the close axiomatic study of euclidean geometry led to the construction of the lambert quadrilateral and the saccheri quadrilateral. these structures introduced the field of non-euclidean geometry where euclid's parallel axiom is denied. gauss, bolyai and lobachevski independently constructed hyperbolic geometry, where parallel lines have an angle of parallelism that depends on their separation. this study became widely accessible through the poincaré disc model where motions are given by möbius transformations. similarly, riemann, a student of gauss's, constructed riemannian geometry, of which elliptic geometry is a particular case.