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Questions tagged [spherical-geometry]

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

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25 views

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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11 views

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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39 views

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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22 views

“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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1answer
20 views

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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27 views

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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2answers
35 views

Bijection between spherical and planar triangle surfaces

I subdivide a unit sphere, centered at origin, onto 20 spherical triangles. For the sake of argument let's take one such triangle $Ts$, in $\mathbb{R}^3$, that has vertices $Normalize(-1,0,g), ...
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36 views

Paramterizing a walk from $(R, \phi, \theta)=(a,\pi /2, 0)$ to the North Pole, walking North-West

Assuming that the earth is a perfect sphere with radius a, we start our journey in the point $(R, \phi, \theta) = (a, \pi/2, 0)$ (Spherical coordinates) and we travel North-West with constant speed v. ...
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52 views

Discrete Spherical Symmetry Group

Take two spheres each having a certain number (say 5) of identical dots on them. What is the approach to proving/disproving that they are equivalent under the set of spherical rotations? One could ...
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1answer
48 views

Area of a circle segment on sphere, given radius (meters) and central angle (degrees)

Situation I have a circle segment and some information about the circle it belongs to. Given Information: radius of the circle in meters central angle in degrees lat/long of all three points on the ...
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1answer
81 views

on the limit of the finite representation of harmonics

Let $Y_n^j, \, -n\leq j \leq n$ be a spherical harmonic. Let $f$ be an even function on the sphere. Then, $f$ can be introduced as $$ f=\sum_{j=-n}^A\hat f(j)Y_n^j, $$ where $\sum_{j=-n}^A|a_j|>0$ ...
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Euclidean N-D Space to Model Great-Circle Distances

Assuming you have a graph and know the distances between the points. You know that they can roughly be modeled as points on the 2D surface of a sphere by treating the distances as great-circle or ...
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286 views

Packing regular tetrahedra of edge length 1 with a vertex at the origin in a unit sphere

I'm currently venturing through Paul Sally's Fundamentals of Mathematical Analysis. This is an unusual textbook in terms of the difficulty of exercises. I've already been stunned by the very first one:...
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10 views

Measuring angle on positively curved space

Suppose you are a 2D being, living on the surface of a sphere with radius R. An object of width $ds << R$ is at a distance $r$ from you. What angular width $d\theta$ will you measure for the ...
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35 views

Intersection points between a circle and a straight line on a sphere

I have a circle on the surface of a sphere. I need to check whether the circle intersects with a given straight line or not. The center of the circle $c$ is given in terms of latitude and longitude $(\...
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52 views

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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1answer
30 views

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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38 views

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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2answers
41 views

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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46 views

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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30 views

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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1answer
34 views

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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37 views

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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2answers
65 views

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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45 views

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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96 views

Laplacian Green's function on the $n-$sphere

I was looking for the explicit expression for the Green's function of the Laplace operators on the Euclidean $n-$sphere, $S^n$, namely the distribution $G(x, x_0)$, with $x$, $x_0$ unit vectors in $\...
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1answer
34 views

Solving stereographic projection for central latitude $\phi_1$ and central longitude $\lambda_0$

For a given longitude and latitude and their projected point I want to know the central longitude and latitude of the stereographic projection. I used the formulars from wolfram alpha: http://...
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2answers
92 views

Is the Line-Of-Sight Bearing equal to the Great Circle Path Initial Heading?

If you are at a known location (you know your precise latitude and longitude for example) and have an unobstructed view of another known location you can: A: Take a precise visual bearing to the other ...
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3answers
111 views

relationship between a great circle arc and a latitude circle arc at a given latitude

My spherical geometry is a very rusty but looking at the figure below: ... my intuition tells me that angles $\phi$ and $\theta$ (measured in radians) are connected with the following equation: $\...
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1answer
35 views

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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2answers
27 views

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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1answer
78 views

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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40 views

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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36 views

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'...
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1answer
129 views

Angle between two 'small circles' on the surface of a sphere

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", ...
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1answer
134 views

Deriving the Euler-Lagrange equations for the arclength of a curve on the unit sphere

I'm trying to derive the extremal solutions to the Lagrangian for arclength on the unit sphere by setting up the Euler-Lagrange equations. Starting from $$L = \sqrt{\dot \theta^2 + \sin^2 \theta \ \...
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17 views

Minimal surface contained in a hemisphere

I am looking for examples of closed, orientable minimal surfaces of the sphere $\mathbb{S}^3$ that are contained in a hemisphere. Do you know any? Thanks!
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Decide whether each triangle below lies in the Euclidean plane, on a sphere (of radius 1), or in the hyperbolic plane

A triangle has sides of length $a$, $b$, $c$ (given correct to $2$dp) and a right angle opposite $c$. Decide whether each triangle below lies in the Euclidean plane, on a sphere (of radius $1$), or in ...
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38 views

Area of Polygon on plane in $XYZ$

I've got WGS84 coordinates that are converted to ECEF ($X, Y, Z$). How can I compute the area of a convex polygon given its vertices in ECEF? One thought I had was that I could rotate the plane the ...
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0answers
23 views

area of a concave spherical polygon with radius 1

In class today our professor showed us the general area formula for the area of a convex spherical polygon with radius 1, which is $\text{area}(\text{spherical polygon}(\theta_{1},...,\theta_{n}))=\...
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1answer
100 views

Pull Constant from Atan2 Function

I am trying to calculate intersection Latitude value based on given coordinates and intersect Longitude. $\begin {align} b &= \text {is bearing in radians} \\ Lat_1 &= \text {is Latitude 1} \\...
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1answer
185 views

Proofs for the Spherical Laws of sines and Cosines

I am looking for "classical" proofs for the spherical laws of sines and cosines. A proof that relies only on knowledge that was common to the ancient greek geometers, not containing analytical ...
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1answer
37 views

Calculate center point of spherical octant

If I had a sphere and an octant like this, how would I find point $P$? I can already calculate the other portions of the sphere using the parametric form equations, but I am unsure how I would ...
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1answer
159 views

Finding the equation of a geodesic passing through two given points

I am trying to find the equation of a geodesic (otherwise known as a great circle or great circle arc) on the surface of a sphere of given radius $a$ through two points on the sphere. I am given the ...
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2answers
131 views

Determine orientation of spherical polygon without trig functions

Is there a way of testing the orientation of a spherical polygon given an ordered list of its vertices that doesn’t involve computing (inverse) trigonometric functions? The polygon is not necessarily ...
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2answers
125 views

a question on the geodesic distance on the sphere

Let us consider (in spherical coordinates) the expression Great arc distance between two points on a unit sphere $$d({\bf v}_1,{\bf v}_2)=\cos^{-1}\left(\cos\theta_1\cos\theta_2+\sin\theta_1\sin\...