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 ...
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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 ...
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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 ...
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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 ...
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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 $n$ ...
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What is the parametric path of an equalateral spherical triangle in the first octant on the unit sphere with the edges equadistance from each plane?
I want to clarify a bit. The vertices need to be the same distance a path that follows from (1,0,0), (0,1,0), and (0,0,1) and converges at ($\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}...
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The length of shortest curve dividing topological sphere into two parts of same area
$(S^2,g)$ is a Riemannian manifold, where $S^2$ is homeomorphic to the 2-dimensional sphere. And the area of $(S^2,g)$ is $4\pi$. $\gamma$ is the shortest curve dividing $S^2$ into two parts of equal ...
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Diameter of Voronoi cells of regular simplex in sphere
Let $\mathbb{S}^n\subseteq\mathbb{R}^{n+1}$ be the sphere with geodesic distance, and let $p_1,\dots,p_{n+2}\in\mathbb{S}^n$ be distinct points forming a regular simplex in $\mathbb{R}^{n+2}$. For $i=...
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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 ...
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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 <= ...
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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 ...
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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 $s_1$ ...
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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 ...
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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 $p=2$. ...
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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 ...
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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 ...
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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 ...
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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, ...
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Girard's Formula for Arbitrary Spherical Polygons (concave, self-intersecting, etc.)
In every reference I've seen, the area of an $n$-sided spherical polygon is given by the sum of all interior angles, minus some amount of spherical excess, or
$$\text{Area}(\text{polygon}) = \sum_{i} ...
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Extract Spherical coordinates of panorama from camera pose information and panorama (+pixel) size for correct position in a 3D space.
I am working with spherical image representation. I have the following camera pose information:
[pose.rotation.x, pose.rotation.y, pose.rotation.z, pose.rotation.w],
[pose.translation.x, pose....
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How do you divide surface area?
I am having difficulty trying to scale the surface area of a rectangle on a sphere.
I have tried many different things and have gotten many different answers. I am doing this with a worldbuilding ...
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Measure of Cylinder Intersecting Sphere in $\mathbb{R}^n$
Please check my work.
Let $\mu$ be the uniform measure on $\mathbb{S}^n$, the unit sphere with radius $1$, and $$A=\{x\in\mathbb{S}^n:x_1^2+x_2^2\le \sin^2\alpha\}$$
where $\alpha$ is constant.
Claim $...
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Dirac delta function in spherical cordinates
I am studying Dirac Delta Function in Spherical Polar Cordinates. I found this expression
$$\delta\ ^3 (\vec r -\vec r_{o}) = \frac{ \delta\ (r -r_{o})\delta\ (θ -θ_{o})\delta\ (φ -φ_{o} )}{r^2 \sin ...
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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 ...
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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 ...
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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 ...
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Why is sine-squared used in the haversine formula?
The traditional "haversine" formula for great-circle distance is traditionally expressed as
$$
2R \arcsin\left(\sqrt{
\sin\left(\frac{\Delta\varphi}{2}\right)^2 +
\cos\left(\varphi_1\...
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Derivation of solution for the first geodetic problem on a sphere
Some version of the following formula is often quoted for use in solving the "first geodetic problem" (aka "direct" or "forward" geodetic problem) on a spherical Earth:
$$...
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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 ...
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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 ...
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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 ...
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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 ...
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Find the image of spherical surface x^2+y^2+z^2=1 under the composite transformation
\begin{array}{l}
( 5-1) \ From\ the\ question,\ the\ normal\ vector\ of\ plane\ x+y+z=0,\ \vec{u} =( 1,1,1)^{T}\\
According\ to\ Householder\ Transformation,\ g=I-2\vec{u}\vec{u} =\begin{pmatrix}
1 &...
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Convolution on a sphere using generalised fast Fourier transform
I'm reading this paper, which is about spherical convolutions and defining equivariance on a spherical signal. Similar to planar signals where we compute the convolution using FFT by doing $f \ast ψ = ...
