Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [noneuclidean-geometry]

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

0
votes
1answer
37 views

Scalar multiplication and vector addition with hyperbolic vectors

I'm representing hyperbolic vectors using the Minowski hyperboloid: $x_0^2 - x_1^2 - x_2^2 = 1$. I understand that the distance between two hyperbolic vectors of this form is $acosh(B(u, v))$ where $...
2
votes
1answer
35 views

distance function in context of elastic energy of non euclidean thin bodies

I am currently reading a paper about elastic energy of non euclidean thin bodies, about which I might want to write my thesis for my bachelors degree. You can find it here: https://arxiv.org/abs/1801....
0
votes
0answers
35 views

Find the area of the hyperbolic (geodesic) triangle with vertices (0, 0),(0, 1) and (1, 0).

I'm not really sure where to start for this question. I'm aware of the Gauss-Bonnet theorem $Area = \pi -(A+B+C)$ where $A,B,C$ are the interior angles of the triangle. However I am not sure how to go ...
0
votes
1answer
16 views

Proof with d-quadrilateral

This is the question that I am working on currently I am just stumped on how to go about solving it. I don't really know how I can use the Euclidean triangle to solve it either. Any advice or tips ...
0
votes
0answers
15 views

Parallels and meridians on a pseudosphere (tractricoid)

I am trying to visualize and understand parallels and meridians on a pseudosphere with respect to the usual parallels and meridians on the sphere. On a sphere of radius $r$, using the usual $\theta, ...
1
vote
0answers
22 views

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 ...
0
votes
0answers
20 views

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 ...
3
votes
3answers
138 views

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. ...
0
votes
0answers
33 views

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 ...
2
votes
0answers
73 views

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 ...
0
votes
0answers
15 views

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 ...
2
votes
1answer
90 views

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 ...
2
votes
1answer
80 views

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 ...
3
votes
1answer
103 views

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 ...
0
votes
0answers
57 views

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 ...
1
vote
1answer
59 views

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 ...
1
vote
0answers
19 views

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 ...
2
votes
1answer
35 views

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 ...
1
vote
1answer
64 views

Isometries on the Hyperbolic Plane

This paper states in definition 1.12 that a function $\phi:\mathbb{H}\to\mathbb{H}$ is an isometry of the hyperbolic plane if for all $z\in\mathbb{H}$ and $v,w\in T_z\mathbb{H}$, $$\langle v,w\...
1
vote
1answer
128 views

Is Non-Euclidean geometry really “Non”?

The definition of a straight line according to google. I do not understand why I call these geometries "non-Euclidean". In my view, both hyperbolic and elliptical geometry are just a dimensional ...
0
votes
2answers
122 views

Why is the hyperbolic plane homeomorphic to $\mathbb{R}^2$, whilst the 2-sphere isn't?

Geometrically speaking, one might consider the hyperbolic space $H^2$ and the sphere $S^2$ as manifolds which deviate from the euclidean plane $\mathbb{R}^2$ in exactly opposite ways – $H^2$ having ...
1
vote
0answers
75 views

How does a polar triangle characterize/ define a hyperbolic ellipse?

I am reading a paper stating the following: "In the Cayley-Klein model, hyperbolic ellipses are conics that lie in the interior of K. The ellipse center is the unique vertex c of the common ...
4
votes
1answer
129 views

Why is the mirror descent considered a “non-Euclidean” algorithm?

I am perplexed by the fact that many sources online such as this website refer to mirror descent as a non-Euclidean gradient descent. Why non-Euclidean? The constraint set $\mathcal{C} \subset \...
0
votes
1answer
41 views

Construction of Parallel Lines in Absolute Geometry

Let $l$ be a straight line, and $P$ be a point not on it. We do not assume the parallel postulate. Then from one of the equivalent versions of the absolute geometry, it is not clear that we can draw a ...
6
votes
3answers
159 views

Why don't we have many non euclidean geometries out there?

In the current semester I've taken a course about Non-Euclidean Geometry. During the course, we presented two types of non-euclidean geometries: the spherical geometry and the hyperbolic geometry. So ...
0
votes
1answer
74 views

Sharing endpoint at infinity [closed]

I could not solve this problem. Can you help me please? Prove that if m and n are hyperbolic lines that share an endpoint at infinity, then there does not exist a hyperbolic line perpendicular to ...
2
votes
1answer
110 views

Does a geometry exist where the circumference can be the square of the radius, $C = R^2$?

In normal, Euclidean space, the circumference of a circle is a linear function of the radius: $$f(R) = 2 \pi R$$Does a geometry exist where the locus of points equidistant from a common point, $P_0$, ...
1
vote
0answers
116 views

Explanation for Schlafli's differential formula calculation of volumes.

