Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [spheres]

For geometrical problems involving spheres. Use the tag (geometry) as well. For intrinsic geometry of spheres, see (spherical-geometry).

0
votes
1answer
25 views

Parametrizing the curve of intersection between a elliptic cylinder and a sphere

How can I form the parameterization of a curve of intersection given a sphere $x^2+y^2+z^2 =1$ and an elliptic cylinder $2x^2 + z^2 = 1$ ?
2
votes
1answer
25 views

calculating new 3D position on sphere with angular velocity vector

I feel like this is actually pretty simple but still could not find any solutions so far... I'm trying to calculate the movement of a point in a rigid rod with the equation $ \dot P = [ v + \omega \...
0
votes
1answer
42 views

Can we prove that two circles lie on a sphere?

There is a problem in my book In the question the circles are given in the form $S+kP=0$ where $S$ is plane and $P$ is a plane and $k$ is real number. The question asks us to prove that the circles ...
2
votes
0answers
17 views

distribution of bisecting great circles

For any point on the globe, I believe there is (by the mean value theorem) at least one great circle containing that point and dividing the world's land area (or water mass, or population, whatever) ...
1
vote
0answers
27 views

A minimizing property of the Sphere?

From the wiki “Sphere” entry: “The sphere has the smallest surface area of all surfaces that enclose a given volume, and it encloses the largest volume among all closed surfaces with a given surface ...
0
votes
1answer
31 views

Spherical cap problem - trigonometry / circle theorems problem / surface area

Graph here I am trying to derive the following equation from a paper I am studying, which the author has derived from the graph above. The two slightly curved lines here are modelled as the surfaces ...
1
vote
0answers
41 views

Calculate the integral of the following function over the sphere in $\mathbb{R}^4$

Calculate the integral of $f(x,y,w,z) = |y||z|^2$ over $S=\{(x,y,w,z)|x^2+y^2+z^2+w^2=1\}$, i.e. calculate: $\int_S|y||z|^2d\sigma_3$. My attempt - I'll define a parameterization as follows - $\phi(...
0
votes
3answers
50 views

Area of part of the surface of a sphere

Calculate the area of sphere $x^2+y^2+z^2 = 1$ where $a \le z \le b$ where $-1 \le a \lt b \le 1$. So I know I should use the usual parameterization: $r(\phi, \theta)=(cos(\phi) sin(\theta), sin(\phi)...
1
vote
0answers
10 views

Plot projective curves

Do you know any website, app or program that can plot curves in the projective plane? I know the projective plane $\mathbb{P}^2 \mathbb{R}$ can be visualized as a sphere with the antipodal points ...
0
votes
0answers
30 views

Express the volume of $S^n(a)$ in terms of the volume of $B^{n-1}(a)$

I'm working on an exercise which appears in a chapter about integration on manifolds. It asks to express the volume of $S^n(a)$ in terms of the volume of $B^{n-1}(a)$. Here $S^n(a)=\{x\in\mathbb{R}^{n+...
1
vote
0answers
36 views

Smallest close sphere containing $V$

Suppose I have a finite set of vectors $V = \{v_1, v_2, ..., v_n\} \subset \mathbb{R}^3$. What is the best way to find the smallest close sphere containing $V$?
0
votes
1answer
18 views

volume inside a solid solid angle

Is it right to say that in a sphere of radius R, the volume inside a solid angle $\Omega$ is just : $V=\frac{4\pi R^3}{3}\frac{\Omega}{4\pi}=\frac{R^3 \Omega}{3}$ ?
1
vote
0answers
53 views

Odd version of 24 squares formula?

The Leech lattice is related to the formula: $$1^2+2^2+3^2+....+24^2=70^2$$ This is related in turn to 26D bosonic string theory. Is there another known formula that involves the integers 1..8, ...
2
votes
1answer
30 views

Deriving the formula of the surface of a sphere using triangles.

My friend tried to find the formula of the surface of a sphere using the following reasoning, but we can't see the mistake: Let's first take half a sphere and divide the sphere into infinitely small ...
1
vote
1answer
54 views

What will a sphere look like if it's unwrapped?

I actually google alot, but all results are related to 3D design apps like blender bla bla bla, No direct answers or even something to help. I tried to imagine it like some triangles arranged and all ...
1
vote
3answers
24 views

Sphere with shell, so that the volume equals two times the original sphere

I was wondering about the following: Consider a sphere with radius $r$, then the volume equals $\frac{4}{3}\pi r^3$. Now consider to cover this sphere with a shell of thickness $h$. Then the new ...
0
votes
0answers
16 views

General properties of closed curves on a sphere

This is a question from a physicist trying to understand space curves. In 2D, a closed curve equidistant from a point is a circle, and has constant curvature and zero torsion. What can be said for ...
2
votes
2answers
98 views

Prove following points lie on a circle.

