What is Non-euclidean geometry: Definition and 14 Discussions

In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either relaxing the metric requirement, or replacing the parallel postulate with an alternative. 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 the same plane):

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.

View More On Wikipedia.org
  1. Ahmed1029

    I What's the definition of angle in a curved space embedded in a higher Eucledian space?

    I don't want to post this in a math forum because it's very basic and I just want a straightforward answer, not something math heavy . What's the definition of angle in a cuved space embedded in a higher eucledian space? Like when I have a spherical surface in 3d eucledian space and want to work...
  2. B

    A How Does Non-Euclidean Geometry Differ from Euclid's Postulates?

    In the book: The Elements Euclid defined 5 postulates: 1) A straight line segment can be drawn joining any two points. 2) Any straight line segment can be extended indefinitely in a straight line 3) Given any straight line segment, a circle can be drawn having the segment as radius and one...
  3. SamRoss

    B Is my intuitive way of thinking about non-Euclidean geometry valid?

    I always tend to get confused when thinking about non-Euclidean geometry and what straight lines and parallel lines are. If I think of a sphere, I get how two people driving north would almost mysteriously intersect at the North Pole and how the angles of a triangle would not add up to 180...
  4. nomadreid

    I What Lobachevski meant by parallel lines

    I am not sure that this is the right rubric for this question, as it is historical, but as it is part of the history of Model Theory, I am putting it here. I will not be offended if the moderators decide that it doesn't belong here. In https://arxiv.org/pdf/1008.2667.pdf, the author states...
  5. A

    Geometry Any good books on non-Euclidean geometry?

    As the title implies, I'm looking for books on non-euclidean geometry. I'm not looking for very advanced thing, more on some book with a good introduction to this topic.
  6. D

    Non-Euclidean geometry and the equivalence principle

    As I understand it, a Cartesian coordinate map (a coordinate map for which the line element takes the simple form ##ds^{2}=(dx^{1})^{2}+ (dx^{2})^{2}+\cdots +(dx^{n})^{2}##, and for which the coordinate basis ##\lbrace\frac{\partial}{\partial x^{\mu}}\rbrace## is orthonormal) can only be...
  7. 24forChromium

    Non-Euclidean area defined by three points on a sphere

    A sphere with radius "r" has three points on its surface, the points are A, B, and C and are labelled (xa, ya, za) and so on. What is the general formula to calculate the area on the surface of the sphere defined by these points?
  8. NihalRi

    What topic should I base a non-Euclidean geometry paper on?

    Homework Statement A short paper 12-16 pages I'm also fairly new to this topic Homework EquationsThe Attempt at a Solution I tried to ecplain it's application using the Robertson walker mrttuc but it ended upmlooking too much like a physics psper:/
  9. J

    A circle in a non-euclidean geometry

    Homework Statement Consider a universe described by the Friedmann-Robertson-Walker metric which describes an open, closed, or at universe, depending on the value of k: $$ds^2=a^2(t)[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+sin^2\theta d\phi^2)]$$ This problem will involve only the geometry of space at...
  10. EuclidPhoton

    Complex Geometric Theories and Molecules

    If QM is a statistical model to approximate something underlying space time we don't quite understand yet, and there is a complex geometry underlying space time, is it possible to find other ways to simplify molecular optimizations and electron interactions in computational chemistry using...
  11. Y

    Dimensionality of Non-Euclidean geometry

    In introductions to non Euclidean geometry, examples are often given in the form of measuring angles on a 2D surface embedded in a 3D space, such as the surface of a sphere or on a saddle surface. This gave me the initial impression that 3D non Euclidean geometry would have to be embedded in 4...
  12. Ƒ

    Circles in Non-Euclidean Geometry

    Are circles considered straight lines in Non-Euclidean Geometry?
  13. R

    Non-Euclidean Geometry question

    I'm trying to prove that the interior angles on a spherical triangle sum to \pi + (A)/(a)^2 where A is the area of the triangle and a is the radius of the spherical space. I think I know how to prove it, but there is one part that has me stumped. I'm using the General Relativity book...
  14. D

    Suggestions for non-euclidean geometry (analytical) books?

    1089 and all that. A Journey into Mathematics. David Acheson Oxford University Press (www.oup.com) ISBN 0-19-851623-1 Introduction: You like maths don't you? Of course you do, because you would be making haste to get out of my sight before I damn your eyes with a strategically and...