Questions tagged [noneuclidean-geometry]

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

Filter by
Sorted by
Tagged with
0 votes
0 answers
15 views

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 ...
intravertig0's user avatar
1 vote
0 answers
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 ...
Yueqi's user avatar
  • 11
0 votes
0 answers
20 views

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 ...
Anders Gustafson's user avatar
2 votes
0 answers
89 views

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 ...
juwa's user avatar
  • 29
1 vote
0 answers
74 views

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. ...
NikS's user avatar
  • 739
0 votes
0 answers
50 views

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 ...
lcthaha's user avatar
1 vote
1 answer
76 views

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 ...
user avatar
1 vote
0 answers
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 ...
John's user avatar
  • 161
0 votes
0 answers
41 views

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 ...
RocketTwitch's user avatar
0 votes
1 answer
45 views

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 ...
Atsu's user avatar
  • 21
0 votes
0 answers
45 views

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 ...
Zoe.peace's user avatar
0 votes
0 answers
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 ...
pcZhang's user avatar
  • 51
0 votes
0 answers
25 views

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 ...
Teddyzander's user avatar
1 vote
2 answers
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, ...
Diana's user avatar
  • 15
0 votes
1 answer
109 views

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: ...
Shaun's user avatar
  • 44.5k
0 votes
0 answers
29 views

How long are the diagonals of a unit quadrilateral?

Suppose we have a normed plane. If it helps, assume the norm is strictly convex. Take your favourite point on the unit sphere $(X,Y)$ and take two points $(a,b)$ and $(c,d)$ where the unit sphere at $(...
Daron's user avatar
  • 10.3k
1 vote
1 answer
47 views

Do hyperbolic rotations/lorentz transformations behave like this or did I implement them wrong?

(not sure if this belongs here, in StackOverflow or in the physics StackExchange, but I believe my issue is with the math so I'm posting here.) I've recently been reading the book Dichronauts by Greg ...
The Zip Creator's user avatar
0 votes
0 answers
35 views

How to render .obj files in a non-euclidean way?

Apologies in advance if this question is misguided or foolish; I'm learning. My goal is to write a non-euclidean renderer in the Rust programming language, and I'm wondering if anyone could tell me if ...
Kai's user avatar
  • 109
0 votes
0 answers
45 views

Where do I get started with rendering curved space?

I want to program interactive renderings of spaces of arbitrary curvature (such as spherical or hyperbolic), though I have no idea how to begin. For instance, I have no idea whether I should use ...
Kai's user avatar
  • 109
0 votes
0 answers
40 views

Isometries of hyperbolic space $\mathbb{H}^n$ [duplicate]

First of all, let's fix some notation: $$g^n_{\nu}="\text{standard nondegenerate bilinear form on $\mathbb{R}^n$ with index of negativity $\nu$}"$$ $$\mathbb{R}^n_{\nu}="\mathbb{R}^n \text{ equipped ...
Kandinskij's user avatar
  • 3,699
1 vote
0 answers
86 views

Imagining alternatives to the Cartesian coordinate system

First, I should disclose that I am no mathematician, I am an architect and artist with a crush on geometry :) So please be lenient if my question is non-sense! I'm interested in exploring the ...
Gio's user avatar
  • 19
2 votes
0 answers
83 views

An alternate 3D geometry where there is a single origin where parallel lines intersect?

I'm a programmer trying to get my head around the math I need for a problem and was hoping it has been covered in one branch of mathematics or another. My model is a continuously expanding space (like ...
norlesh's user avatar
  • 125
2 votes
2 answers
131 views

In spherical geometry, are disks convex, and are chords internal to a circle?

In Euclidean geometry, a chord of a circle is internal to the circle. However, the proof of this relies on the External Angle Theorem, which does not hold for spherical geometry. In spherical ...
SRobertJames's user avatar
  • 4,224
0 votes
0 answers
114 views

Are Pasch's axiom and plane separation postulate equivalent?

What is the relationship between Pasch's axiom, plane separation postulate, and betweeness? Specifically, are any of the following derivable from each other: Pasch's axiom In the plane, a line ...
SRobertJames's user avatar
  • 4,224
0 votes
0 answers
54 views

Why are balls in the taxicab metric always hyperoctahedrons?

