# Questions tagged [spherical-geometry]

geometry as on the surface of a sphere, where "lines" are great circles and any pair of lines must intersect

595 questions
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### Equilateral/Equiangular Quadrilaterals on a Sphere

I have recently been taking a Geometry class and I am a little bit confused about spherical geometry. I know there can be equiangular spherical quadrilaterals but does this also imply that the ...
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### Is there a conformal mapping from the surface of a cube to the surface of a spherical cube that preserves edges?

Is there a conformal mapping (with certain singularities noted below) from the surface of a cube to the surface of a spherical cube that preserves edges? Note that this also implies that vertices and ...
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### Special case for Giraud's Theorem

I was wondering how Giraud's Theorem would work for spherical polygons. Do we know a proof?
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### “Height” of an equilateral spherical triangle

consider an equilateral spherical triangle (living on a unit sphere) defined by the interior angle of each of its corners. How can I compute the arc length of one of its vertices to the mid-point of ...
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### Asymptotic behaviour of the intersection volume of balls with the same radius

Let $x,y \in \mathbb{R}^n$ be two fixed points. Is there an easy proof of the fact that $$A(r):=\frac{ \text{Vol}(B(x,r) \cap B(y,r))}{\text{Vol}(B(x,r))}$$ tends to $1$ when $r \to \infty$. I ...
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### Spherical triangles and congruence criteria

I read from different sources that the usual criteria of congruence of triangles work for spherical triangles. However it seems to me that there is a counterexample. Consider two poles on the sphere A ...
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### dependence of angles of a geodesic line on a sphere

Is it possible to prove a unique dependence of $\theta$ and $\phi$ angles for a geodesic line, knowing only the definition of a geodesic line? Otherwise, this task can be set as follows: we can find ...
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### Spherical Triangles: Area and mapping to Euclidean space

If we take a sphere of radius 1 and travel a quarter-circumference south from the north pole, turn 90 degrees, travel another quarter circumference, then return North, we form a triangle with an angle ...
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### Spherical cap problem - trigonometry / circle theorems problem / surface area

Graph here I am trying to derive the following equation from a paper I am studying, which the author has derived from the graph above. The two slightly curved lines here are modelled as the surfaces ...
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### Three points on a sphere define eight spherical triangles

reading a book of spherical astronomy I've read this: Three great circles pass through three points on a sphere. If for each great circle we consider only one of the two parts in which it is ...
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### Spherical Triangle: Law of Sines with Clairaut's theorem

The spherical law of sines states that On the sphere $\mathbb{S}^2$ we consider a triangle, i.e. three points connected by geodesics. We denote the side lengths with $a, b, c < \pi$ and the ...
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### How to find the domain of a function using spherical coordinates?

Hey guys so I was wondering how to find the domain of a function using spherical coordinates.. For example, I take two functions: F(x,y,x)=√x+√y+√z+ln(4-x²-y²-z²) whose domain is D=((x,y,z):x≥0,y≥...
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### Feuerbach's Theorem in Spherical Triangle?

So first of all, just incase you don't know, Feuerbach's Theorem states the nine-point circle of a triangle in Euclidean space is tangent to the three excircles and the incircle of the triangle. ...
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### Electron Groups and Platonic Solids

In my Chemistry class, lately I have been learning about Lewis Structures for molecules, and how the arrangements of groups of electrons on each molecule repel each other to form the molecule into a ...
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### Spherical harmonics of a non-negative function of the two sphere

I am working on a data analysis project, as a part of which I want to express a probability distribution as a spherical harmonic expansion on a 2-sphere. Imposing the condition of realness of the ...
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### Angles in a spherical triangle.

Just seeking advice here! I have 3 coordinates; $A(-0.52992,0.84805,0),\\ B(0.84805,0,0.52992),\\C(0.15461,0.47553,0.86603)$. I want to find the angles at $A$, $B$ and $C$. Hence, I find the normal ...
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### Distance from great circle to North pole

Suppose a great circle passes through a plane with normal $[a,2a,3a]$, so by converting it to a unit vector, the normal of the plane is $\frac{1}{\sqrt{14a^2}}[a,2a,3a]$. My teacher explained that ...
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### Closest point on line segment of a great circle

If I have a sphere of radius R, and two points $A$ and $B$ on its surface, at $(R, \theta_A,\phi_A)$ and $(R, \theta_B,\phi_B)$ respectively in spherical coordinates. Call $AB$ the geodesic from $A$ ...
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### Normal vector to ellisoid surface

How do I obtain a normal vector to the surface of an ellisoid, at a given latitude and longitude. Getting the same normal vector for a sphere is trivial.
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### Perpendicular bisectors of great circles

I want to prove the following statement Given two points $P$ and $Q$ $\in S^{2}$ define $X=\{x \in S^{2} \mid d(x,P)=d(x,Q)\}$ where d denotes the spherical metric. Show that $X$ is a ...
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### Finding the right grid tessellation, for a chess board with elliptical geometry

I came up with the idea of non-euclidean chess(the chess board I'm working on will have 2D elliptical geometry) but I came across a problem. The question: What kind of grid do I put on it(what shapes ...
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### Mapping points on a sphere: many to one [closed]

Is there a finite sequence of rotations that maps any point on a sphere's equator to the north pole (or at least very close to the pole, it would be sufficient for the purposes of my question to reach ...
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### Precalculus in a Nutshell, Geometry, Question 11

"An equilateral triangle and a square are inscribed in a circle, with a side of the triangle being parallel to a side of the square. The entire figure is revolved about that altitude of the triangle ...
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### How many ways to arrange $x$ spherical caps of area $a$ on a sphere with area $A$ to cover a given area. [closed]

I have a sphere which I cover with spherical caps that can overlap in any way (fully or partially...or not at all). The caps are all of equal size. I am looking for a way to compute the number of ways ...
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### Totally geodesic surfaces of $\mathbb{S}^3$

Let $\Sigma$ be a closed, connected and oriented surface embedded in $\mathbb{S}^3$. Denote by $\overline{K}$ and $K$ the sectional curvatures of $\mathbb{S}^3$ and $\Sigma$, respectively. Then, Gauss'...
This seems like it should be fairly simple, but it has me completely stumped. Imagine I have a latitude line at angle $\theta_1$ on the surface of the unit sphere in 3D. This is a "small circle", ...