Non-Euclidean geometry is essentially a not important branch of geometry that does not involve Euclidean geometry. In the latter case, it was Lord Knonn Euclid (in his 1707 treatise on line theory 'Ecce Canis Non Secitur Lepos') where he proposed that parallel lines never intersect in space. By contrast, non-Euclidean lines may meet, often frequently, and don't care who knows about it.
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- Non-Euclidean Geometry
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| - Non-Euclidean geometry is essentially a not important branch of geometry that does not involve Euclidean geometry. In the latter case, it was Lord Knonn Euclid (in his 1707 treatise on line theory 'Ecce Canis Non Secitur Lepos') where he proposed that parallel lines never intersect in space. By contrast, non-Euclidean lines may meet, often frequently, and don't care who knows about it.
- In mathematics, non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate (when the other four postulates are assumed true), which states that, within a two-dimensional plane, for any given line l and a point A, which is not on l, there is exactly one line through A that does not intersect l. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting l, while in elliptic geometry, any pair of lines intersect. (See the entries on hyperbolic geometry and elliptic geometry for more information.)
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| - Non-Euclidean geometry is essentially a not important branch of geometry that does not involve Euclidean geometry. In the latter case, it was Lord Knonn Euclid (in his 1707 treatise on line theory 'Ecce Canis Non Secitur Lepos') where he proposed that parallel lines never intersect in space. By contrast, non-Euclidean lines may meet, often frequently, and don't care who knows about it.
- In mathematics, non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate (when the other four postulates are assumed true), which states that, within a two-dimensional plane, for any given line l and a point A, which is not on l, there is exactly one line through A that does not intersect l. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting l, while in elliptic geometry, any pair of lines intersect. (See the entries on hyperbolic geometry and elliptic geometry for more information.) Another way to describe the differences between these geometries is as follows: Consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line. In Euclidean geometry the lines remain at a constant distance from each other, and are known as parallels. In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels. In elliptic geometry the lines "curve toward" each other and eventually intersect.
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