Mappings preserving the area equality of hyperbolic triangles are motions

Abstract

It is shown that a mapping ϕ : A → B between models A and B of elementary plane hyperbolic geometry, coordinatized by Euclidean ordered fields, that maps triangles having the same area and sharing a side into triangles that have the same property, must be a hyperbolic motion onto ϕ(A). The relations that Tarski and Szmielew used as primitives for geometry, the equidistance relation ≡ and the betweenness relation B are shown to be positively existentially definable in terms of the quaternary relation Δ, with Δ(abcd) standing for "the triangles abc and abd have the same area." Mathematics Subject Classification (2000). Primary 51M10; Secondary 51M25, 03C40.

A New Hyperbolic Area Formula of a Hyperbolic Triangle and Its Applications

“…Demirel [12], Yang and Fang [13,14] gave characterizations of Möbius transformations by use of hyperbolic triangles or Lambert quadrilaterals. Pambuccian [15] showed that mappings preserving the area equality of hyperbolic triangles are motions. One can see [16][17][18] for more characterizations about quasiconformal mappings and harmonic quasiconformal mappings in the sense of hyperbolic metrics.…”

A characterization of Möbius transformations by use of hyperbolic triangles

“…In addition to this, in literature, there are many characterizations of Möbius transformations via some hyperbolic polygons; see [1][2][3][4][5][6][7]. For other characterizations of hyperbolic isometries (in fact, these are Möbius transformations), we refer the reader to [8,9] and [10]. The main purpose of this paper is to present a new characterization of Möbius transformations.…”