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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31 views

Find the rate at which distances decrease in stereographic projection

I want to map a 3D space onto the inside's surface of a sphere. The 3D space is represented by points (x,y,z) where the z axis is the height. The first thing I did was to use the following equation ...
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10 views

Given a matrix nxd, what is the radius of its hypersphere?

Given the matrix: Centered on μ=0 What would be the radius of its hypersphere?
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28 views

Fun thing to think about: Parametrize a sphere from the outside

Suppose I have a sphere of radius R centered at $(x_0,0,0)$ where $x_0 > R$. I would like to parametrize the "face" of the sphere closest to the origin, using the angles $\theta, \phi$, where $\...
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Reconstruct a sphere from 6 patches

Let's say that I need to reconstruct a surface that could be cloned 6 times to create a perfect sphere. It is sampled in some finite N elements per side so the patch will have NxN elements (vertex) ...
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1answer
120 views

Green's theorem (integration by parts) on the unit sphere

List item I am missing something here and I need help to find it: Since the unit sphere $\mathbb{S}^{n-1}$ in $\mathbb{R}^{n}$ has no boundary, then given a smooth function $\phi$ and a smooth ...
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1answer
35 views

Is it easy to see that $S^2\times S^1$ does not admit Euclidean, Hyperbolic or Elliptic geometry?

It is easy to see that $S^2\times S^1$ as a Riemannian manifold is not Euclidean, hyperbolic or elliptic. Is it also easy to see that the topological manifold $S^2\times S^1$ does not admit one of ...
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84 views

Four Orthogonal Circles on a Sphere

Let $b$ and $c$ be two circles on a sphere, and $A$ be one of their intersections. We shall call $b$ and $c$ "orthogonal to each other" as the tangents of $b$ and $c$ at point $A$ are perpendicular to ...
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34 views

$\min_v {\frac{u^Tv}{\|u\|_2\|v\|_2}}$ subject to $v_i\geq 0,\forall i$ with fixed vector $u\in R^n_+$?

This problem comes to my thinking inspired from the Cauchy-Schwarz inequality. Given a vector $u$ in the nonnegative orthant $R^n_+$, what is the largest angle we can get between $u$ and vectors in $...
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39 views

How would we perceive space from within the hypersphere $S^3$?

Imagine being in a relatively small 3-sphere. Since geodesics behave differently from euclidean space, light will follow different trajectories, and I am wondering what the impact would be on our ...
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21 views

Fundamental forms of a sphere are proportional

I'm trying to prove that the first and second fundamental forms of the sphere are proportional to each other, regardless of the parametrization. I was trying to exploit the fact that the normal to a ...
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1answer
32 views

Area inside a curve on the surface of a sphere

Suppose there is a surface $\Sigma$ on the surface of a sphere with unit radius, the surface is bounded by a curve $\Gamma$. The curve is closed and has no wiggles (sorry, im a physicist I forgot ...
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23 views

Legendre Expansions for Derivatives of Delta Function

Expansions of the delta functions, of following type which are called completeness relations is very useful in many problems in physics. $\delta(x-y) = \sum_{n=0}^\infty P_n(x)P_n(y) \frac{2n+1}{2} $ ...
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1answer
28 views

Project a rectangle onto a sphere

As by the title, I would like to project a finite rectangular object onto a sphere. In particular, as depicted in Img1 there are two different kinds of projections that I would like to perform (A and ...
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27 views

For what reason straight lines must be on planes that go through the origin and how were the centers (origin) of the different geometries defined?

I've asked this to many mathematicians but I don't get a conclusive answer. Regarding origins (centers): - I understand that the origin in spherical geometry is the equidistant to all the points on ...
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47 views

Given two points on a sphere, how can I calculate a third point in the same “direction”?

Given the vector points A and B on a sphere, relative to center, I want to find point C, which would be on the other side of B seen from A, equal in distance and in the same direction. I don't know ...
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1answer
27 views

How to calculate the area of a spherical triangle on a globe using spherical trigonometry?

I'm studying spherical geometry and was having some trouble solving an exercise problem. More specifically, this is in the section for the law of cosines. The exercise problem is a navigation/mapping ...
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How to generate a set of uniform points on a given sub-region of hypersphere surface by a given number

I am searching for a solution for generating a set of uniform points on a given sub-region of hypersphere surface by a given number. There are several conditions to describe the set of generated ...
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1answer
43 views

Proof that a spherical triangle is congruent with its dual

I'm currently studying spherical geometry and ran into an exercise problem that I'm having trouble understanding. The book first defines a dual spherical triangle $\triangle^* ABC$ of original $\...
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1answer
19 views

Why do we divide the cross product by sine when obtaining the points for the dual triangle on a sphere?

