# Questions tagged [spherical-geometry]

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

891 questions
Filter by
Sorted by
Tagged with
55 views

### 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 $(S^{n},g)$ ...
1 vote
30 views

### 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 ...
• 292
25 views

### 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 ...
• 4,429
1 vote
72 views

### 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 ...
1 vote
37 views

### 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 ...
• 197
47 views

### 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 ...
• 1,753
50 views

### 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 ...
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 ...
• 203
32 views

### 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 ...
• 1
73 views

### 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 ...
1 vote
91 views

### 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 ...
• 11
37 views

### 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 ...
37 views

### 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 ...
89 views

### 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 ...
1 vote
74 views

### 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 ...
• 11
1 vote
130 views

### 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 ...
• 25
1 vote
127 views

### 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 ...
• 25
209 views

### 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 ...
121 views

### 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 ...
39 views

### 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 ...
• 101
78 views

### 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--...
• 19
60 views

### 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°. ...
• 11
1 vote
125 views

### 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 ...
• 1,044
1 vote
157 views

### 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 ...
• 43
1 vote
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 ...
• 2,407
185 views

• 23
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 ...
• 556
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 ...
• 9
1 vote
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 ...
• 67
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 ...
• 314
1 vote
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 ...
• 770
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 ...
• 712
1 vote
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 ...
1 vote
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 $n$ ...
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}}...