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

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Trouble with understanding a spherical coordinate system.

We have a sphere with $r=1$, and we want the coordinates of $C$. $A$ is the north pole, and $AB$ is our prime meridian. See picture: I'm familiar with an $(x,y,z)$ coordinate system, but not so ...
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88 views

How I cut my orange - spherical volume integral

I cut my orange in six eatable pieces, following some rules. My orange is a perfect sphere, and there is a cylindrical volume down through my orange, that is not eatable. In the diagram, the orange ...
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1answer
34 views

What are the umbilic hypersurfaces in a sphere?

It is a well-known result that all umbilic hypersurfaces (complete and connected, say) of $\mathbb{R}^n$ are spheres or planes. But what can we say about umbilic hypersurfaces of a constant curvature ...
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49 views

Intersection of Two Parabolas on a Sphere

I'm trying to implement the algorithm in this paper which describes an implementation of Fortune's algorithm on a sphere, and I'm getting hung up on the math explaining how to calculate the ...
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32 views

Implicit Representation of Surfaces - Basic Quesion

I am reading on implicit representations of surfaces and cant quite come around the following example. Take $F : \mathbb{R^3} \rightarrow \mathbb{R}$, where $F(x,y,z)=x^2+y^2+z^2$. Now we want to ...
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37 views

Curving points to a sphere

I think math.stackexchange is the right place to post this, but if not, feel free to tell me. I have a series of points to be plotted on a sphere (Each one has a latitude and longitude value). These ...
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1answer
41 views

Closed Operations in Spherical Space

Euclidean vector arithmetic (addition/subtraction) is not closed under a spherical space. For example: $\mathbb{S}^2=\{v \in{\mathbb{R}^3}|\|v\|=1\}$ We have $(1, 0, 0)\in\mathbb{S}^2$ and $(0, ...
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80 views

Computing distance from line to point in geodetic environment

Supposing to be in a cartesian plan and that I have the following point: $$A(x_{1},y_{1}), B(x_{2},y_{2}), C(x_{3},y_{3}), D(x_{4},y_{4})$$ $$P(x_{0},y_{0})$$ Now immagine two lines, the fist one ...
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387 views

How to construct three mutually orthogonal circles in stereographic projection?

I'm new to spherical geometry and I enjoy doing ruler-and-compass constructions, so I'm trying to teach myself to do them in stereographic projection. I'm finding it challenging, to put it mildly. ...
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1k views

Different ways for calculating distance between two geodetic points give me different results

I'm trying to calculate the distance between two geodetic points in two different ways. The points are: A:(41.466138, 15.547839) B:(41.467216, 15.547025) The ...
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71 views

Given 3 Vertices of a Tetrahedron, Find the 4th

A regular tetrahedron is circumscribed by the Earth (assume spherical). You are given 3 of the 4 vertices (as latitude and longitude in decimal format), and asked to find the 4th. Any help is most ...
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22 views

Verify whether the centre of a sphere is outside a pyramid whose vertices are given

I have a maths question where the equation of a sphere $S$ is said to be $x^2+y^2+z^2-12x-6y-4z = 0$, and I'm asked to show that the centre is outside the pyramid whose vertices are $A(12,0,0)$, $B ...
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1answer
20 views

Equation of a sphere passing 3 points and tangential to a line

I got a maths question that gives you 3 points, $A (6,0,0)$ and $B (6,6,0)$ and $C (0,6,0)$ and a line DG, D being $(0,0,6)$ and $G (0,6,6)$ so the equation of DG is $\vec r$ = 6$\hat i$ -$6t\hat j$ . ...
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54 views

Internal angle formula of a generalized polygon as a function of side length and apothem

I am looking to compute the internal angle of a generalized regular polygon (spherical, euclidean, or hyperbolic) as a function of its apothem and side length. I know the equation for a euclidean ...
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1answer
41 views

Suppose that a triangle T on a unit sphere has area equal to π/2. Which of the following is necessarily true?

Suppose that a triangle T on a unit sphere has area equal to π/2. Which of the following is necessarily true? ...
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73 views

Volume of a spherical tetrahedron

In the paper Jun Murakami, The volume formulas for a spherical tetrahedron a formula for the volume of a spherical tetrahedron is given. I am trying to work through the details for the specific ...
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1answer
90 views

Any $1$-form (only on $S^3$?) can be uniquely written as a sum of a closed $1$-form and a coclosed $1$-form?

