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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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Covariant derivative of a function along a curve

I am trying to find the covariant derivative of the function $f(x) = \arccos(\langle x, y\rangle)$ with respect to $x$ along a smooth curve using Christoffel symbols. My working space is $(S^{n},g)$ ...
vanilla-objective's user avatar
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What is the proper term for the hemisphere of an $n$-sphere?

The question is in the title. My intuition would tell me that it should either be called an $n$-hemisphere, a hemi-$n$-sphere, or a semi $n$-sphere. However, I am primarily curious if there has been ...
David G.'s user avatar
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Question from the Proof of the Classification of Finite Subgroups of SO(3)

On page 14 here, the author sets up the following situation: say we have an order 12 subgroup $G$ of $SO(3)$ acting on the poles of $G$ (i.e. on the set of vectors on $S^2$ fixed by some non-identity ...
Johnny Apple's user avatar
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Derivative on a submanifold of $\mathbb{R}^n$ through a curve

I am a beginner in differential geometry and have been trying to find the gradient of the great arc distance $f(x) = d(x,y) = \arccos(\langle x, y\rangle)$ with respect to $x$, but getting two ...
vanilla-objective's user avatar
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How to prove that in a geometry with no parallel lines, all the lines must be finite in length?

In the Veritasium video on the fifth Euclidean postulate, there was a statement (link with a timecode) that mathematicians (before discovering spherical geometry) managed to prove that in a geometry ...
g00dds's user avatar
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In expectation, what is the area of a triangle formed by three points chosen randomly on the surface of the earth?

Three points are randomly chosen on the surface of a sphere. They are connected to each other by great arcs to form a (curved) triangle. What is the expected value of the area of the triangle? We know ...
Thomas Delaney's user avatar
2 votes
1 answer
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Regular spherical quadrilateral tiling for a game board

Are there constructions for tiling a sphere with mostly regular spherical quadrilaterals, but with correction spherical polygons whose number and total area are minimized? In other words, a corrected ...
David Spector's user avatar
2 votes
1 answer
42 views

Determine Rotation Angles for Overlaying Regular Polygons on a Sphere

On a sphere of radius $R$, suppose I have regular polygons $P$ and $P'$ with vertices $(v_1, v_2, \dots, v_n)$ and $(v_1', v_2', \dots, v_n')$ with the polygons positioned so their centroids are on ...
BoCoKeith's user avatar
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Is the projective sphere a finite projective plane?

I am working on an introductory project on projective planes. I'm aware of finite plains such as the order 2 and order 3 projective plane. Wikipedia mentions the complex projective plane as an example ...
Josh's user avatar
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Show that a linear fractional transform is a rotation

I have a linear fractional transformation given by: $$F_K (z) = \frac{\cos{\theta}z + i\sin{\theta}}{i\sin{\theta}z + \cos{\theta}}$$ And I am supposed to find the fixed points and then verify that ...
Alex Lott's user avatar
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Volume integration spherical cap cut by plane

When a spherical cap is cut by a plane $y=y_1$ that is perpendicular on the plane that made the sperical cap $z=z_1$, I managed to calculate the volumes of the resulting two pieces of the spherical ...
Klavier's user avatar
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Calculate the intersection volume of two spherical caps on the same sphere

My question is the same as this question except that I am looking for the intersection volume of two caps instead of the area. Given a sphere with radius $R$ that has two spherical caps with base ...
Danny's user avatar
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Bath towel on the spheric rope: minimize the area of self-intersection of a 'folded' spheric rectangle

Some time ago I was curious about a question related to my bath towel, which I hang on a rope to have fun (you can use your own towel to do this experiment in bath-o if you want): 'There is this ...
Mikhail Gaichenkov's user avatar
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Is the complement of a triangle on the sphere also a triangle?

In the sphere above, the shaded area defined by the points A, B, C clearly makes a triangle. My question is, can the complement of this area, that is everything on the sphere that is white, also be ...
Christofer Ohlsson's user avatar
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Does this shape have a name? (A 'spherical circular triangle' ???)

This shape is formed by 3 'small' circles on the surface of a sphere, each touching the other 2. On a plane, the shape is called a 'circular triangle' (refer to Wikipedia). In this particular example ...
Parsley's user avatar
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1 answer
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Equation for 2-Axis Rotation to Point to Target Azimuth & Elevation

I previously asked this question the other way around (given the angles of rotation, what are the resulting angles of azimuth and elevation), thinking that it would be simple to reverse it, but it's ...
Fate24's user avatar
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2 answers
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Equation for Elevation & Azimuth after Rotation in Two Axes

Ideally I'd like a general equation but I've drawn a specific example of rotating a board $45^\circ$ from horizontal then $45^\circ$ on its own axis. I haven't been able to come up with an equation ...
Fate24's user avatar
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4 votes
0 answers
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How many ways can a cube fit into a sphere through its vertices?

