=D. The set D is c...The Poincaré disk model for hyperbolic geometry is the pair (D,H) where D consists of all points z in C such that |z|<1, and H consists of all Möbius transformations T for which T(D)=D. The set D is called the hyperbolic plane, and H is called the transformation group in hyperbolic geometry.