What is meant by saying that any $1$-form (only on $S^3$?) can be uniquely written as a sum of a closed $1$-form and a "co-closed" $1$-form? [...Since $H^1$ of $S^3$ is trivial it follows that the ...
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56 views

What is the difference between the sphere and projective space?

I know about the antipodal mapping. What I want to know is what the most significant differences between the sphere and projective space are, and how to think of each of them and their relationship ...
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1answer
104 views

Cover the n-sphere with sub-hemispherical caps

Original Question (answered): Define a cap (x,Phi) to be the set of all points of the sphere that are within an angle Phi of the point x. $ 0 \le \phi < \frac{\pi}{2} $. (define the angle ...
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1answer
117 views

Angled Spherical Sector - formulas?

I am glad to be here. First off, please excuse my almost saddening lack of knowledge. Math (in general) isn't my strong point. I might ask some really basic questions, you have been warned :) The ...
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53 views

How to rotate point on a spherical coordinate

I am looking for formula that shows hot to find the spherical coordinates of a point after the axis rotated. Assume that I am on earth and I have the coordinate of a point in lan/lot (which I believe ...
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1answer
173 views

Spherical Harmonics expansion of a Dirac Delta at the North Pole

I think all the coefficients for the spherical harmonic expansion of a delta function at the north pole should be a constant (presumably 1), but I'm having difficulties calculating them. Could someone ...
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1answer
58 views

Energy of particles on sphere and uniform rotation

I have a computer program containing some particles on a unit sphere, characterized by their positions $\{(\theta_i,\phi_i)\}$. They have a total energy given by the arc distance between particle ...
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58 views

Precalculus in a Nutshell, Geometry, Appendix B, Section 5, Question 14.

A sphere is circumscribed about a cube. Find the ratio of the volume of the cube to the volume of the sphere. So I drew this diagram: Next, I want to relate s and r. I apply Pythagorean theory to ...
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37 views

Extremal constant width curves on sphere

Definition Given some length $w\in\mathbb R$, I'm interested in closed convex sets $S$ of points with the following properties: For all pairs of points from $S$, the distance between them will be ...
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1answer
108 views

Project point onto line in Latitude/Longitude

Given line AB made from two Latitude/Longitude co-ordinates, and point C, how can I calculate the position of D, which is C projected onto D. Diagram:
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1answer
56 views

Which icosahedral triangles projected onto sphere's surface contain points in P?

I am working on a Python script to: Compute the vertex coordinates of a geodesic sphere/icosahedron, Project the triangles onto a sphere, then Find which spherical triangle contains an arbitrary ...
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2answers
948 views

How to find the intersection of three spheres (full solutions)?

The three equations of spheres are given $(x-x_{1})^2+(y-y_{1})^2+(z-z_{1})^2=a^2$ $(x-x_{2})^2+(y-y_{2})^2+(z-z_{2})^2=b^2$ $(x-x_{3})^2+(y-y_{3})^2+(z-z_{3})^2=c^2$ How do I find $(x,y,z)$ ...
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19 views

Angle to Object on Earth from Airplane

I am trying to find an angle from an airplane flying at 600ft to predetermined distances, namely 100, 200, 300, 500, and 700 meters. I want to incorporate a spherical earth model into calculating ...
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2answers
110 views

IS ASA applicable on triangles on the sphere?

$ASA= \text{Angle-Side-Angle}$ I was wondering if $ASA$ still worked on triangles for the sphere. I have a pretty hard time visualizing triangles on the sphere because I know the sum of their ...
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1answer
52 views

Law of cosine in spherical trigonometry

I found from a book of mine the formula $\cos a=\cos b\cos c+\sin b\sin c\cos\alpha.$ Can this be true? If for example $a=1m,b=1m,c=1m,\alpha=1$, $m$ denotes by meter, then $\cos m=\cos^2 ...
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56 views

Is it possible to convert meters to angles along the sphere?

I apologize for my scant knowledge of mathematics, but I'm developing a small web app, which will detect intersection points of circles on Google Maps. I used a lot of info from the internet to ...
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2answers
98 views

can a great circle route be predicted from initial condition?