I mean, we know that every cube has 8 vertices. Now imagine a sphere with a fixed radius. Cubes can have arbitrary sides, one way is this. that there are no vertices on the sphere and the entire cube ...
Rockstar M's user avatar
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Intersection points of two spherical circles

I'm trying to find the intersection points of two spherical circles on a unit sphere. Each circle is defined by a normal vector n pointing to its center, and an angle θ used as radius, meaning that a ...
Nicolas Repiquet's user avatar
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How can I curve a rectangular plane to match the surface of a sphere?

I have a flat rectangular plane representing an area of the earth. I am trying to bend this plane on all three axes to fit the surface of a sphere (in this case I am modeling the earth as a perfect ...
John's user avatar
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Spherical Trig: Finding A Missing Angle w/ The Sine Law

I've been working on these spherical trig questions that I initially thought were rather straightforward. Instead, my attempts to finish this assignment have descended into madness because I cannot--...
upas's user avatar
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1 answer
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Given latitude and longitude and facing north, how can I calculate the rotation needed to face another latitude and longitude (namely 0,0)? [closed]

As the title says, given I'm somewhere on earth facing north, I can determine the magnitude of the distance to get to 0°,0°, but I am not sure how to calculate the rotation needed to be facing 0°,0°. ...
bclax5's user avatar
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1 answer
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Proving a remark of Gauss on quaternions and spherical triangles in a more transparent way.

In the fragment "Rotations of Space" (1819) in which Gauss outlined the general properties of a quaternions algebra, Gauss stated the following: Given three consecutive scales with the ...
user2554's user avatar
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Expressing a circle in 3d space in terms of its extrema

I want to define a circle in 3D-space by its extremal points (maxima and minima) with respect the three coordinate axes. I have tried unsuccessfully to express ...
Pat Ward's user avatar
1 vote
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35 views

Using the Chern–Gauss–Bonnet theorem to calculate the hypervolume of a spherical simplex

The area formula for a spherical triangle is often stated as a consequence of the Gauss-Bonnet theorem: $$\int_MKdA+\int_{\partial M}k_gds=2\pi\chi(M)$$ The idea is that in a spherical triangle $T$ on ...
Jacob's user avatar
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2 votes
1 answer
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Proving a connection noticed by Gauss between lemniscatic functions and spherical geometry.

In p.415 of volume 3 of Gauss's werke one can find the following remark of Gauss: [Later note]: I. $$\alpha+\delta+\gamma = \pi [=\varpi]$$ set $$\mathbb{sinlemn}(\alpha) = \mathbb{tang} (a), \...
user2554's user avatar
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1 vote
1 answer
99 views

$SO(n+1)$-invariant metrics on the $n$-sphere [closed]

Fix $n \geq 1$, and let $\mathbb S^n$ denote the sphere of radius one centred at zero in $\mathbb R^{n+1}$. Let $SO(n+1)$ denote the Lie group of $n+1$ by $n+1$ orthogonal matrices with determinant ...
Joseph Kwong's user avatar
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1 answer
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How would I calculate the probability that rays pointing out of a sphere hit a square of some area? [closed]

In this case I am imagining a light source which shoots rays in all directions and forms a sphere and I would like to know how to calculate what percentage of the sphere's surface area is intercepted ...
spicy-lemonade's user avatar
1 vote
1 answer
59 views

Can i define a two variable function from the plane to the semisphere?

I'm wondering if i can define a function $S^2: \mathbb{R}^2\rightarrow\mathbb{R}^3$ such that a subset $P\subseteq\mathbb{R}^2$ of the cartesian plane has the unit 2-semisphere $S^2(P)$ as image? I ...
Simón Flavio Ibañez's user avatar
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Intersection of two straight lines in a spherical geometry

I have two spheres with origins $O_1 =(a_1,b_1,c_1)$ and $O_2 =(a_2,b_2,c_2)$ in cartesian co-ordinates respectively . A radial line is drawn with azimuth and elevation given by $\theta_1$ and $\phi_1$...
wanderer's user avatar
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Do two points on the surface of a sphere uniquely describe a great cirlce?

