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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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1answer
90 views

If two medians are congruent… is the triangle isosceles in a Hilbert plane?

If $ABC$ is a triangle for which two medians are congruent... is it true that the triangle $ABC$ is isosceles in a general Hilbert plane? I am having a little bit of trouble trying to prove this, if ...
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0answers
17 views

Projective group PGL(2) and projective collineations

I'm reading Euclidean and non-euclidean geometry by Patrick J.Ryan and I'm stumped at the chapter 5: The projective group. Precisely , I don't understand the following statement: Each invertible ...
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1answer
43 views

If a rectangle exists then rectangles with arbitrary length of sides exist in neutral geometry.

Suppose one given rectangle exist, how can we use the Archimedian property to lengthen or shorten the sides and obtain a rectangle with sides of any prescribed length in neutral geometry (that is, ...
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3answers
208 views

Are the following statements equivalent to the parallel postulate?

In one of my Elementary Geometry previous exams, one of the questions was the following: Study if the following statements are equivalent to the paralellism axiom: $(i)$ Any three straight ...
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0answers
14 views

Tessellations of 3-sphere

How many are there regular geodesic tessellations of the 3-sphere? What kind polyhedrons are used in those?
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0answers
36 views

Trigonometry equivalent

$\sin (A) = \sin 3x$ is equivalent to $\sin(A) = 3 \sin(x) - 4\sin^3(x)$, then $-\cos 3x$ is $-4\cos^3x + 3\cos$. is that correct? I just want to make sure I'm distributing the negative sign ...
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0answers
15 views

Transitive action of SL2(R) on the hyperbolic lines in Poincare upper half plane [duplicate]

I'm trying to prove that the action of SL2(R) is transitive for the hyperbolic lines in H (Poincaré's Half-Plane Model). Thank you in advance for your answers!
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1answer
31 views

Condition on the existence of a spherical triangle

It is known that $a,b,c>0$ are the sides of a triangle in the Euclidean plane if and only if $$a+b>c,\hspace{0.3cm} a+c>b,\hspace{0.3cm} b+c>a.$$ I would like to give a similar condition ...
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1answer
21 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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0answers
25 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
17 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
46 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(...
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1answer
40 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
75 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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0answers
141 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
86 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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0answers
38 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
62 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
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1answer
44 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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70 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
19 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
34 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
22 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
148 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
36 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
78 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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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 ...
3
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1answer
119 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
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1answer
112 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
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1answer
166 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
69 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
72 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
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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
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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
78 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
166 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
209 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
86 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
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1answer
179 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
214 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
134 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
120 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
81 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
88 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?