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Mapping a hemisphere onto the unit circle
How can I approach attempting to map a hemisphere onto the unit circle such that the meridians become arcs of a circle through $(0,1)$ and $(0,-1)$ and are evenly spaced around the x-axis (which is ...
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How to find the parameters necessary for drawing the lines of latitude and longitude of a spherical orthographic projection as partial ellipses?
I recently asked this question, which is a preface to this one. In that one I explain roughly what I'm attempting to achieve, but I realized that once I found that cutoff angle I still didn't have a ...
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Solid angle on a sphere of the intersection of multiple hemispheres
Consider the following scenario where I have a unit sphere cut by $N$ halfplanes that all contain the center of the sphere, thus forming several hemispheres. How can I calculate the solid angle ...
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Is the surface of a sphere 2D or 3D
I'm reading an article and it describes signals on a sphere. I dont understand how SO(3) isn't considered a signal on the sphere. And why is the sphere referred to as S2, when spheres are 3D manifolds?...
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Integration over angular spherical coordinates
I am trying to solve the following integral in spherical 3D coordinates
$$\mathcal{I}(\theta,\phi)=
\int_0^{2\pi}d\phi_{\Delta}\int_{\delta}^{\Delta/2}\sin\theta_{\Delta}d\theta_{\Delta}
\frac{1}{[1-\...
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In spherical geometry, are disks convex, and are chords internal to a circle?
In Euclidean geometry, a chord of a circle is internal to the circle. However, the proof of this relies on the External Angle Theorem, which does not hold for spherical geometry.
In spherical ...
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Is it possible to uniformly subdivide a sphere into arbitrarily small cells?
I am not a geometer, so I might be misusing some terms. So let me try to be more explicit regarding what I mean.
"Subdivide a sphere into cells" means to partition the set of all points on a ...
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Do Schläfli symbols unambiguously represent gemetric shapes?
According to Wikipedia, the tesseract is the four-dimensional analogue of the cube and has the Schläfli symbol {4,3,3}, and Wikipedia features the following visualization:
However, when looking at it,...
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Understanding the spherical means on the sphere
I was studying spherical means on the sphere for some purpose, when I had this question. Many sources of literature give the spherical means in a geometrical way, and I want to understand them through ...
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Finding final bearing of a ship given its initial bearing, initial coordinate, and final longitude
So I’m new to this spherical trig and great circle route stuff. The question says:
There is a ship departing from point A (84° W, 15°S) on a great circle route. If its initial bearing is 70°, then ...
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Convex hull of points on sphere: open hemisphere condition
I've read in several places that the convex hull of a subset of the sphere is not the whole sphere if and only if the subset is contained in an open hemisphere, but I'm not sure how to prove it. Does ...
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What is the closest equivalent of an elliptic Gaussian distribution on the hypersphere?
The van Mises Fisher (vMF) distribution ($p(x;\mu,\kappa) \propto e^{-\kappa\mu^\top x}$ can be considered a close equivalent of a symmetric Normal Distribution $\mathcal{N}(\mu, \mathbf{I})$ on the ...
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Sum of angles of triangles on a sphere
I've forgotten too much math to do this myself:
A guy in a general forum proposed a regular tetrahedron inscribed inside a sphere (specifically, the Earth), and then the borders of the sides are ...
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Intersection of 2 circles on the surface of a sphere?
I'm out of my depth playing with non-euclidean geometry, so if terminology is off or I'm missing critical things, leave a comment and I'll edit.
Given the radii ($r_1, r_2, r_3$) and centerpoints of 3 ...
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Distributing points uniformly on a 4D surface
I am trying to distribute points onto a 4D surface. I have an implementation that evenly distributes data points onto a 3D surface octant.
...
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Why is $\pi_1 (S^n) = 0$ for $n \ge 2$?
You can geometrically show why it's like that for $S^2$, as every curve based in a point $x$ can be retracted to that point $x$, so the fundamental group is trivial hence equal to the point $0$.
But ...
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