Models for Euclidean and equiform triangle geometry (Q1268023)

Associate to a triangle \(\Delta(s_1, s_2, s_3)\) of the Euclidean plane \(E^2\) (or of the equiform plane (where a triangle is an equivalence class of similar triangles in \(E^2\))) with sides \(s_1, s_2, s_3\) (in a fixed numbering of the sides) the point \(S(\Delta): = (s_1^2, s_2^2, s_3^2)\) of the Euclidean space \(E^3\) (or of the projective plane \(P^2\), respectively). The object of this paper is to investigate the relationship between triangle properties and properties of the image of the mapping \(S\), which is the part of the interior of a rotation cone in \(E^3\) which lies in the first octant (or, in the equiform case, the interior of a conic section in \(P^2\), which may be thought of as the hyperbolic plane \(H^2\) with the conic section as Absolute). In the Euclidean case the image sets of equilateral, right-angled, fixed area, and fixed perimeter triangles are described. In the equiform case it is shown that one can establish a natural isomorphism via \(S\) between the Klein and the Poincaré (half-plane) models of hyperbolic geometry. The usefulness of the method is tested on two practical applications: reconstruction from three orthographic projections and the trilateration problem (describe the possible configurations of six points \(P_1, P_2, P_3, Q_1, Q_2, Q_3\) given the nine distances \(d(P_i, Q_j)\)).