a. Elliptic Geometry One of its applications is Navigation. {\displaystyle z=x+y\epsilon ,\quad \epsilon ^{2}=0,}  There are Euclidean, elliptic, and hyperbolic geometries, as in the two-dimensional case; mixed geometries that are partially Euclidean and partially hyperbolic or spherical; twisted versions of the mixed geometries; and one unusual geometry that is completely anisotropic (i.e. In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, but did not realize it. parallel lines is established with the aid of the assumption that a straight line is infinite, it comes as no surprise that there are no parallel lines in the two new, elliptic geometries. So circles are all straight lines on the sphere, so,Through a given point, only one line can be drawn parallel … Euclidean geometry:Playfair's version: "Given a line l and a point P not on l, there exists a unique line m through P that is parallel to l." Euclid's version: "Suppose that a line l meets two other lines m and n so that the sum of the interior angles on one side of l is less than 180°. Elliptic geometry is a non-Euclidean geometry with positive curvature which replaces the parallel postulate with the statement "through any point in the plane, there exist no lines parallel to a given line." 3. h�bf������3�A��2,@��aok������;:*::�bH��L�DJDh{����z�> �K�K/��W���!�сY���� �P�C�>����%��Dp��upa8���ɀe���EG�f�L�?8��82�3�1}a�� �  �1,���@��N fg\��g�0 ��0� Above, we have demonstrated that Pseudo-Tusi's Exposition of Euclid had stimulated borth J. Wallis's and G. Saccheri's studies of the theory of parallel lines. Through a point not on a line there is more than one line parallel to the given line. + In three dimensions, there are eight models of geometries. Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry several years before, though he did not publish. to a given line." A line is a great circle, and any two of them intersect in two diametrically opposed points. The difference is that as a model of elliptic geometry a metric is introduced permitting the measurement of lengths and angles, while as a model of the projective plane there is no such metric. They are geodesics in elliptic geometry classified by Bernhard Riemann. x An important note is how elliptic geometry differs in an important way from either Euclidean geometry or hyperbolic geometry. The perpendiculars on the other side also intersect at a point, which is different from the other absolute pole only in spherical geometry , for in elliptic geometry the poles on either side are the same. He worked with a figure that today we call a Lambert quadrilateral, a quadrilateral with three right angles (can be considered half of a Saccheri quadrilateral). There are NO parallel lines. 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