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Questions tagged [noneuclidean-geometry]

For general questions about non-Euclidean Geometry. Consider using more specific tags, like (projective-geometry), (hyperbolic-geometry), (spherical-geometry), etc.

Non-Euclidean geometry is the study of geometries with different versions of the parallel postulate. Intuitively you can think of this as doing geometry on surfaces besides the plane.

304 questions
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Can elliptic space be infinite?

The go-to example of elliptic space is a sphere where geodesics turn into great circles of finite length. But is it possible to have an elliptic space which doesn't 'merge' with itself once it's made a full turn? ie. infinite, unbounded,…
David Rutten
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What is the definition of a straight line and parallel lines in non-euclidean geometry?

Recently, I've been trying to learn a bit of Non-euclidean geometry. I hear people talking about parallel straight lines meeting. But I don't understand this. I mean I get the idea visually, but I am confused about the definitions. I've learnt so…
Chandrahas
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Does there exist a general formula relating the diameter of a circle to its radius in the case of a non-euclidean geometry?

It is well known that the ratio of the circumference of a circle to its diameter in Euclidean geometry is the constant $\pi$. I also understand that in the case of non-euclidean geometry this ratio is in general not a constant. What I would like to…
John Gould
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Angle measure in non-Euclidean geometry

I'm reading Donal O'Shea's The Poincare Conjecture, a nontechnical book for mainstream audiences. It's reminded me of a question I've long had -- which this book hasn't answered. In non-Euclidean space, triangles don't necessarily measure…
DBS
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Is there any space in which circles can be tiled without gaps?

Octagons can't be tiled in flat space but they can in hyperbolic space. Likewise pentagons can be tiled on a sphere. Imagine you had some flat circles then you glued them by their edges to create a honey cone structure. You'd have to bend the…
zooby
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Distance for Non-Euclidean Space

In the book "In pursuit of the Unknown" by Ian Stewart, page 19 of chapter "Pythagoras' Theorem" shows the equation for the distance between two points in a non-euclidian space from point $(x,y,z)$ to $(x+dx,y+dy,z+dz)$ as: $ds^2 = X dx^2 + Y dy^2 +…
paul0207
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How is Euclid's 5th Postulate wrong exactly? Why do we think that parallel lines and geodesics are the same?

I am wondering about the axiomatic method here and Euclid's 5th often comes up. I don't understand why they said that parallel lines could meet after all in Non-Euclidean Geometry. Surely Euclid did not have this exact definition of a line in mind…
Pineapple Fish
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Why is the fourth angle of a Lambert quadrilateral obtuse in elliptic geometry?

I know that fourth angles of Lambert quadrilaterals are acute in hyperbolic geometry and right in Euclidean, but why are fourth angles only able to be obtuse in elliptic geometry? Edit: Some background information. We are learning about Lambert and…
Dawson
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Sum of external angles in non-euclidean geometry

When Gauss-Bonnet thm is used on a constant Gauss curvature surfaces of $ (~ K=-1/a^2,K=0,K<=+1/a^2)$ along a closed radial contours, the exterior angle rotation sum can be shown to be respectively: $$ \Sigma \psi= 2 \pi (1+\Delta r/a),~~2 \pi (1+…
Narasimham
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Given lengths of sides and and angles between the sides of a triangle find the surface that makes it possible

For example given a tringle with each side equal to $1$ and angles between each side being perpendicular, then this triangle is on a quarter of sphere with radius $\dfrac {2}{\pi}$. Is this example of triangle in elliptic geometry? what would…
jimjim
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Families of planar curves that are neither parallel nor intersect

A family of rectangular hyperbolas of the form $xy = k$ (e.g. $xy = 2$, $xy = 3$ etc) consists of curves that are neither parallel nor intersect. $k$ is any constant. Similar families seem to be $xy^2 = k$, $y = e^{(k/x)}$ and so on $1)$ Is there…
phil342
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Parallel postulate of euclid

How euclid's 5th postulate transforms in non euclidean geometry? How to visualize that there are more than one lines that are parallel to the point?
user448047
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