I found this in a textbook without a solution and I wasnt able to solve it myself. Let ABCD be a tetrahedron with all faces acute. Let E be the mid point of the longer arc AB on a circle ABD. Let F ...
1
vote
1answer
43 views

Number of hypercubes that fit into a hypersphere with radius 1 and centered at the origin

There is a hypercube with sides of length 2 and a hypersphere with a radius of 1. Both of them are centered at the origin (the hypercube surrounds the hypersphere). Each of the d dimensions in the ...
0
votes
1answer
31 views

Are a sphere cross a circle ($S^2 \times S$) and the 3-dimensional projective ($\textbf{RP(3)}$) space homeomorphic?

I have some questions about the space $\textbf{SO(3)}$ (special orthogonal transformations in 3-dimensions). I understand that this transformations represent rotations in 3D space fixing the origin ...
1
vote
0answers
23 views

Using randomly selected points uniformly distributed on the interval (0,1) find the volume of the unit sphere

I've been given this extra credit assignment to do in Matlab, but I don't understand the question. The part that I'm not clear about is where I have to satisfy the equation x+7+z>1. Can someone ...
3
votes
2answers
122 views

$I = \int_{- \infty}^{\infty} \delta (n - ||\mathbf{x}||^2) \mathrm d \mathbf{x} $ should not diverge

I'm having trouble evaluating this integral: $$I = \int_{-\infty}^\infty \delta (n - ||\mathbf{x}||^2) \mathrm d \mathbf{x} $$ where $\delta(x)$ is Dirac delta function, $\mathbf x$ is the real $n$-...
0
votes
1answer
23 views

Can higher dimensional spheres be regularly partitioned/discretized?

A circle can be partitioned into $n\in\mathbb{N}$ congruent 1-spherical line segments similar to the regular polygons. A sphere can be partitioned into $n\in\{4,6,20\}$ congruent 2-spherical ...
4
votes
5answers
79 views

If there are $2$ linearly independent vectors $x,y \in X$ such that $||x+y||=||x||+||y||$, then the unit sphere $S(X)$ contains an interval

Let $S(X)= \{x \in X: ||x||=1\}$ be the unit sphere in $X$. Assume that there are $x,y\in X$ linearly independent such that $||x+y||=||x||+||y||$. Prove that $S(X)$ contains the following set:$[x,y]=\{...
1
vote
1answer
66 views

Why does this vector field approach zero near the north pole?

In this question, Raziel's answer builds a vector field over $S^2$. The vector field is built from the push forward of the stereographic projection on $N$. Let $p \in S^2 \setminus \{N\}$, and let's ...
0
votes
1answer
79 views

Proving that cuboid of maximum volume in a sphere is a cube.

I was preparing for my maths test . And preparing application of derivative (theory based question ) there I saw a problem of proving rectangle of maximum area in a circle is square . So there were ...
0
votes
1answer
54 views

What is equivalent of $\Gamma(n/2)$?

I should solve $\pi^{n/2} / \Gamma(n/2 + 1) = 1$. Therefore, I need to know other forms of $\Gamma(n/2)$ or $(n/2)!$. I have already checked the Mathematica and MathWorld, very well. But unfortunately,...
0
votes
0answers
19 views

Area of the intersection between a sphere and a cone (located in the center of the sphere)

Please, how do I calculate the area of the intersection between a sphere and a cone, as shown in the image below? The beginning of the cone is located in the center of the sphere, and both geometric ...
0
votes
0answers
32 views

How to calculate arc distance on a sphere

I hope my question makes sense. I just don't know how to describe it using math lingo. Please bear with me. Let's say on a globe I'm traveling from point A to point B that is exactly opposite side of ...
0
votes
0answers
9 views

Closed Form Distance From Great Arc to Point Derivation

I'd like to derive a closed form solution for the arc length between a point along a great arc and an arbitrary point on the sphere. The angles $\theta$, $\Gamma$, and $\alpha$ are all measured from ...
1
vote
1answer
30 views

Symmetry group of a sphere in $\mathbb{R^3}$

It's obvious that the sphere is an absolutely symmetric surface. It remains the same for any reflection over the planes which include its diameter and also for any rotation of arbitrary angle $\theta$...
0
votes
2answers
26 views

What is the angle from (0,0) to (longitude, latitude)?

Starting on the crossing between the equator and the Greenwich meridian (0,0), one can proceed to any point on Earth with a given longitude and latitude along a great circle. Seen from the centre of ...
3
votes
3answers
93 views

Non-vanishing volume form on $S^2$

I have been given the form $\mu=x\,dy\wedge dz+y\,dz\wedge dx+z\,dx\wedge dy\in\Omega^2(S^2)$. I am asked to prove that this form is never vanishing. That is, $\nexists p\in S^2$ such that $\mu_p=0$. ...
3
votes
3answers
75 views

Definition of sphere without using a metric

In differential geometry the $n$-sphere $S^n$ seems to be always defined as the set of points in $\mathbb{R}^{n+1}$ with distance $1$ from the origin. I am interested in a more topological definition, ...
7
votes
4answers
66 views

Proving that $S=\{ x \in (X,\| \cdot \|) : \|x\| =1 \}$ is a closed set.