Assuming the world is a sphere with no wind, can the great circle route of a vessel be predicted from the current position $\{\phi_i,\lambda_i\}$ and the current true course $\theta_i$? Presently, ...
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48 views

spherical geometry

A mobile, on the surface of the earth, is at a point A. Travels 200 km south arriving at a point B. Later moves 200 km west arriving at a point C. Finally moves over 200 kilometers to the north, back ...
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369 views

General Formula for Volume of Spherical Triangle

I'm working on a problem for which I'm trying to divide a sphere into layers defined by integer radius values ($r\in{1, 2, ...}$) such that the segments in each layer all have the same volume. Doing ...
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1answer
109 views

Formula for surface measure of spherical cap on $S^n$.

Can you show me an easy to use exact formula, or good lower and upper estimates, for the measure of a spherical cap of height $h$ on the sphere $S^n$?
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Map Earth surface so straight line distance is great circle distance?

Is there a function $f:$ (latitude, longitude) $\longrightarrow \mathbb{R}^n$ (for any finite $n$) such that the linear distance between $f(x)$ and $f(y)$ is the great circle distance between $x$ ...
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Make a balloon with least total length of seams

I'd like to build a small hot-air balloon out of flame-retardant plastic sheeting to suspend a camera. The plastic sheeting (plastic film) is commonly sold in a long roll in a width of 20ft (6 ...
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1answer
224 views

Ratio of Circumference to Diameter on a sphere

I was listening to an audiobook of Einstein when they started discussing spherical geometry and how Pi was no longer the ratio of a circle's circumference to its diameter, so I set out to find the ...
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1answer
924 views

Ratio of volume to radius of a sphere

If I have two spheres of a known volume $V$ and radius $r$ and I make one bigger sphere out of them, what will the new sphere's radius be? For example: Sphere $1$: $r_1=2$, $V_1=8$. Sphere $2$: ...
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432 views

The unsolved mathematical light beam problem

I have the following problem: Imagine that you have a sphere sitting at the interface of two media(like water and oil). And the position(the heigth) of the interface to the center of the sphere is ...
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1answer
134 views

surface area of a slice of a hemisphere

Imagine a hemisphere on it's base in the horizontal plane (center at the origin). Imagine another plane P which passes through the center (origin) and inclined at alpha to the horizontal (the base of ...
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0answers
45 views

integral over two spherical Bessel function

I am now having a problem regarding the integral over two spherical Bessel function. If anyone can give any help, it would be so nice of you. Thank you so much for any help. Specifically, I intend to ...
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2answers
112 views

How can I “move through a hypersphere?”

A man walking along a 2 dimensional circle will take a periodic path that begins and ends at the same point. Since he can travel in only a single direction, let's say how far along he is in his ...
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“Great Circle” distance [duplicate]

Given two points on a sphere, then the "great circle distance" between two points is the length of the smallest arc of a great circle containing both points. Assume that $\Sigma$ is a sphere of radius ...
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Are there exotic symplectic structures on $ S^2 $?

Besides the obvoius symplectic structure on $ S^2$ given by the area element in the standard embedding $ S^2 \to \Bbb R^3$, are there any other closed 2-forms on $ S^2$ which produce nonisomorphic ...
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Algorithm to generate an uniform distribution of points in the volume of an hypersphere/on the surface of an hypersphere.

I am searching two simple/efficient/generic algorithms to generate a uniform distribution of random points: in the volume of a n-dimensional hypersphere on the surface of a n-dimensional hypersphere ...
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59 views

Finding the coordinates of the corners of an aligned pole-centered spherical square

Given a spherical square of radius $1$, with edge midpoints at $(1, x, 0)$, $(1, x, \pi/2$), $(1, x, \pi)$ and $(1, x,3 \pi/2)$ (in the spherical coordinate system of (radial distance, polar angle, ...
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152 views

What is the maximum entropy distribution of points on a sphere that has a fixed non-zero average cosine of the polar angle?

Suppose we have a unit vector in 3D space whose orientation has some unknown distribution $p(\theta,\phi)$. All we know about this distribution is the average value of $cos(\theta)$: ...
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1answer
137 views

Laplacian on Sphere of Function Only Depending on Angle Between Points

Consider a function $f:S^2 \to \mathbb{R}$ , with $S^2$ the unit $2$-sphere in $\mathbb{R}^3$. Let's say that $f$ depends only on the polar angle $\theta$ from the north pole (e.g., $f(r,\theta,\phi) ...