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Any system of geometry that is not based on the parallel postulate of Euclid, which says that one and only one line can be drawn through a point outside a given line and parallel to that line. Euclidean geometry deals with the geometry of our everyday world. The parallel postulate of Euclid seems intuitively clear, but nobody has ever been able to prove it.
If we replace the parallel postulate of Euclid with an assumption that there is more than one line parallel to a given line through a given point, we have a non-Euclidean geometry called hyperbolic geometry. If we assume that there are no parallel lines, we have a non-Euclidean geometry called elliptic geometry.
