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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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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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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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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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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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51 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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26 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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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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1answer
32 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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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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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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33 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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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
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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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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
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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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65 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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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'...
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1answer
102 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
98 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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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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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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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
91 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
148 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
36 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
96 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
128 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
80 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\...
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2answers
65 views

Path on a sphere

I am trying to solve a exercise problem in GR on a "triangle" whose sides are great circles of a sphere of radius $R$. So this is the triangle that I chose (coordiantes are written as $(r, \theta, \...
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Defining the winding number on a sphere

This is for “point-in-polygon” testing on the sphere. I’ve defined a spherical polygon as a list of points on the sphere, coupled with whether the corresponding great circle arcs on the sphere are ...
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1answer
59 views

Understanding differences in Geodetic (WGS84) to ECEF equations?

Methods suggested in this, that and there all recommend the following: $$x = R \cos(\theta) \cos(\phi)$$ $$y = R \cos(\theta) \sin(\phi)$$ $$z = R \sin(\theta),$$ where latitude is $\theta$, ...
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How to calculate angle between line segment and line passed through the middle of the segment

There are three points on sphere with latitude/longitude coordinates: ${a}_1, {a}_2, {b}_1$. Let ${a}_m$ is a center of spherical line segment $\overline{{a}_1{a}_2}$. How to calculate an angle ...
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153 views

How to discretize a sphere?

I would like to discretize a sphere into icosahedra whose vertices are equidistant, i.e., I want to plot $n$ equidistant points on the surface of a sphere. I am familiar with R, Python, and Matlab. ...
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Are there any complete orthogonal basis functions inside unit sphere?

I went through so much literature, but couldn't find any orthogonal complete functions within the boundary $0 < r <1, 0 < \theta <\pi, 0 < \phi <2\pi$ other than 3D Zernike ...
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Finding points on the surface of a sphere following a set of conditions

I have a number of locations on the earth and I need to find out if they obey a certain set of conditions. I have one point, say P1=(a,b), a and b being the longitude and latitude of the point ...
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1answer
97 views

How to calculate angle between two directions on sphere

There are four points on sphere with latitude/longitude coordinates: ${a}_1, {a}_2, {b}_1, {b}_2$. How to calculate an angle between two vectors on sphere: $a = \overline{{a}_1{a}_2}, b =\overline{{b}...
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1answer
122 views

Projecting Circles of a Sphere on a Mercator Projection

Recently, I have come across a map showing the coverage of a missile given its radius on a Mercator projection. In a 3-D space, the boundary of the missile would be a circle of a sphere. However, I am ...
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1answer
84 views

Interpolating two spherical coordinates (theta, phi)

I have two points on a unit sphere, given as tuples $(\theta, \phi)$. I need an efficient way to interpolate them, for example given $t$ between zero and one, to generate a point that lies (on the ...