# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...
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-\...