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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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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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1answer
13 views

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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21 views

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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1answer
12 views

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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0answers
23 views

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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0answers
43 views

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(...
0
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1answer
39 views

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?...
3
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1answer
59 views

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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136 views

Modular geometry: The parabolas of quadratic residues modulo $p$

[For using the available space better, I rotated the function graphs by 90 degrees.] For the quadratic function $f_1(x) = x^2$ (with $x \in \mathbb{R}$) there is only one parabola all integer points $...
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1answer
33 views

In neutral geometry construct a transversal through two parallel lines and a point between them

Is this possible to do? Here is what I have tried. Draw a line that is perpendicular to either of the parallel lines that go through the point between the parallel lines, but then there is no way to ...
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1answer
72 views

Summary and understanding of non-euclidean Geometry

I'm trying to understand the 'paradigm shift' from Euclidean to non-euclidean geometry. Though I can understand simple models like why the angle in a triangle on a sphere would not add to 180 degrees ...
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2answers
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Does rotation in plane involve Euclid's parallel postulate?

It is given a line $l$, a point $P$ not on $l$, and $PQ$ perpendicular to $l$, i.e., $Q$ is on $l$. Let $R$ be a point on $l$ with $QR = PQ$. Therefore, $<PQR = 90^{\rm o}$. I can't figure out if ...
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36 views

Troublesome theorem in hyperbolic geometry

I want to prove the following theorem without using Dedekind's axiom(i.e. only with axioms of hyperbolic plane) Given arbitrary line $\mathcal l$ and a point $\mathcal P$ which does not lie on $\...
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1answer
48 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 $...
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1answer
40 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....
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45 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 ...
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1answer
18 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 ...
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0answers
26 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, ...
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0answers
23 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 ...
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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 ...
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3answers
145 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. ...
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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 ...
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0answers
75 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 ...
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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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1answer
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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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1answer
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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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1answer
130 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 ...
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0answers
62 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 ...
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1answer
65 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 ...
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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 ...
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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 ...
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1answer
74 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\...
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1answer
153 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 ...
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2answers
174 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 ...
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0answers
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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 ...
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1answer
161 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 \...
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1answer
48 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 ...
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3answers
192 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 ...
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1answer
76 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
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1answer
112 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, $P_0$, ...
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0answers
126 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 ...
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0answers
115 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?
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0answers
79 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 ...
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0answers
80 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?
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0answers
136 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 $a\notin L$, ...
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0answers
228 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\...
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0answers
104 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 ...
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1answer
314 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." ...
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0answers
318 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 ...
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0answers
219 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 ...