# 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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### How can I smoothly connect two points on the surface of a sphere with a pair of arcs passing through the points in certain directions?

Given two points on the surface of a sphere $\vec a$ and $\vec b$ and directions tangent to the sphere at those points, $\vec {a'}$ and $\vec {b'}$ how can I connect them with a pair of circular arcs ...
1 vote
32 views

### Seeking Cartesian to Hyperspherical Coordinates Conversion Examples in Higher Dimensions

I've recently coded a function that transforms Cartesian points into hyperspherical points. While I've verified its accuracy for 2D and 3D points, I'm finding it challenging to test for higher ...
58 views

### Using feet latitude/longitude and flat distance between to approximate spherical distance between two points, huge error why?

Setup: you 2 have points on the surface of the Earth that are close to each other, say 30N100W and 30.1N100.2W, in decimal degrees. Step 1: find the feet latitude/longitude for each point. For ...
1 vote
44 views

### Degenerate Schläfli symbols involving 1

I understand that Schläfli symbols with integral elements $\{p,q\}$ both greater than or equal to $2$ represent planar graphs (multigraphs if $p$ is $2$), and these graphs, if finite, represent ...
1 vote
33 views

### What's the equivalent for spheres to homogeneous coordinates for projective spaces?

Projective or homogeneous coördinates are an extremely useful parameterization of projective spaces (indeed often used to define them!), but they are redundant -- a projective space of dimension $n$ ...
1 vote
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1 vote
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### Spherical Triangle with right-angle calculating an angle with one side and one angle [closed]

Given the following spherical triangle, is it possible to calculate B given side b, angle A and angle C = 90 degrees? If so, which formula is it. I've tried the sine Rule and cosine rule but I need to ...
27 views

### Finding the center of a series of points on the surface of a sphere

I have a series of points on a unit sphere that are given in azimuth, elevation coordinates, where azimuth has a domain of -180 < azimuth <= 180 degrees, and elevation has a domain of -90 <= ...
50 views

### Volume of intersection of two partial spheres having origins at different coordinate frames using spherical coordinates

Assume I have a world coordinate frame $\mathbf{w}$. Assume I have a second coordinate frame that can be parameterized as a $4\times4$ homogenous transformation matrix with respect to the world ...
44 views

### What is the volume of intersection of a spheres along the contour lines of a spherical volume and a different spherical volume?

Assume I have one spherical volume $s_0$ with origin $(x_0, y_0, z_0)$ and radius $r_0$ represented by $(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 \leq r_0^2$. There is another spherical volume $s_1$ ...
26 views

### Number of vertices in spherical embeddings with large stars.

Suppose I am given a spherical embedding of a graph, specifically a triangulation, onto the unit sphere such that all edges are short geodesics (lengths strictly smaller than $\pi$). Let us further ...
13 views

### Parameterization of closed spherical elastic curve

This paper provides an expression of the spherical $p$-elastic curves (bottom of page 9) and claims that this expression is obtained by adapting the results of this paper which treats the case $p=2$. ...
1 vote
30 views

### Isoperimetric inequality on $(S^2,g)$

After reading the Wikipedia page on the spherical isoperimetric inequality, I came up with the following inequality by calculating some examples. Consider a Riemannian metric $g$ on the topological ...
13 views

### Set of bounded diameter is contained in a spherical ball

In the following I consider spherical distances. Say a subset $X$ of the sphere $\mathbb{S}^n$ is convex if it contains all geodesic segments between its points. Let $k<\frac{2\pi}{3}$, and suppose ...
31 views

### Longest distance between spherical segments

In the following, I use the spherical distance in $\mathbb{S}^n$. Let $a,b,c,d\in\mathbb{S}^n$ and suppose we know the pairwise distances between them. Are there well known formulas for the maximal ...
108 views

### Converting between two spherical coordinate systems with an application to astronomy

I am currently working on a sub-problem of a larger problem which involves being able to efficiently align an equatorial telescope mount with greater precision than the conventional amateur method, ...
40 views

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### Stereographic Projection From an Arbitrary Point on the Sphere

In the usual stereographic projection, the projection point is taken to be $(0, \ldots, 0, \pm 1) \in S^n$ and we project to the plane $x_{n+1} = 0$. Is there a formula for the projection from an ...
216 views

### What is the radius of a given circle of latitude?

I have a sphere with some latitude parallels in its northern hemisphere. I know the radius of the sphere, and I know the radius of each of the latitude parallels. I also know the distance between all ...
36 views

### Finding latitude values that bound spherical segments of a desired surface area [closed]

Let's say we have a perfect sphere with surface area equal to 1. In the following diagram (not to scale), I need to calculate the latitudes of parallels A through G (based on the following ...
1 vote
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### What is the mathematical shape of a projected spherical ellipse on the base of the hemisphere?

The projection of a disk, with an arbitrary orientation, onto an sphere is an spherical ellipse(right image), I believe. Now, what is the projection of this shperical ellipse on to the base plane of ...
1 vote
68 views

### Rotation along geodesic in n-dimensional sphere

Suppose I have two unit-length vectors $a, b$ in n-dimensional Euclidean space - so they are two points on the n-dimensional unit sphere. Now there is a geodesic between these two points, which ...
1 vote
47 views

### Find coordinate for the third point of a triangle on the unit sphere, given the coordinates of the two points and their angles

Suppose the Cartesian coordinates of the three points are $A,B,C,\|A\|=\|B\|=\|C\|=1$. Given $A$,$B$ and $b=dist(B,C)$, $c=dist(A,C)$, how to find $C$? Here, $dist(\cdot,\cdot)$ is the distance on the ...
1 vote
147 views

### Calculating the intersection area for circles on spheres

How would one calculate the intersection area of two circles on the surface of a unit sphere, defined by its direction and angle. In the pictures there are three possible problems. One where the ...