The Elements of Non-Euclidean Geometry

A₁ absolute polar antipodal points axioms axis B₂ Bolyai bundle of lines called centre Chap circle circumcircle coincide collinear common perpendicular conic constant coordinates corresponding cosh coth cross-ratio curve cut BB determine dihedral angle distance draw elliptic geometry equal equation equidistant equidistant-curve Euclid euclidean geometry exterior angle fixed point formulae Gauss given line Hence homographic horocycle horosphere hyperbolic geometry hypothesis ideal points imaginary infinity or ideal inversion line at infinity lines and planes Lobachevsky locus marginal images non-euclidean geometry opposite orthogonal pair parallel lines passes pencil of lines point of intersection points at infinity projective geometry proof prove quadrilateral radius represented right angles Saccheri sides sinh space sphere spherical geometry straight line surface tangents tanh theorem theory of parallels transformation triangle ABC trigonometry vertex vertices y₁ zero