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Questions tagged [spheres]

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

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1answer
19 views

A geometric question about spherical planet and being obscured by the horizon.

Is there a general formula to use to determine the amount that a distant object is obscured by the horizon? Here are a couple of examples: I can view the mountains that are 70 km away. I am at or ...
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0answers
35 views

Trying to calculate the normal derivative of a function on the sphere but getting an 'inverted' representation?

The normal derivative of a function $u(x)$ on the surface of a sphere of radius $r$, i.e. $r = |x|$ can be represented in spherical polar coordinates as $$ \begin{align} \frac{\partial u}{\partial \nu}...
2
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0answers
58 views

Measure of Sphere Surface inside Concentric Ellipsoid (exact or lower bound)

Consider the space $\mathbb{R}^n$, a sphere $\sum_{k=1}^n x_k^2=1$ and an ellipsoid $\sum_{k=1}^n a_k x_k^2 \leq 1$. Note that both sets are origin centered, and the ellipsoid is axis-aligned. Suppose ...
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0answers
55 views

A sphere and tangent planes - based on a geometry problem from Viktor Prasolov Problems in plane and solid geometry

Consider a $S$ sphere, planes $p_1,\dots,p_6$ and points $A,B,C,P,Q$ in Euclidean space. We know that each of the six planes is tangent to the sphere. Planes $p_1,p_2,p_3$ have a common point, $P$, ...
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1answer
55 views

Radius of hypersphere for higher dimensions

So while trying to improve my intuition about higher dimensions I found this video on youtube: Specific Part Visualized for 2D(from the Video): I would formulate the problem as follows: d-cube $[-...
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1answer
38 views

Reasonably-Good Bounding Sphere of an Affine-Transformed Sphere

I have a unit sphere transformed by a given $4\times 4$ affine transform (it is a 3D transform; using homogenous coordinates with $w=1$ allows representing translation). I now need to calculate a ...
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0answers
48 views

Parametrisation of $\mathbb{S}^n$ and smooth atlas.

Two questions regarding the $n$-sphere: construct a parametrization what is the minimum number of charts needs for a smooth manifold of $\mathbb{S}^n$ With regards to the first question: $$\varphi \...
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0answers
20 views

How to measure the value of density near a center point?

If I have M points distributed around a center-point, X, in the n-sphere, how to measure the value of density near the center-point, in the selected area? For example: In the image1 and image2, I ...
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1answer
25 views

Rotating two vector randomly keeping the relative orientation between them unchanged

Let's consider we have two vectors $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$. Now want to rotate these two vectors in $3D$ space (such that the relative orientation between them is always same). How ...
6
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3answers
448 views

Surface optimization for a volume

One of my children received this homework and we are a bit disoriented by the way the questions are asked (more than the calculation actually). This is exactly the wording and layout of the homework: ...
0
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1answer
115 views

How to calculate the volume of intersection of a cone and a hollow sphere

This question is based a little on orbit mechanics, so the sphere has the Earth at the centre (but isn't the Earth) - not relevant but helps me explain it. Given a hollow sphere I can calculate the ...
2
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0answers
39 views

Shortest path that avoids a ball

Given an open ball $B=B(\xi,r) \subset \mathbb{R}^3$, and two points $x,y \notin B$, how long (by Euclidean metric) is the shortest path from $x$ to $y$ that does not intersect $B$? I am in a context ...
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0answers
115 views

Why does no minimal surface in $\mathbb{R}^3$ exist that is diffeomorphic to the $2$-sphere? [duplicate]

I stumpled upon the following question in one of my exercise-sheets: Justify that there are no minimal surfaces in $\mathbb{R}^3$ that are diffeomorphic to the 2-sphere $S^2$ I have no idea ...
2
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1answer
102 views

Surface Area of a $p$-norm Sphere in $\mathbb{R}^n$

There are many various ways to derive the formula for the volume of a $p$-norm ball in $\mathbb{R}^n$ $$V_{n,p,R}=\frac{(2\Gamma(1+\frac{1}{p}))^n}{\Gamma(1+\frac{n}{p})}R^n$$ And I found many ...
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0answers
55 views

Symmetric fibre product and Euler classes

Let $\mathbb{S}^2\to E\to B$ be an $\mathbb{S}^2$-bundle. We may assume that it is the fibre-wise one-point compactification of a complex line bundle, so there is a zero section $s_0:B\to E$ and an $\...
2
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2answers
97 views

why a particular map $f: \mathbb{S}^2 \times \mathbb{S}^1 \to SO(3)$ is not a homeomorphism?