I had a debate with a buddy about this. He said you could get a chord by drawing the triangle formed by the two points and the center of the sphere and that chord corresponds to a single great circle ...
Frank Conry's user avatar
2 votes
1 answer
41 views

Radius of a sphere given the circumferences of 3 concentric circles and the distances between them along the surface of the sphere

Let's say that we know the circumferences of 3 concentric circles on the surface of a sphere, as well as the distances between the concentric circles and the order of the circles in a given direction, ...
Anders Gustafson's user avatar
1 vote
0 answers
172 views

Formula to project a 3D point onto a sphere, when centre of projection is not necessarily at the centre of the sphere

People sometimes talk about "projecting onto a sphere" analogously to projection onto a plane, but how is this defined (a formula would be nice)? It's not obvious to me how we should handle ...
Silverfish's user avatar
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1 vote
1 answer
256 views

Metric on S2 in Cartesian coordinates

I saw a stack post a couple of years ago about the metric on the two-sphere, S2 in cartesian coordinates, x, y and z. I can’t find the post. Below I sketch out what I remember – my apologies for these ...
GaryW's user avatar
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derive a depth of plate contacting a particle by using a curvature

I am learning physics theory called Hertz contact theory which considers a small particle contacting a plate. But I have been stuck in the geometry problem. I would like to get $z$ in the figure. When ...
Kinnikuman's user avatar
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What would be the new Equation of motion if the magnetic field's origin is shifted from the origin of a co-rotating spherical polar coordinates?

The equation of motions due to the dipole magnetic force of a planet in a frame corotating with the planet and origin at the centre of planet assumed to be sphere components wise are given as below: \...
Lunthang Peter's user avatar
1 vote
2 answers
101 views

How to find the launch direction to intercept an object moving on a sphere?

For my game, I have a projectile moving around the surface of a sphere, along a great circle of the sphere. I want to be able to launch an other projectile from a different object such that the two ...
Giorgio Guttilla's user avatar
2 votes
2 answers
110 views

Generalising Thales theorem for points on a sphere to form a 3-orthoscheme (tetrahedron.)

I am trying to find the condition that four points $p_1,p_2,p_3,p_4$ on the unit sphere $\mathbb{S}^1$ need to statisy in order to form a 3-orthoscheme (Tetrahedron with all faces as right angled ...
Vishesh's user avatar
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3 votes
1 answer
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Solid angle of human field of vision

This is a question about solid angles. According to Wikipedia, the central/binocular field of human vision is about $2\pi/3$ in the horizontal plane, and $\pi/3$ in the vertical axis. Roughly, this ...
JP McCarthy's user avatar
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How many balls to cover a hyperbolic or spherical disk?

Definitions: Let $D_k$ be the disk with constant curvature $k$ and with volume $1$. We call a finite subset $X\subset D_k$ a $\delta$-covering of $D_k$ if $D_k=\bigcup_{x\in X}B_\delta(x)$, where $B_\...
Alex's user avatar
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0 answers
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Dyadic partitions of $n$-spheres

For Euclidean space $\mathbb{R}^n, \ n\geq1,$ the $k$-level dyadic partition $\Delta_k, \ k=0,1,...,$ is defined as the set of cubes in $\mathbb{R}^n$ of sidelength $2^{-k}$ and corners in the set $$\...
ILLIA's user avatar
  • 23
0 votes
2 answers
121 views

Explain how shall we get a direction at a point on the surface of earth other than north pole using magnetic compass [closed]

The Qibla Compass can give the direction towards "some points on earth" other than north pole, eg : Mecca. Wikipedia first paragraph, last two lines : To determine the proper direction, one ...
lorilori's user avatar
  • 556
-1 votes
1 answer
69 views

How do I calculate new point after traveling a distance along great circle?

I'm brand new to geo processing/mapping, and I'm struggling with spherical trig, trying to understand what's going on without even knowing the terminology of what I'm looking for. After a lot of ...
Luke's user avatar
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1 vote
1 answer
81 views

Simplifying $\arcsin\frac{\sin a}{\sin c}+\arccos\frac{\tan a}{\tan c}-\frac\pi2$, the area of a spherical right triangle w/hypotenuse $c$ and leg $a$

Given an unit sphere and on this sphere a right angled triangle and given the length of the hypotenuse as $c$ and a leg as $a$. The area $E$ can calculated via the Napier rules as: (using radians as ...
Whogius's user avatar
  • 67
10 votes
2 answers
285 views

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 ...
Cirdec's user avatar
  • 314
1 vote
1 answer
46 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 ...
Peyman's user avatar
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0 votes
0 answers
122 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 ...
danzibr's user avatar
  • 712
1 vote
1 answer
54 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 ...
user1211016's user avatar
1 vote
1 answer
66 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 $n$ ...
wnoise's user avatar
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1 vote
0 answers
31 views

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}}...
Digital Greenery's user avatar

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