I'm studying spherical geometry and had a question regarding triangles on spheres. The book that I'm using says that if we have triangle $\triangle ABC$ and we want to obtain the dual triangle $\...
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24 views

How to get the coordinates of a point on a sphere when we only know the value of the side of a triangle?

To elaborate a bit more on the title, I'm currently learning about spherical geometry and had a question regarding getting the coordinates of a point on a spherical triangle. I have the following ...
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12 views

How can I define a Goldberg polyhedral with nearly-regular hexagons, given a hexagon width and sphere circumference?

I'm trying to create a "hexagonal grid" that covers the planet earth with nearly regular hexagons. I understand that, by including 12 pentagons, the remainder of a sphere can be covered by pseudo-...
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25 views

Spherical Triangle angles

I was asked to observe the relationship of the sum of angles on spherical triangles inscribed on a circle on the sphere, I have found that the sum of angles on an inscribed spherical triangle is ...
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35 views

Find the coordinates of a point in 3D space

I have a sphere with radius $d$ centered at the origin. The $z$-axis is vertical. I take any point $P$ on that sphere, $P(d \cos\phi \sin\theta,\ d \sin\phi \sin\theta,\ d \cos\theta)$. I take a ...
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1answer
87 views

In non-euclidean geometry, can hypotenus of rightangle triangle be shorter than sides?

Disclaimer. This is only a recreational question in geometry... In euclidean geometry, the following picture is definitely inconsistent! Of course, the issue is that, the picture suggests that the ...
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29 views

Average angle between a point on a sphere and the line of sight

I want to know what the average angle is for a random point on a sphere with respect to our line of sight. I simulated it by doing the following (Image link): Randomly sample points on a surface of a ...
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Given a spherical triangle ABC and a point P, find (u, v) for slerping

Hmm. Where am I going wrong? Firstly, I apologise, I'm only a programmer, and skinny also, not a maths bod. Everything is on the unit sphere I have a spherical equilateral triangle ABC and a 2D co-...
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How to define tangential (circumferential) angle on a sphere

I have a cross section which is shown below: I have made hemisphere by revolving the above cross section around Y axis as shown here: My question is: How can I define tangential angle in Cartesian ...
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Can a “biangle” on a sphere be considered a “regular polygon” on the sphere?

I am being asked whether a lune on a sphere —that is, where the sphere is divided into four regions by the intersection of two great circles— can be considered a regular polygon on the sphere. Here, a ...
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Right Spherical Triangle with rational sides in Spherical Trigonometry

What formula relates rational sides and hypotenuse in spherical geometry? I obtained an example quite by chance: $$ \cos \frac{5}{18} \cos \frac{1}{3} = \cos \frac{31}{72} $$ with radius of ...
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1answer
62 views

Parameterization of a spherical spiral

For a spherical spiral curve, parametric representation is given as: $x=r \sin(t) \cos(ct)$, $y=r \sin(t) \sin(ct)$, $z=r \cos(t)$ with $t=[0,\pi]$ and $c$ a constant. How can I translate this to a ...
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1answer
93 views

Formula for the Volume of a Spherical Triangle given the Solid Angle

I'm working on a problem for which I have to calculate the volume of a spherical triangle given the Solid Angle Ω. Specifically, if the Solid Angle Ω (in steradians) span by 3 vectors $\...
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29 views

How to find the distance between two points on a sphere in polar coordinates with fixed $\phi$

I'm trying to calculate the distance between to points on a sphere with fixed $\phi$. For clarity, the sphere has radius $R$ and is centered at the origin. So, if we let $\phi = \frac{\pi}{2}$, then ...
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1answer
36 views

What are some nice, high resolution tessellations of $S^3$?

I'm looking for "nice" tessellations of $S^3$ into as many pieces as possible. Another way to think about this problem is looking for "nice" 4-polytopes with as many faces as possible, since we can ...
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Where does the haversine formula come from?

