# Questions tagged [noneuclidean-geometry]

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

158 questions
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### Rendering of Riemann hypothesis as moot if non Euclidean geometry (hyperbolic or elliptical) is refuted? [closed]

If non Euclidean geometry (hyperbolic or elliptical) is refuted, then does that render the Riemann hypothesis as moot?
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### Right triangle circumscribed by a horocycle

Is it possible that a right triangle is circumscribed by a horocycle? Or, is this statement a theorem in the hyperbolic gemetry? For any horocycle $\gamma$, there are no three distinct ordinary ...
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### Definition of constant-curvature curve embedded on an Ellipsoid of revolution

I am interested in identifying a type of curve so I can do literature review on it. What is the name of a curve embedded on an ellipsoid of revolution in which the curvature of the embedded curve is ...
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### Name of Non-Unique Coordinate systems

Are there some examples (and a name) for non-unique coordinates (non-unique meaning may have multiple ways to represent the same point). Such as the one below.
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### Trajectory of points at infinity

If two points initially starts at zero and travel to infinity,what is the nature of the space if the trajectory where to: 1) converge 2) diverge & what will happen to the trajectory of two points ...
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### Surface area of a sphere segment. [duplicate]

I got this problem and solution in a paper but cannot find how they have solved it. Consider I have a sphere that is equally divided into two different patch(P, S). The sphere can rotate and translate(...
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### Area of touching part of Sphere to the wall.

I believe that it has a very simple explanation but one question stuck in my mind. What is the area between sphere and wall when it touches to it. If it is zero, why it is not occurring in real life?...
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### Proving that the geodesics of $S^n$ are its great circles

I am working my way through John G. Ratcliffes 'Foundations of Hyperbolic Manifolds', and while doing this, I need some help proving a theorem; Theorem 2.1.5 A function $\phi: R \rightarrow S^n$ is a ...
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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 ...
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### What would be the parametric equation of a geodesic, and a non euclidean equivalent of a circle, in this type of space?

In a two dimensional space there is a a line A, and a curve B. Curve B has the equation $y=ax+b$. If the arc length of curve B is very small then the arc length of curve B can be approximated by the ...
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### Is a $3D$ volume possible with only $3$ faces?

I was wondering if a $3D$ volume with only $3$ faces was possible. I know that in the Euclidean space, it is technically not possible (the minimum being $4$ faces), but maybe there was some other way. ...
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### Effects of radius on hypersphere.

I am struggling to understand how the curvature of a hypersphere decreases as its radius increases? The Wikipedia article on hypersphere does not give a mathematical or intuitive reason as to why this ...
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### theories where angles exist without a metric

(moved from https://mathoverflow.net/questions/307703/theories-where-angles-exist-without-a-metric) The underlying basic question, which I'm sure I'm not the first to ask, is what are the possible ...
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### Taxonomic ranks of spaces

Where might I find a good overview of different categories of space? I am most interested in the idea of a taxonomy as described here (multiple levels of classification listed with differentiating ...
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### Neutral geometry: If one triangle has angle-sum $180^\circ$, then all triangles have angle-sum $180^\circ$

One of the more interesting things we can say in neutral geometry (that is, without assuming the parallel postulate, but assuming e.g. the rest of Hilbert's axioms) is the following: Suppose that a ...
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### Similarities between non-Euclidean geometries

So I've looked into Euclidean, spherical/ellpitic, and hyperbolic geometry, and found some possible similarities. I'm not much of an expert, so I can't really verify them for myself. I'd like to know ...
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### Solving for angle of hyperbolic triangle in Poincare disk

I am working out an example problem trying to find the angles of a hyperbolic triangle in the Poincare disk model. I am getting inconsistent answers. For the sake of simplicity, I am using these ...
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### Mapping between the Poincare disk model and a negatively curved surface

I am comfortable using the Poincare disk as a model of hyperbolic geometry, but it has left me with a question for which I don't have a good answer: Is there a negatively curved surface that we could ...
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### Why is the center distorted in the hyperbolic circle on the Poincare Disk?

I know that it is partly because the distances get smaller logarithmically as you get towards the ‘edge’, and I know how to construct a hyperbolic circle on the Poincaré disk. I just don’t have the ...
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### Use scale in projection to solve for curvature

Our original plane has curvature $K$, which can be any real, and may not be constant, though as a function of location it will be smooth. Assume we have some Euclidean projected plane which is a ...
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### Relation between curvature on surface, curvature of surface in space, and curvature of space

Let's say we have some surface embedded in a higher dimensional space. The space has curvature $K_1$. The surface has curvature $K_2$. Call the curvature on the surface $K_3$. Is $K_3=K_1+K_2$? What ...
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### How many platonic solids do exist in non-euclidean space?

The proof that there exists only five platonic solids assumes that the angle between the adjacent sides must be less than 360°, because otherwise the surfaces would be flat or even overlap. However ...
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### Why is sphere non-euclidean space?

There are 5 axioms that define euclidean space and I believe that all hold also for a sphere. The definition of axioms from wikipedia: "To draw a straight line from any point to any point." ...