# Questions tagged [noneuclidean-geometry]

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

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### What are All the Reflections in Minkowski Space $\mathbb{R}^{1,n}$?

All the literature on reflections in minkowski space, that I have found, have defined ways to reflect about an arbitrary planes or lines and they always add the disclaimer eventually that the plane or ...
1 vote
43 views

### Smoothness of the Fréchet Function on Riemannian Manifolds

Suppose $M$ is a compact Riemannian manifold and let $d$ be the induced distance function on $M$. Let $\mu$ be a probability measure on $M$ with continuous density. The Fr$\acute{\mathrm{e}}$chet ...
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### Is there a name for geometries that can be approximated as Euclidean over infinitesimal scales?

I understand that some non Euclidean Geometries, such as Hyperbolic and Spherical Geometry, that are can be approximated with ordinary Euclidean Geometry over very tiny increments of space. For ...
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### How can I get the appropriate coordinates to display 3D-hyperbolic space in the Beltrami Klein model?

I am trying to create a "game" with a hyperbolic world. The goal is to have some objects (for example trees, cars, buildings, cubes, spheres, ...) which are displayed using the Beltrami ...
• 29
1 vote
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### Axiomatizing higher-dimensional geometries

While there are various axiom systems for two and three-dimensional geometries (Hilbert, etc.), it seems not at all clear that this axiomatic approach generalizes well to more than three dimensions. ...
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### Prove the following statement in Mobius geometry

Let $C$ be a cline, $z$ and $z^{*}$ be two distinct symmetric points w.r.t. $C$. Then Any cline $C'$ that is orthogonal to $C$ and passing through $z$, must also pass through $z^{*}$. And its ...
1 vote
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### How does Euclid's Fifth postulate not hold here?

If look at the Latitude lines - blue circles below - and pick one, each point $P$ not in one selected circle/line, will only lie in one other blue circle, circle here taken as line, and will follow ...
1 vote
63 views

### spherical and hyperbolic triangle definitions with geodesics

What is the equivalent of "collinear" for points on a geodesic? Are they just called "cogeodesic" or something? Since the definition of a Euclidean triangle is "any three ...
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### Find equation for surface area of ellipse projected on a sphere

A grad school problem has been haunting me for years and I have never been able to find it on the google machine nor was the professor who assigned the problem able to describe how to actually solve ...
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### Correspondence between Euclidean and non-Euclidean geometries

I would like to ask correspondence between Euclidean and non-Euclidean geometries. In the science and hypothesis, Poincare says that non-Euclidean geometry can be translated into Euclidean geometry ...
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### How to define addition law in hyperboloid model(lorentz space) of hyperbolic space

I know mobius addition and Einstein addition are well defined in Poincaré ball model . But how to define addition in hyperboloid model(lorentz space) of hyperbolic space,and can we define the exact ...
19 views

### Given two points $P,Q$, and a hyperbolic line $l = C\cap \mathbb{H}^2$. How to find another hyperbolic line $m$ such that $\Omega_m\Omega_l(P)=Q$

The hyperbolic line $l=C\cap \mathbb{H}^2$ has center $(\frac{5}{4}, \frac{1}{4})$ and radius $r=\frac{\sqrt{10}}{4}$. Given two points $P=(\frac{5}{8}, -\frac{5}{8})$ and $Q=(\frac{3}{4},0)$, I need ...
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### Creating an (linear) interpolant of points on the surface of d-dimensional hyper-sphere.

Let's say we have a $d$-dimensional unit hypersphere. On the surface of the hypersphere, we have points that have a value of either $\{-1, 1\}$. I wish to create a function $\Phi$ that interpolates ...
1 vote
243 views

### On the Consistency of Non-Euclidean Geometry

I've recently found a really old "Philosophy of Math" book in my University library, and in the book it says that the it has been proven that : if non-euclidean geometry is inconsistent, ...
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### Do there exist (non-Euclidean) equilateral $n$-gons whose angles are all right angles, for $n\geq6$? (via Numberphile)

In the YouTube video "5-Sided Square" from Numberphile, Cliff Stoll states that, if a "square" is a shape with sides of equal length whose angles are all $\pi/2$, then we can find: ...
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