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

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59 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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1answer
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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0answers
49 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
40 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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67 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 ...
2
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1answer
85 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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1answer
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
97 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
106 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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50 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
152 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 ...
3
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1answer
54 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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1answer
44 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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35 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
100 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
55 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
719 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 ...
3
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2answers
101 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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0answers
47 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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44 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 ...
2
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2answers
96 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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1answer
47 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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326 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
103 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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2answers
195 views

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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0answers
34 views

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 ...
4
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1answer
212 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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0answers
156 views

Spherical Coordinates Integral to Great Circle

I need the function $\cos^n(\phi)$ integrated over the unit hemisphere cut with a great circle path. The following diagram shows the area to be integrated and the coordinate system used: The great ...
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1answer
780 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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3answers
419 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
122 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 ...
2
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2answers
106 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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55 views

“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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1answer
167 views

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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2answers
224 views

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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57 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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1answer
135 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 $cos(\theta)$: ...
3
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1answer
136 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) ...
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1answer
181 views

Convert spherical coordinates to Cartesian coordinates for a vector

So let's say I have a normalized vector $N$ given in cartesian coordinates and I have another normalized vector $V$, defined in spherical coordinates relative to the vector $N$. So $\theta_V$ is the ...
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2answers
159 views

Gauss Theorem example

Verify the Gauss theorem for the vector field $F(x)=\frac{x }{\|x\|},$ where $x \in W \subset \mathbb{R}^3$ and $$W=\left\{(x,y,z) \in \mathbb{R}^3 \left/ a^2\right.\leqslant x^2 + y^2 + z^2 \leqslant ...
5
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3answers
201 views

How to understand and create quaternions?

I have to multiply two quaternions to calculate a so called spherical linear interpolation between two $R^3$ coordinate systems within the interval $t = [0, 1]$. I understand how to do the ...
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1answer
216 views

Spherical coordinates of a unit vector around a normal $N$

So if I have a unit normal for a surface $N(x,y,z)$ and an incident unit vector $V(x,y,z)$ to that surface, how would I represent the vector V in spherical coordinates relative to the normal?
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1answer
155 views

Directions in spherical coordinates

Say I have a system with standard spherical coordinates. There's a man on that sphere and he's standing on the equator facing east. He chooses a random angle $0°-360°$ and turns that much in the clock ...
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2answers
712 views

Determine depth of a partially filled hemisphere

Recently came across a question in a Year 9 math book of which there was no "working out" supplied and offers now description on how they obtained the answer. The question goes like this: A bowl ...
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2answers
205 views

Vector Picking on the Unit Sphere

Imagine a vector from the center of a unit sphere to its surface: Now imagine a second vector generated in indentical fashion. Given the first vector, how can I generate vectors to uniformally ...
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60 views

Addition of spherical surface vectors

I'm making a planet simulator, which makes much use of a sphere. I'm trying to figure out how to represent and manipulate vectors on the surface of the sphere. Currently, my coordinates are all ...
2
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2answers
216 views

“Center” of a spherical triangle

I have a very deficient background in geometry, so I come across questions like these and I'm not sure how to verify my intuition. Consider three points in $\mathbb{R}^3$, given by position vectors, ...