Questions tagged [spheres]

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

Filter by
Sorted by
Tagged with
0
votes
0answers
16 views

Nearest intersection point to the center of multiple spheres

I have multiple spheres with the same radius r, and I need to find the nearest intersection point to the center of all the spheres. I also know that all the spheres intersect together. I'm using ...
2
votes
2answers
30 views

Change in volume of sphere given change in radius

Finding the change in volume $$V=\frac{4}{3}\pi a^3$$ of a sphere when the radius change from $a_{0}$ to $a_{0}+da$ What I tried: Using differential formula $$\frac{\Delta V}{\Delta a}=\frac{d V}{da}...
1
vote
0answers
21 views

Unit $d$-sphere looks locally like $\mathbb R^{d-1}\times\{0\}$ under a suitable diffeomorphism

If $k\le d$, then $M\subseteq\mathbb R^d$ is a $k$-dimensional $C^\alpha$-submanifold iff for all $p\in M$, there is an open neighborhood $U\subseteq\mathbb R^d$ and a $f\in C^1(U,\mathbb R^{d-k}$ ...
2
votes
0answers
17 views

Two integrals in d-dimensional spherical coordinates depending only on the relative angle

I have the following problem: I have two d-dimensional integrals, where the integrand has the following dependence: $$\int\mathrm{d}^{d}x\mathrm{d}^{d}y\,f(\vert\vec{x}\vert,\vert\vec{y}\vert,\vert\...
0
votes
0answers
7 views

Derive expression for surface area $(n-1)$-sphere in hyperbolic space

The answer is $A(r) = n \frac{\pi^{n/2}}{\Gamma(n/2)}\sinh^{n-1}(r)$ as is given in (3.2) of this paper or in eqn III.4.2 of Riemannian Geometry: A Modern Introduction. However, neither source offers ...
-1
votes
0answers
24 views

The plane x+2y+3z=12 cuts the coordinate axes A,B,C Find the equation of circle circumscribing the triangle ABC.

The plane x+2y+3z=12 cuts the coordinate axes A,B,C Find the equation of circle circumscribing the triangle ABC. IN my textbook they are using equation of circle as x^2+y^2+z^2+2ux+2vy+2wz+d=0. but to ...
0
votes
0answers
30 views

Trigonometric functions as a ball

I cannot stop thinking how these trigonometric functions look like the surface of a sphere in 2D... could this mean that where functions are not defined or are as infinite or minus infinite, we can ...
0
votes
2answers
56 views

What is the distance between the centre of the sphere and the vertex of the cone?

A hollow right circular cone rests on a sphere as shown in the figure. The height of the cone is $4$ metres and the radius of the base is $1$ metre. The volume of the sphere is same as that of the ...
0
votes
1answer
17 views

The radius of a sphere is measured as 5 cm ± 0·1 cm. Use differentiation to find the volume of the sphere in the form Vcm^3 ± bcm^3.

I have tried: V = 4/3πr^3 If the radius is 5 ± 0·1, then V = 4/3π(5 ± 0·1) ^3 giving V = 555.65 or V = 492.81 The provided solution is (524 ± 31) cm^3 but I am not sure how you derive this result. ...
0
votes
0answers
6 views

Compute ND Triangulation from Pairwise Distance Matrix

Is there a way to compute a triangulation of points (e.g. triangles that approximate the surface of a 2-sphere) from a pairwise distance matrix in n-dimensions? So far, what I've found is from ...
0
votes
1answer
38 views

Stereographic Projection: Proof that circles from the plane goes to circles in the sphere

I'm trying to prove algebraically that for the stereographic projection $p:S^2\to \mathbb{R}^2\cup\{\infty\}$ $$ p(x,y,z)=\left( \dfrac{x}{1-z},\dfrac{y}{1-z}\right) $$ $$ p^{-1}(a,b)=\left( \dfrac{...
0
votes
0answers
24 views

How to prove that this curvature is constant

I am trying to prove the following statement: let $\alpha: I \longrightarrow \mathbb{R}^3$ such as $|\alpha'(t)|=1$ and $\alpha''(t)\neq0$ for all $t$ in I. Prove that $\alpha(I)$ is a circumference ...
0
votes
0answers
17 views

Volume of 4-ball in Minkowski Space

I had an argument about whether or not the Minkowski space is a hyperbolic space. The argument the other party has is: Is the following true in Minkowski space: $$ V_4 \propto e^r $$ $$ V_4 ≔ \...
1
vote
1answer
44 views

volume of intersection of a plane and a sphere [closed]

given a,b,c,d in R. what's the volume of: $ax+by+cz<d; x^2+y^2+z^2<1$. I've tried using polar coordinates, but got a really hard integral. Is there a way to solve it with integrals?
6
votes
5answers
177 views

How to find the distance along a sphere from an angle?

Imagine there are two points on planet Earth and a light is shone from one to the other by reflecting off an object 500km up (think of this as a mirror oriented parallel to the surface right below it)....
0
votes
0answers
13 views

I am trying to find the intersection of the subtraction of 3 spheres.