I want to get a detailed explanation (i.e via special cases and illustrations) of Schlafli's differential formula for calculation of n-dimensional volume. My motivation is to understand a result of ...
0
votes
0answers
99 views

Cayley Salmon Theorem assumptions

I know that the Cayley salmon theorem is defined as a smooth cubic surface in CP^3 contains 27 lines but what are the assumptions of this theorem?
0
votes
0answers
75 views

Natural coordinates of a n-sphere

I stumbled on n-spheres recently, and recalled spherical coordinates. They are very useful for doing computations on spheres embedded in $\mathbb R^{n+1}$, but oddly uneven, if not to say ugly, when ...
1
vote
0answers
70 views

Isometry of hyperbolic plane

I'm trying to prove the following statement: "every isometry of the hyperbolic plane has a fixed point, given that its orbit is bounded". Can someone help me with that?
2
votes
0answers
108 views

Model of Hilbert's plane ordered geometry in which the elliptic parallel property holds

Elliptic parallel property: Given a line $L$ and a point $a\notin L$, there exist no lines parallel to $L$ passing through $a$. Euclidean parallel property: Given a line $L$ and a point $a\notin L$, ...
0
votes
0answers
199 views

Polar and Cartesian coordinates in Poincare disk model.

I wonder if we can say that, similar to Euclidean 2D space, there are two equivalent views on the coordinates of the points, Cartesian $(x,y)$ and Polar $(r, \phi)$ with correspondence: $$x = r\cos\...
3
votes
0answers
98 views

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 ...
0
votes
1answer
251 views

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." ...
1
vote
0answers
280 views

How did Saccheri try to prove the Parallel Postulate?

In order to prove that Euclid's Fifth Postulate was right, Saccheri used the reductio ad absurdum method; he considered the Parallel Postulate was false, thus being allowed the existence of a ...
1
vote
0answers
197 views

Hyperbolic Geometry: Defect of a triangle

I'm working on a problem and I needed an angle I couldn't find, so I looked it up in the solution manual off chegg-com. And I do not understand what they are saying, can you help? I copied the entire ...
9
votes
1answer
251 views

What is the most efficient shape for tiling curved spaces?

In a great video by PBS Infinite Series, the mathematician Kelsey Houston-Edwards argues that bees build their honeycombs into hexagonal shapes because that's the most efficient way of tiling two-...
2
votes
2answers
300 views

Showing that sum of angles of hyperbolic triangle is strictly less than $\pi$

I am hoping that someone can help me see understand a proof that the sum of angles of a hyperbolic triangle $\triangle ABC$ is strictly less than $\pi$. I want to stick to the upper-half-plane model $...
1
vote
1answer
158 views

Manifold-related Question

I'm a little new to this area of mathematics. I understand that manifolds are topological spaces that locally resemble Euclidean space near each point. I also understand that manifolds are very common....
1
vote
0answers
124 views

Finding a non-euclidean geometry model - Howard Eves' Geometry

I was given a set of axioms and the task to find two models. found the first one without any real problem. The second one is proving to be very difficult. Thought I finally had it, but as you can ...
0
votes
2answers
243 views

Equilateral triangle in hyperbolic plane

Show that there is an equilateral triangle with angles $\pi/m$ for any integer $m\ge4$. What is the corresponding result for regular n gon? My attempt: I know that area of triangle in hyperbolic plane ...
0
votes
2answers
38 views

Affine/projective axioms for $\mathbb Z_p^2$ vs. $\mathbb Z_p^3$

Let $p$ be prime. We say that a geometry is an affine plane if it satisfies three properties: (i) Any two distinct points determine a unique line. (ii) There exists a set of four points so that no ...
2
votes
3answers
71 views

Some combinatorial properties of Fano geometry

I am starting to learn about finite geometries and in particular am trying to understand some basic (mainly combinatorial) features of the seven point geometry. (I have not yet reached the context of ...
1
vote
1answer
89 views

How to prove that a non-euclidean shape is infinite or finite?

There are mainly three theories about the shape of the universe: it can have a shape with zero curvature (flat), positive curvature (spherical) or negative curvature (hyperbolic). My question is "How ...
2
votes
0answers
136 views

Are there such a thing as non-rectangular or non-Euclidean matrices?

I was thinking about surface geometry and spherical geometry in particular and I was wondering about whether or not there was such a thing as a matrix (as in vectors and matrices) in the shape of a ...
0
votes
1answer
59 views

Plane Separation Axiom (PSA) and Taxicab Geometry

The PSA states that: A line $l$ separates a plane into two separate disjoint regions such that: i) If points $P$ and $Q$ are in the same region, then the segment $\overline{PQ}$ does is fully ...
-1
votes
1answer
28 views

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?
0
votes
1answer
40 views

Special Distance function

This is my first post. I am not a professional mathematician, but there is a question that is relevant to the work I am doing. (Please forgive me if I make any errors) Say I got a vector $\vec{x}=(...
0
votes
1answer
73 views

Surface Area bounded by Geodesics

Is there a formula for calculating the surface area of a region on a sphere which is bounded by 3 geodesics? Or, given three points on a sphere is there a way to calculate the surface area of the ...