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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35 views

Can Euclidean and non Euclidean geometry be tied together? [closed]

Is there a field of math that contains operations(i.e. tensors/ matrices) which mediate the smooth transition between Euclidean and non Euclidean geometry?
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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 $\ell_1$. $\ell_1$...
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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 ...
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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 ...
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1answer
54 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 ...
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The smallest codimension for isometric immersions

I just read Azov's article in the considered two classes of Riemannian metrics, \begin{align*} ds^2&=du_1^2+f(u_1)\sum_{i=2}^ldu_i,&f>0\\ ds^2&=g^2(u_1)\sum_{i=2}^ldu_i^2 ,&g>0\...
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Textbook Suggestions for Teaching Senior Level Geometry

I will be teaching a senior-level class on geometry next semester. The class consists of both Euclidean and non-Euclidean geometry with an axiomatic approach. Right now, it appears I have the liberty ...
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Need to calculate the 3D volume of a growing 4D hypersphere in Minkowski space

4D geometry and Minkowski space are areas of expertise which I fundamentally lack, so I'm hoping people are able to help me with this. The problem is this, if you had a hypersphere in Minkowski space ...
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117 views

Real numbers vs. the real number line

I don't know how to formulate this question precisely, so let me explain where I am coming from, noting that I know little about nonEuclidean geometry. I was thinking about how to explain how ...
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Newton's Shell Theorem in non-Euclidean spaces

Newton's Shell Theorem for gravity states, in two parts, that The gravitational field of a sphere outside the sphere is equal to the gravitational field of a point mass at the sphere's center. (This ...
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Confusion regarding composition of two spatial rotations of a sphere.

I am currently studying Complex analysis from the book "Visual Complex Analysis" by Tristan Needham. In the chapter "Non-Euclidian Geometry" on page 282 the author says that the ...
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Does classical mathematical logic consider Euclidean Geometry and Non-euclidean Geometry to be distinct object languages?

From the basis of mathematical logic (specifically classical logic), would we consider Euclidean Geometry and Non-euclidean Geometry as distinct object languages? Is that what it means to be an object ...
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Coordinate transformation that would transform an doughnut inside out (in 2D)

Sorry for the non-precise lingo, but not really a mathematician here. I am looking for a transformation that given a set of points in a carthesian plane, would move the outermosts points closer to the ...
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Determining point of intersection with real axis in hyperbolic upper-half plane

Here is the question I am working on: For $s>2$ ,let $Q_s$ be the hyperbolic quadrilateral in $\mathbb{H}^2$ with vertices $−1+i$, $−1+2i$, $1+i$,and $1+si$.Determine the values of $s$ for which $...
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Is there a name for an infinite spherical plane?

I was dabbling in hyperbolic/spherical geometry when I had the thought, "Why does an ant walking on a spherical plane have to come back to the same point it started on?" I knew that the ...
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Are $v,u$ orthogonal on a surface if $v^{\alpha}u^{\beta}g_{\alpha\beta}=0$?

I already know that for euclidean vectors, $u\cdot v=u^{\alpha}v^{\beta}\delta_{\alpha\beta}$, and that if this is zero then the two vectors are orthogonal. If we let the metric tensor describe a ...
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Visualising non-euclidean geometry

Apologies in advance if this question is too simple! I'm learning about non-Euclidean geometry & am wondering about how the practical visualizations work. Quite often textbooks show diagrams where ...
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What is meant by "cross ratio of four points" in Weyl's discussion of the Klein Disk?

The following is from Hermann Weyl's Space-Time-Matter. Although the structure was thus erected, it was by no means definitely decided whether, in absolute geometry, the axiom of parallels would not ...
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What would be the tools in the compass box of a geometer living in a curved 3-dim space?

To do Euclidean geometry, we only need a compass, protractor and a scale (maybe a set square to make our life easy) to make all possible geometrical constructions. Suppose a geometer living on a ...
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What is the maximum "alive density" of cells in Conway's Game of Life when played on a torus?

I've read that Conway's Game of Life (CGOL) can have unbounded growth from a finite initial number of alive cells (e.g. a glider gun). However, if CGOL is played on a torus, space (the number of cells)...
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57 views

Non-Euclidean distance in the upper half plane between two complex numbers

I’m going through Stillwell’s Pillars and trying my hand at finding non-Euclidean distance between two complex numbers in the upper half plane not on the positive y-axis, namely $1 + 2i $ and $2 + i$. ...
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Third vertex of equilateral SPHERICAL triangle

I’m trying to solve Fermat’s problem on sphere for the given triangle ABC using wolfram.I already made out,that in order to find a Fermat’s point i need to build three equilateral triangles on each ...
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Is there a conformal map that preserves the Poincaré disc and maps two points $z_1, z_2$ to different points $z_1', z_2'$?

Or does such a map not even exist? I think the wanted map has to merely translate/rotate/maybe reflect the underlying lattice. I would like to tessellate the hyperbolic plane. My idea is to draw the ...
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Properties of pseudospherical circles and geodesics

I am reading the note "Beltrami's Models of Non-Euclidean Geometry" (PDF link via unibo.it) and I have some questions about pseudospherical circles at page 10: the author says that Beltrami ...
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Why do isometries of Beltrami-Klein model correspond to projective transformations fixing the circle at infinity?

I am reading the note "Beltrami's Models of Non-Euclidean Geometry" (PDF link via unibo.it) by Nicola Arcozzi, and I am not quite understanding what is happening at page 9, in discussing ...
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Study of Strange Attractors

I have a background in Mechanical Engineering and I am currently studying non-linear systems with a focus on 'strange attractors.' What suppplementary reading should I do or what courses should I ...
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Is there a name for generalized manifolds?