Exercise : Show that the unit sphere $$S=\{x \in (X,\|\cdot \|) : \|x\| =1\}$$ of a normed space, is a closed set. Attempt : For a set to be closed, its complement must be an open set. Define the ...
0
votes
0answers
17 views

What is the best way to tessellate sphere into equal area in any level of detail? HEALPix or Geodesic Grid or another method?

I want to tessellate sphere into a grid in my 3D world map. There was 2 ways that I was consider right now, HEALPix and Geodesic If I use it specifically for world map that could be zoom into any ...
0
votes
3answers
47 views

Finding a 4th point in 3D space knowing 3 other points and 2 distances to the 4th point from them

I have 3 points in space A, B, and C all with (x,y,z) coordinates, therefore I know the distances between all these points. I wish to find point D(x,y,z) and I know the distances BD and CD, I do NOT ...
2
votes
0answers
79 views

Symmetric powers of odd spheres

Given a sufficiently nice space $X$, say a connected and compact polyhedron, one has a nice formula for the Poincaré polynomial of the orbit space $SP^n X:= X^n/S_n$ of the $n$-fold Cartesian product ...
1
vote
3answers
54 views

Rotating a Circle in $\mathbb R^3$ using $(\theta,\phi,\psi)$ and projecting onto $\mathbb R^2$

Description I am working on creating a wireframe sphere that showcases latitude and longitude lines. A visual representation of what I'm attempting to achieve can be found here. To be clear, I'm ...
0
votes
1answer
24 views

Moving a point on a sphere

I have a sphere and a point on it. Now I want to move this point in the direction of a randomly choosen angle "a" for for the arc length "l", over the surface of the sphere. What are the mathematical ...
3
votes
1answer
126 views

Locus of boundary when shadow is taken

For the first part (i), I could solve by taking images and I got the answer as ellipse, but for the second part I don't know how to take shadow. Can I get exact equation of locus or do I get just ...
0
votes
1answer
40 views

Connectedness of the sphere

There is a theorem in "Lecture notes on elementary Topology and Geometry" by Singer and Thorpe, that states: Theorem: Let $S,T$ be topological spaces and $f:S \rightarrow T$; be continuous and ...
0
votes
1answer
24 views

The region of integration

So I was given this question $$x^2 + y^2 + z^2 ≤ 4$$ where $x≥0$, $y≤0$ and $z≤0$. I was told to sketch the $3D$ plot of the region $A$ and the projection region $R$ of $A$ in the $xy$-plane. How do I ...
2
votes
3answers
89 views

relationship between a great circle arc and a latitude circle arc at a given latitude

My spherical geometry is a very rusty but looking at the figure below: ... my intuition tells me that angles $\phi$ and $\theta$ (measured in radians) are connected with the following equation: $\...
1
vote
1answer
86 views

A property on the unit sphere

I found this lemma in: [Furi, M., Vignoli, A.: On a property of the unit sphere in a linear normed space. Bull. Pol. Acad. Sci. Math. 18, 333–334 (1970)] Lemma: Let $S^{n-1}$ be the unit sphere ...
0
votes
1answer
54 views

Distribution of $x^2+y^2$ vs. $x^2+y^2+z^2$ where $x, y$, and $z$ are each uniform random ~$ (0,1)$

I'm trying to figure out why $x^2+y^2$ is uniform distributed while $x^2+y^2+z^2$ appears to be distributed as $\sqrt(x)$. Both distributions drop off once $x^2+y^2 $ is bigger than 1 or $x^2+y2+z^2$ ...
1
vote
0answers
20 views

Projection of portion of equirectangular image

I am working on a project and a doubt arises when thinking on image projections. I have and equirectangular image created using a cubemap like the following: Equirectangular projection If I draw a ...
2
votes
1answer
54 views

Uniqueness of Lebesgue measure on $S^n$

I am trying to prove that the Lebesgue measure on $S^n$ is the unique countably additive, rotation-invariant measure of total measure 1 defined on Lebesgue-measurable sets. I know the proof of the ...
2
votes
1answer
40 views

How to draw lines and circles on cylindrical projection map?

I am trying to draw a circle with known radius around a coordinate on a cylindrical projection map. Which is a circle around equator and egg shaped closer to the poles. And also trying to draw a line ...
0
votes
1answer
23 views

How do you determine the distance from a projected point and its point of projection on a sphere having a unit radius

Let a sphere with a unit radius lie on the x-y plane, centered (touching) the point zero. Let a single point be somewhere on the surface of the sphere, but we can only see its projected point on the ...