I understand that the haversine formula is used to calculate the great circle distance between two points, but I haven't been able to find a satisfactory explanation of the mathematical foundations of ...
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Spherical radius for local oblate spheroidal geometry

I have some geometry to do on the surface of the Earth, where the scale is sufficiently large to justify spherical geometry. That's implemented and working fine. I'm playing with the idea of using ...
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1answer
41 views

“Sphere Ellipse” locus on surface of sphere

Been attempting again to find a neat equation on a unit radius sphere for Sphere Ellipse locus in 3D conceptually similar to a plane ellipse. Geodesic arc distances between point P, in spherical ...
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21 views

Fractional winding numbers on S3/Z2

I am trying to understand what happens to the winding number of a mapping $$\omega: S^3_1 \rightarrow S^3_2$$ when antipodal points on $S^3_2$ are identified. Suppose I take the identity map between ...
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1answer
44 views

How to compute the DFT of a radially symmetric function in 3D?

Let $\rho: \mathbb{R}^3\rightarrow\mathbb{R}$ be a radially symmetric function, i.e., $\rho(x,y,z)=\rho(r)$, where $r^2=x^2+y^2+z^2$. Then the Fourier transform can be computed. $$ \begin{equation} \...
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1answer
89 views

Showing a curve must lie on a sphere.

Let $\mathbf{r}(t)$ be a parametrisation of a curve $\mathcal{C}$ in $\mathbb{R}^3$ such that $\mathbf{r}(0) = (R, −R, R)$, where $R \in \mathbb{R}$. Suppose $\mathbf{r}(t)\neq \mathbf{0}$ and $\...
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1answer
39 views

Does the shape of a tangerine (a sphere pinched at the poles) have a name?

I have in mind a modified sphere which, if all longitudinal radii were equal, would just be a sphere however instead, each longitudinal radius has an increasingly smaller value than those of the ones ...
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1answer
252 views

cyclical points on a conic section

$P_1, P_2, P_3, P_4$ are four arbitrary points on $xy = 1$. $P_1P_4$ intersects $P_2P_3$ at $D_1$, and similarly define $D_2, D_3$. Prove that $O, D_1, D_2, D_3$ are cyclical, where $O$ is the origin. ...
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69 views

the angle between two great circles on the sphere

I am trying to learn spherical geometry, but I have difficulty resolving a simple issue. Let's define a sphere's equator and it's poles N, S. if we create a great circle by tilting the equator circle ...
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1answer
48 views

Equation for surface of intersection between cone and sphere

Given a sphere of radius $r$ centered at the origin $(0,0,0)$ and a cone with an apex $> r$ from center of the sphere, with a cone angle of $\theta$, pointing at the center, how do I define the ...
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distinguishing positive and negative angle derived from spherical law of cosines

I have the coordinates of the endpoints of two geodesics on a sphere (in latitude longitude) coordinates. I've used the spherical law of cosines to determine an angle between the geodesics. ...
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1answer
59 views

How to calculate area of these overlapping rings?

I want to calculate the surface area of what the brown paint stripes cover on the sphere. $r = 5$ cm.
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100 views

intersections of line segments (geodesics) on a sphere

I am given two geodesics on a sphere, each designated by longitude and latitude of the endpoints. How can I calculate whether they intersect each other? I.e. whether the great circles intersect ...
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150 views

How many $\mathbb R^n$ vectors are needed to guarantee a large correlation between at least a pair of them - an open problem in mathematics?

Suppose you had $m$ vectors each of dimension $n$x$1$, and you compute a Pearson correlation coefficient for every unique pair of them. For example, for one such pair of vectors $x$ and $y$, the ...
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69 views

Maximum number of vectors in $\mathbb R^n$ that are at least an angle $\theta$ apart

In a space $\mathbb R^n$, how many vectors can you fit that are at least an angle $\theta$ apart, where $\theta>0$? For example, in $\mathbb R^2$, the maximum number of vectors that are at least ...
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1answer
29 views

Spherical Triangle: Derive law of sine

Prove the law of sines for the spherical triangle PQR on surface of sphere. BACKGROUND Suppose we have a sphere of radius 1. Let vectors $\vec{A}$, $\vec{B}$, and $\vec{C}$ be drawn from the center ...
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1answer
22 views

Number of vertices in a convex uniform polyhedron

Given any convex uniform polyhedron, denote all angles formed at a vertex $\alpha_1, \alpha_2, \dots$, I find that, \begin{equation*} \text{the number of vertices in the polyhedron}=\frac{4\pi}{2\pi-\...

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