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On Lobachevsky`s trigonometric formulae
On Lobachevsky`s trigonometric formulae

... the family of non-intersecting lines was used by the three founders of hyperbolic geometry. 14In using the transitivity property, it is necessary to keep track of the direction of parallelism for pairs of parallel lines involved. ...
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Duality (projective geometry)

In geometry a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept. There are two approaches to the subject of duality, one through language (§ Principle of Duality) and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a duality. Such a map can be constructed in many ways. The concept of plane duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry.
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