Let X be an unknown point in 3D space and let $ A_1, A_2, and\ A_3 $ be a set of known points. $\ X = [x, y, z] $ $\ A_n = [x_n, y_n, z_n]$ Find X from the set of equations where $\phi_n \ and \ \...
3
votes
3answers
44 views

How to create a non-square 2D grid with spherical topology.

When programming Conway's Game of life on my computer. A problem arises; how to deal with the borders on the board? Do the cells at the border have to take into consideration less neighbours than the ...
0
votes
1answer
41 views

Flux of the vector field through the surface of a sphere?

Find the flux of the vector field $F=(x^3, y^3, z^3)$ through the surface of a sphere $x^2+y^2+z^2=x$ Can someone please show me how to calculate this? Thank you in advance.
0
votes
1answer
13 views

Proving postulate about a property fo spherical vectors

Assume we have $X, Y$ constant unit vectors of $\mathbb{R}^3$ I postulate that the maximum of the function: $(V \cdot X) (V \cdot Y)$ I reached by the halfway vector between $X,Y$ i.e at the vector ...
0
votes
0answers
32 views

Integrate over $D^2$, polar coordinates and other problems

Consider in $\mathbb{R}^3$ the unit sphere $$S^2 = \{ (x_1, x_2, x_3) \in \mathbb{R}^3 | x_1^2 + x_2^2 + x_3^2 = 1 \}$$ with inclusion map $i: S^2 \hookrightarrow \mathbb{R}^3$. Let $\alpha := x_1dx_2 ...
0
votes
1answer
36 views

Manipulating the product of the dot product of multiple vectors is producing a paradox

Unless otherwise indicated all defined vectors lie on the surface of the unit sphere (i.e their norm is 1). Let's take the following expression: $(V\cdot Z_1)(V\cdot Z_2) = V^TZ_1V^TZ_2$ Given that ...
-1
votes
2answers
22 views

About points in the circle of sphere surface

here is the image Hey guys! I’m so sorry for the silly question but my math skills are very poor and I just need this problem fixed. I made a simple image about it and I hope it won’t confused you. ...
2
votes
0answers
35 views

Distance between points maximally distributed on n-dimensional unit sphere?

This problem arose in some of my own personal data science research and I am wondering if anyone has encountered this before. Consider $k$ points that lie on an $n$-dimensional unit sphere such that ...
1
vote
1answer
34 views

Hairy ball theorem for $S^2$ [duplicate]

It is well-known that there is "no" nowhere vanishing continuous tangent vector field on $S^2$, by the so-called Hairy-ball theorem. But then, is there a continuous tangent vector field on $S^2$ which ...
0
votes
0answers
8 views

Number of points on a sphere with known average diameter and distance between points

I try to find the best solution of a mathematical problem. I know the diameter of a sphere, lets say 10 +- 1 cm. I would like to estimate the number of points on the surface of this sphere, with a ...
0
votes
1answer
49 views

A geodesic on a unit sphere

Points $A(\cos\alpha,0,\sin\alpha)$ and $B(0,\cos\beta,\sin\beta)$, $(0<\alpha$ and $\beta<\pi/2)$ are on a unit sphere and $l$ is the shortest line (geodesic) between $A$ and $B$ on the sphere. ...
0
votes
1answer
14 views

Angle of horizontal polarization on a sphere

Given the following geometrical problem: linear polarized waves arrive with a certain angle on a sphere. Dependent of the location on the sphere, the angle of the incoming plane wave varies (see ...
0
votes
1answer
107 views

Curse of dimensionality $2^d +1$ hyperspheres inside a hypercube

Consider a $d-$dimensional hypercube $Q$ of side length $l ∈ R, i. e. |x_i − y_i | ≤ l$ for all $x, y ∈ Q$ and all $i ∈ [d]$. Note that $Q$ has $2^d$ corners. We fill $Q$ with ($L_2-)$hyperballs the ...
0
votes
1answer
27 views

Minimizing costs of a specific geometry shape

I have geometry mathematical problem. I have a shape that is made of a cylinder and two half spheres as to top and bottom. How can I minimize the cost of this shape when the volume is known and the ...
0
votes
2answers
20 views

How many parameters does the set of all spheres, which satisfy the given condition, depend on?

How many parameters does the set of all spheres, which satisfy the given condition, depend on? (i) Spheres that pass through the given point. (ii) Spheres that touch the given line (iii) Spheres that ...
0
votes
1answer
52 views

Why is the surface area of a sphere not $ 2 ( \pi r)^2 $?

I was trying to derive the formula for the surface area of a sphere and thought of deriving it this way. If we have a circle with radius $r$ and we rotate it along its center by $180$ degrees, the ...
1
vote
0answers
25 views

How many points on a sphere whose distance is greater than $\epsilon$

I have a "simple" question, but my first researches are not very successful. Given a dimension $d$ and a distance $\epsilon$, is there a standard lower bound on the number of points $N(d,\epsilon)$ ...
0
votes
1answer
37 views

Find longitude intersection of known latitude at z=0 for a globe?