Let $\ell_1^n$ be the space $\mathbb R^n$ equipped with the $L_1$, i.e., taxicab metric. When $n=2$, a unit ball is a square with sides parallel to $45^\circ$. When $n=3$, we get an octahedron, which ...
pyridoxal_trigeminus's user avatar
1 vote
0 answers
91 views

Non-Euclidean geometry which is the union of Euclidean geometries?

While reading on Wikipedia about non-Euclidean geometry, I tried to think of very simple geometries that invalidate Euclid's first postulate. This lead me to the following question. I do not have any ...
user1136247's user avatar
0 votes
0 answers
68 views

Intersection of 2 circles on the surface of a sphere?

I'm out of my depth playing with non-euclidean geometry, so if terminology is off or I'm missing critical things, leave a comment and I'll edit. Given the radii ($r_1, r_2, r_3$) and centerpoints of 3 ...
belkarx's user avatar
  • 101
0 votes
0 answers
71 views

Under what conditions could you (theoretically) reach the centre of a 2-sphere by traveling away from it?

Comments have helped me hopefully sumarize the question better: Is there a type of geometry under which center connects to periphery? Edit: Based on mr_e_man's comment below, I now believe what I'm ...
HRW's user avatar
  • 61
1 vote
0 answers
43 views

Can someone check if this “proof” is correct?

I posted about this a few weeks ago, but I am trying to determine if there is some point south of Seattle (along the same longitude) in the continental United States with a larger geodesic distance ...
bennykuttler's user avatar
2 votes
2 answers
596 views

How does Earth’s curvature affect flight times?

I have heard people say that the flight time from Fort Lauderdale to Seattle is the longest possible flight time within the continental United States. However, upon further consideration, I realized ...
bennykuttler's user avatar
1 vote
0 answers
29 views

Closed-form expression for the value of $\underset{\|x\|_1 = 1}{\min }r\|x\|_2 + a^\top x$

Let $a$ be a fixed vector in $\mathbb R^n$ and let $r \ge 0$. Question. Is there a closed-form expression for the value of $\underset{\|x\|_1 = 1}{\min }r\|x\|_2 + a^\top x$ ?
dohmatob's user avatar
  • 9,450
0 votes
0 answers
22 views

Simultaneous equations in non-Euclidean space?

As I recall, one visualizes equations as lines or planes in Euclidean space and the solutions are intersections among these lines, planes or higher-dimensional equivalents. Is there some use to ...
releseabe's user avatar
  • 213
2 votes
0 answers
63 views

What is the rigorous definition of a cycle in non-Euclidean geometries?

The cycles of Euclidean geometry are: Points, lines, circles. The cycles of hyperbolic geometry are: Points, lines, circles, hypercycles, horocycles. There is a commonality between these: These are ...
wlad's user avatar
  • 8,175
2 votes
0 answers
70 views

Why are all non-Euclidean geometries quadratic?

The non-Euclidean geometries are both based on conic sections: Elliptic and Hyperbolic. Conic sections are described with 2nd-degree polynomials. Are there any geometries based on higher degree ...
Koen Van Damme's user avatar
2 votes
1 answer
177 views

Non-Euclidean geometry: any practical use at the times of Gauss? [closed]

I'm making a historical research on the origins of differential geometry, starting with non-Eculidean geometry introduced by Gauss. Reading different historical accounts, what I don't understand is ...
Mark's user avatar
  • 139
1 vote
0 answers
126 views

Does $c^2 \leq a^2 + b^2$ hold for right triangles on the sphere?

I was hoping for a proof of something which appears to be intuitive to me, but which I can't prove. Let $a, b$ & $c$ be lengths of the sides of a triangle. We know that on a plane, $c^2 = a^2 + b^...
ranban282's user avatar
2 votes
2 answers
249 views

In the Poincare disk, there exist points $A$ and $B$ on the same side of line $L$ such that no circle through them lies entirely on that side

I am attempting this problem using Poincare disk. I want to show for hyperbolic plane (using this Poincare disk), there exists 2 points $A$, $B$ lying on same side $S$ of line $L$ such that no circle ...
koolaids's user avatar
2 votes
0 answers
61 views

Does the Hinge Theorem hold for non-Euclidean metrics?