A topological manifold is a topological space which locally resembles real $n$-dimensional Euclidean space. Here I consider removing `Euclidean' from manifold: Suppose that $X$ is a topological space, ...
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1answer
59 views

Does non-Euclidian geometries induce euclidian geometry locally?

This is essentially a soft question. We know the world is a sphere, following spherical geometry, yet at local levels, we can observe euclidian geometry. This brings me to my question, Does non-...
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Non-Euclidean Geometry: Correspondence of inversion in spherical lines and reflections in great circles under stereographic projection

I am trying to prove the statement: Reflection in a spherical line (i.e. the image of a great circle under stereographic projection) in the extended complex plane corresponds (via stereographic ...
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Analogous Euler line of a hyperbolic triangle

What curve passes through the circumcentre C, orthocentre O and centre of gravity G of a hyperbolic triangle? Would it be hyperbolically straight? Any ratio of hyperbolic distances among them? If the ...
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Beltrami's Essay on the Interpretation of Non-Euclidean Geometry

I am reading the Essay of the title written by Beltrami in Italian and I found a specific point of the essay which in my opinion could be fully clarified only if compared with its translations. At the ...
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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 ...
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Proving that the definition of measure of a dihedral angle is well-posed

One can define the measure of a dihedral angle $\alpha$ as follows: let $P$ a point of the line of intersection of the two planes limiting $\alpha$ and let $S$ the surface of the sphere centred in $P$....
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Euclidean angles on curved surfaces

So usually when defining angles between two curves on some curved surface, we essentially take the angle between their unit speed tangent vectors where they meet. This definition leads to nice things ...
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Surfaces formed by infinitesmal equilateral triangles at common vertices

Say we have equilateral triangles coming together at a common vertex with each angle is $ 2 \pi/n= 2 \pi/6$. As the hexagon side tends to zero we have a flat plane defined with hexagonally packed ...
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Assigning Tiles on a Hyperbolic Grid with Unique Coordinates

I am creating a location in an RPG campaign that deals with non-euclidean space, and I'm currently toying with the idea of a forest with a finite border that takes up infinite space. The idea is that ...
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1answer
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Proving the excess of two split triangles is the same as the original

The excess of a triangle is the number of degrees over $180$ a triangle has in spherical geometry. Given spherical $\triangle ABC$ that is split into two triangles by an arc from the vertex going to ...
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135 views

Vector Addition/Translation in Hyperboloid model

I have problems understanding vector addition in Hyperbolic space. In the Poincaré ball model, vector addition/translation is the Möbis addition and defined as: $$ x \oplus_c y = \frac{(1+2c\langle x,...
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Inner product in hyperbolic space

I am trying to learn about Hyperbolic Spaces. I can't find information about the inner product in hyperbolic spaces. So the paper "Multi-relational Poincaré Graph Embeddings" from Balaževic ...
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Why bending some rules is meaningful, while bending others is fruitless?

This might be a purely philosophical question, but still... Up to some points in geometry, Euclid's axioms were accepted, even though the 5th one was causing headache. Then someone comes up and say &...
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How to verify the Hyperbolic Axiom from the Poincare model?

I was wondering how to verify that the Poincare model satisfies the hyperbolic axiom (i.e. that there are at least 2 lines through a point A parallel to line L?
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Simplest Discrete 3D Model of a Regular 2D Hyperbolic Tiling

I only have a beginners level understanding of hyperbolic geometry, and I am afraid that the following question might be too vague, but here goes. I know one can make real 3D models of regular tilings ...
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Equiconsistency of euclidean, hyperbolic, and elliptic geometry

Pretty much every text about non-euclidean geometries talks about the various models by Beltrami, Riemann, Poincaré, Klein, and others which demonstrate that if euclidean geometry is consistent, then ...
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56 views

Prove a certain quadrilateral is actually a Saccheri Quadrilateral

Question: "Show that a quadrilateral ABCD, which has angle C = angle D = right angle; and angle A is congruence to angle B, is a Saccheri Quadrilateral." My attempt: As by definition, a ...
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389 views

What exactly are Cayley-Klein geometries?

Everywhere I see Cayley-Klein geometries discussed, they're presented in a manner that seems wholly disconnected from the rest of geometry. Just what sorts of mathematical object are Cayley-Klein ...
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Showing that if $P,Q \in \mathbb{S^2}$ are equidistant from $A,B,C \in\mathbb{S^2}$, then $P=Q$

We are given that the equidistant set of two points on $\mathbb{S^2}$ is a "line" (great circle) on $\mathbb{S^2}$. Using this fact I must show that: If $P,Q \in \mathbb{S^2}$ are ...
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Query on how to study well in non-Euclidean/ hyperbolic geometry

I m currently taking a course on non-Euclidean/ hyperbolic geometry, mainly consists of using axioms in these geometries, etc... to proof certain theorems or results. I am a beginner in these sorts of ...
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138 views

Using cross ratios to show how an old projection technique used by artists was wrong

These questions are taken from Stillwell's The Four Pillars of Geometry. For $Q 5.7.3$, I am confused about what the question is asking. Is at asking me to: 1) Compute the cross ratio's between four ...
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167 views

Show that $\mathbb{RP}^2$ has four "lines", no three of which have a common "point"

I'm going through Stillwells The Four Pillars of Geometry and this is one of the questions from an exercise. I tried searching for this both here and other places online but all of them contain ...
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164 views

If all lines on a sphere converge, what happens to latitude lines?

I mean, on an image like this, how do latitude lines not converge? Everywhere i read says that there can be no parallel lines on a sphere. Is this related to the different definition of 'line' in ...

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