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  2. Elliptic geometry - Wikipedia

    en.wikipedia.org/wiki/Elliptic_geometry

    Elliptic geometry may be derived from spherical geometry by identifying antipodal points of the sphere to a single elliptic point. The elliptic lines correspond to great circles reduced by the identification of antipodal points. As any two great circles intersect, there are no parallel lines in elliptic geometry.

  3. Non-Euclidean geometry - Wikipedia

    en.wikipedia.org/wiki/Non-Euclidean_geometry

    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. In the elliptic model, for any given line l and a point A, which is not on l, all lines through A will intersect l.

  4. Elliptic curve - Wikipedia

    en.wikipedia.org/wiki/Elliptic_curve

    In projective geometry this set is simply the point = [::], which is thus the unique intersection of the curve with the line at infinity. Since the curve is smooth, hence continuous , it can be shown that this point at infinity is the identity element of a group structure whose operation is geometrically described as follows:

  5. Line (geometry) - Wikipedia

    en.wikipedia.org/wiki/Line_(geometry)

    In elliptic geometry we see a typical example of this. [1]: 108 In the spherical representation of elliptic geometry, lines are represented by great circles of a sphere with diametrically opposite points identified. In a different model of elliptic geometry, lines are represented by Euclidean planes passing through the origin. Even though these ...

  6. Clifford parallel - Wikipedia

    en.wikipedia.org/wiki/Clifford_parallel

    In elliptic geometry, two lines are Clifford parallel or paratactic lines if the perpendicular distance between them is constant from point to point. The concept was first studied by William Kingdon Clifford in elliptic space and appears only in spaces of at least three dimensions.

  7. Foundations of geometry - Wikipedia

    en.wikipedia.org/wiki/Foundations_of_geometry

    The summit angles of a Saccheri quadrilateral are obtuse in elliptic geometry. The sum of the measures of the angles of any triangle is greater than 180° if the geometry is elliptic. That is, the defect of a triangle is negative. [80] All the lines perpendicular to a given line meet at a common point in elliptic geometry, called the pole of ...

  8. Elliptic coordinate system - Wikipedia

    en.wikipedia.org/wiki/Elliptic_coordinate_system

    In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci F 1 {\displaystyle F_{1}} and F 2 {\displaystyle F_{2}} are generally taken to be fixed at − a {\displaystyle -a} and + a {\displaystyle +a} , respectively, on the x ...

  9. Ellipse - Wikipedia

    en.wikipedia.org/wiki/Ellipse

    An ellipse (red) obtained as the intersection of a cone with an inclined plane. Ellipse: notations Ellipses: examples with increasing eccentricity. In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant.