Wiktionary
alt. (context geometry English) Any system of geometry not based on the set of axioms of Euclidean geometry, which is based on the three-dimensional space of common experience. n. (context geometry English) Any system of geometry not based on the set of axioms of Euclidean geometry, which is based on the three-dimensional space of common experience.
WordNet
n. geometry based on axioms different from Euclid's
Wikipedia
Behavior of lines with a common perpendicular in each of the three types of geometry
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In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry.
The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, 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 line through A intersects ℓ.
Another way to describe the differences between these geometries is to 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 (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) 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 intersect.
Usage examples of "non-euclidean geometry".
Your booby-trapped car, an arson case in Logan, Professor Brain's convenient disappearance, my cousin's death in Sumatra - and your six-dimensional non-Euclidean geometry.
Your booby-trapped car, an arson case in Logan, Professor Brain's convenient disappearance, my cousin's death in Sumatra -- and your six-dimensional non-Euclidean geometry.
Your booby-trapped car, an arson case in Logan, Professor Brain's convenient disappearance, my cousin's death in Sumatra-and your six-dimensional non-Euclidean geometry.
I think I saw this land of curve once, on a blackboard, when a class in non-Euclidean geometry had used the room before my own class in Eng Lit Pope to Swinb.
The development of non-Euclidean geometry led to the recognition of the fact, that we can cast doubt on the infiniteness of our space without coming into conflict with the laws of thought or with experience (Riemann, Helmholtz).
For example, Euclidean geometry is useful on Earth, but out in the great depths of space a non-Euclidean geometry is more practical.
These representations would quickly have gone from solid geometry through non-Euclidean geometry to bewilderment, had not a computer simultaneously developed the appropriate equations.
Janos Bolyai, one of the independent inventors of non-Euclidean geometry.
Maybe the peculiar features of the Eye, like its non-Euclidean geometry, are there solely as a puzzle-box for us.
It is a commonplace that any equation can be expressed in the figurative language of non-Euclidean geometry and represented in three dimensions.
It is one of the technical expressions used in non-Euclidean geometry.