I have a globe image that is centered at lat,lon. At this lat,lon, by definition x=0, y=0, z=1. A line of latitude drawn on this map can either be completely visible (in which case it would be drawn ...
0
votes
0answers
14 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?
0
votes
1answer
15 views

When rotation a sphere $90^{\circ}$ away from you and then $90^{\circ}$ counter-clockwise, what point is either fixed or sent to its antipode?

There's a theorem in Algebraic Topology that says any continuous map $f \colon S^{2n} \rightarrow S^{2n}$ has either fixed points or sends a point to its antipode. Let $f$ be the function described ...
3
votes
2answers
54 views

Homotopy groups of $S^\infty$

I have seen that it is possible to see $S^\infty$ is contractible, which gives trivial homotopy groups $\pi_k(S^\infty)=0$ for all $k\geq1$. Are there different proofs to show the homotopy groups are ...
1
vote
1answer
47 views

Curvature and torsion of coordinate curve on the sphere [closed]

find the curvature and torsion of a $v=v_0$ (= constant) coordinate curve on the sphere $x(u,v)= (a.\cos u.\sin v$, $a.\sin u .\sin v $, $a.\cos v$), $\;0 < u < 2\pi$ , $0 < v < \pi$ I ...
0
votes
0answers
7 views

sphere-circle related problem clarification.

I am trying to figure out an equation of a cylinder and I am facing some doubts. First of all here is the question for clarity: Find the equation of the circular cylinder whose guiding circle is $x^2+...
7
votes
2answers
120 views

Does an analogue of the sphere theorem hold in higher dimensions?

The sphere theorem of Papakyriakopoulos states that if $X$ is a 3-manifold with non-trivial $\pi_2(X)$, then some non-zero element of $\pi_2(X)$ is represented by an embedding $\mathbb S^2 \to M$ (...
0
votes
0answers
38 views

Why does the Gromoll-Meyer Sphere have dimension $7$?

In exercise 10.33 of the book "Matrix Groups for Undergraduates" the Gromoll-Meyer sphere is described by taking the quotient of a smooth left action of $Sp(1) \times Sp(1)$ on the manifold $Sp(2)$ ...
1
vote
1answer
26 views

Two-point equidistant projection of the sphere

According to this Wikipedia article, there is a projection from the $2$-sphere to a region in the plane that preserves distances to two given points. The article says that the projection was first ...
0
votes
1answer
23 views

Calculate $\int_{S^{n-1}} x_1x_2 dS$

How do I calculate $\int_{S^{n-1}} x_1x_2 dS$ (where $S^{n-1}$ is the $n-1$ dimensional sphere in $\mathbb{R}^n$)? At the first part of the question I needed to calculate $I=\int_{S^{n-1}} x_1^2$ and ...
0
votes
0answers
15 views

Can we find a point in the unit sphere that maps to a point who's projection onto the unit sphere is 'close' to the original point?

Given an non-zer0 $n\times{}n$ symmetric matrix $A$ and an $\epsilon$-net $\cal{N}$ of $S^{n-1}$. Is it possible to find a $x'\in\cal{N}$ s.t. $\bigg\|\frac{Ax'}{\|Ax'\|_2}-x'\bigg\|_2\le\epsilon$? I ...
0
votes
3answers
44 views

Finding the dimension of the sphere cube

If you take an $2r\times 2r\times 2r$ cube, and divide it to 27 equal cubes, and then remove all the "axis" cubes (all the cubes which are straight left, straight right, straight up, etc. from the ...
2
votes
3answers
53 views

Volume and surface area of $3/4$ of a sphere

Take for example a 3-D sphere cut horizontally into quarters: How would I identify the volume and surface area of top $3$ horizontal cuts? Would it just be $\frac34\cdot$volume of complete sphere ...
1
vote
0answers
46 views

Haar measure on unit sphere

I am reading a paper where weak solutions to the Euler equations should be found by using the concept of convex integration. Therefore the proofs are very short and I've got some problems ...
1
vote
1answer
38 views

A sphere can be everted, but a spherical vector field can not - why?

As can be seen here a sphere can be smoothly and continuously turned inside-out by a process called "sphere eversion". Let's call this scenario A. On the other hand a 3d-unit-vector-field defined on ...
1
vote
1answer
46 views

A different perspective of sphere eversion?

Usually the process of sphere eversion starts with a sphere whose normals are pointing outward, undergoes some spatial transformation without creasing or pinching (a homotopy), usually using a halfway-...
4
votes
2answers
91 views

Why is the euler characteristic of a sphere 2?

When calculating the Euler Characteristic of any regular polyhedron the value is 2. Since a sphere is homoeomorphic to all regular polyhedrons, the sphere ought to have a Euler Characteristic of 2 as ...
0
votes
1answer
23 views

Figuring out the bounds the triple integral over region inside x^2+y^2+z^2=1 and above the cone z = sqrt(x^2+y^2)

I'm currently taking a Calculus III course and we're learning about triple integrals. So far I am having a lot of trouble with them. I need help figuring out how I should solve the following integral. ...

1
2 3 4 5
21