We define the unit circle as the collection of all vectors with length 1 centered at some point. (The one below specifically defines the unit circle centered at the origin) $$ \mathscr{C}_2 = \{x\in\...
CobaltDev's user avatar
1 vote
0 answers
74 views

Showing Euclid's Proposition 30 ("Lines parallel to the same line are parallel to each other") is equivalent to the 5th Postulate.

I am trying to show that the 30th Euclid's proposition, "Straight lines parallel to the same straight line are also parallel to one another." is equivalent to the 5th Postulate: "If ...
Mathematican's user avatar
1 vote
1 answer
33 views

Property of hyperbolic rotation matrix with entry 1

I am considering the group of hyperbolic rotation matrices $G=\{A\in M_{3\times 3}(\mathbb{R}): A^TDA=D \}$, where $D=\begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&-1\\ \end{bmatrix}$. ...
James Cheung's user avatar
5 votes
0 answers
68 views

How would you describe the symmetries/group actions on the $\ell_p$ circle?

We define the unit circle as the collection of all vectors with length 1 centered at some point. (The one below specifically defines the unit circle centered at the origin) $$\mathscr{C} = \{ x\in\...
CobaltDev's user avatar
1 vote
0 answers
158 views

How to get a conformal factor between the Poincare Disk's metric and the Euclidean Metric?

Conformal Factor. As you see in the linked post, the conformal factor is $\frac{2}{1-\left\|x \right\|^2}$. But I'm more interested in how the outcome turns out. According to what I've heard, this may ...
john's user avatar
  • 1,268
2 votes
0 answers
59 views

Can we do regular geodesic tiling on a quadric surface, such as hyperboloid and paraboloid?

Can we do regular tiling on a quadric surface, such as hyperboloid and paraboloid? Assume that the straight lines are the geodesics. Can you show the pictures of the tiling?
Bik Kuang Min's user avatar
0 votes
0 answers
61 views

Is there a finite boundariless pseudosphere without edges and vertices?

A sphere represents finite boundariless surface with constant positive curvature, while a Clifford's torus represents finite boundariless surface with constant zero curvature. Both of these surfaces ...
Prido1024's user avatar
6 votes
1 answer
148 views

If proposition $\text{I}, 17$ in Euclid's Elements does not depend on the parallel postulate, how can elliptic geometry be consistent?

I'm from Italy, so I was reading an Italian translation of Euclid's Elements; and while I was doing just that, something about the introduction—written by Attilio Frajese—really caught my attention. ...
Labba's user avatar
  • 1,001
1 vote
1 answer
285 views

Is the hyperbolic plane convex?

I'm attending a lecture series about introduction to non-Euclidian Geometry, but it is focused on the intuition of that topic without giving me the tools to analize the following question: Is the ...
Douglas's user avatar
  • 394
0 votes
0 answers
55 views

If $\ell_1$ is a hyperbolic line, show there exists a perpendicular line $\ell_2$ such that $p\in \ell_2$ where $p \in \mathbb{H}\setminus\ell_1$

I am working within the upper half plane. I broke this question into three categories: $\ell_1$ is a vertical line. $\ell_1$ is a semi-circle and $p$ is directly above the center of $\ell_1$. $\ell_1$...
WaterDrop's user avatar
  • 600
2 votes
0 answers
367 views

Is spherical geometry "infinite" in the same sense that a Euclidean plane is?

This seems like a pretty straightforward question (assuming I worded it well), but I've never been able to find an answer anywhere. So, in Euclidean geometry, a plane extends infinitely in all ...
e4494s's user avatar
  • 475
1 vote
1 answer
240 views

Projecting a band of a sphere onto a 2D surface

For a craft project, I want to take a "band" of a sphere (i.e. the area between two latitudes) and project it onto a plane, so that I can fold the 2d shape onto the sphere and recreate the ...
Suudsu2200's user avatar
2 votes
1 answer
695 views

Spherical Pythagorean Theorem

In class, we came across the following relation for a right triangle on the surface of the sphere. $cos(\frac{c}{R})=cos(\frac{a}{R})cos(\frac{b}{R})$ where R is the radius of the sphere. Here a, b, c ...
Ook's user avatar
  • 211

1
2 3 4 5
7