Let $SO(3)$ denote group of rotations of the unit sphere $\mathbb{S}^2$. It is a well known fact that the fundamental groups $$ \pi_1 \Big( SO(3)\Big) \cong \frac{\mathbb{Z}}{2\mathbb{Z}} \not\cong ...
1
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1answer
219 views

Showing the $2$-sphere with antipodal points identified is homeomorphic to the upper hemisphere with antipodal points identified.

Let $S_+$ be the closed upper hemisphere of the $2$-sphere $S^2$. We can define an equivalence relation $\sim_+$ on $S_+$ as follows: $x\sim_+ y\:\:\Leftrightarrow\:\:\begin{cases}x=x^\prime,&\...
2
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0answers
64 views

All the tetrahedra that can be inscribed in the unit sphere up to isometry

I'm interested in restricting the set, $A$, of all tetrahedra whose vertices lie on the unit sphere to a set, $B$, such that every element of $A$ can be rotated/reflected into a unique element of $B$. ...
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0answers
58 views

Find a point of tangency in a plane from a point to a sphere using spherical coordinates

I'm facing a problem very similar to this one, but I can't resolve mine: find ANY point of tangency from a point to a sphere using spherical coordinates Here's the situation: Illustration I know ...
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2answers
244 views

Related rates - Melting snowball

Assume that a snowball melts in such a way that its volume decreases at a rate proportional to its surface area. If half the original snowball has melted away after 2 hours, how much longer will it ...
3
votes
2answers
203 views

Volume of a high dimensional cone

I would like to choose arbitrarily two vectors $a$ and $b$ $\in \mathbb{S}^{n-1}$ and I would like to calculate the probability they are at least some distance $\delta$ apart. This probability should ...
0
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1answer
95 views

Divide sphere surface into proportional parts

I am trying to make an algorithm which will calculate N zones on sphere surface which are related with given sequence (f.e. $4$ zones with $1:3:2:4$ sizes) and returns all $\theta$ and $\phi$ values ...
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1answer
98 views

What is the relationship between the intersectional area of two circles, compared to two spheres? [closed]

For example, if the intersectional area between two circles was a square cm. Is it possible to say what the intersectional volume be if those same circles were spheres as a function of a? To clarify ...
2
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1answer
61 views

Area sphere integral trapezoid elements

I would like to compute the area of the sphere of radius $R$ with the following method: I consider section of the sphere in an orthonormal frame with Cartesian coordinates $(x,y,z)$ delimited by the ...
0
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1answer
492 views

How to calculate the intersect of two spheres in a 3d space?

Right now, I have the equation for working out the two intersects of two circles in 2D space, however when I was studying http://paulbourke.net/geometry/circlesphere/ and reading about the 3D ...
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3answers
483 views

Nonexistence of a homeomorphism between a open set and the unit ball

Let $U\subset\mathbb{R^n}$ be a open set and $\mathbb{S^n}$ the unit sphere of $\mathbb{R^{n+1}}$(i.e. $\mathbb{S^n}=\{x\in\mathbb{R^{n+1}}:||x||=1\}$). How can I show that there's no homeomorphism ...
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2answers
44 views

Finding Sphere radius given radius of intersecting circle and distance from the top of the sphere

How can I find the radius of a sphere if I am given the radius of an intersecting circle- 24cm. The only other unit I am given is the distance from the center of the circle to the top- most point of ...
4
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2answers
91 views

Intuition for spheres in high dimension

In particular, I'm interested in the property that the surface area of a sphere in D dimensions gets concentrated near the equator. I know it can be shown with some integrals, for example done in ...
2
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1answer
113 views

How to calculate dimensions of a pyramid consisting of spheres?

I have found an image that shows my problem pretty well. At the top, you can see a single phere stacked ontop of three other ones. Let's say the radius of all the spheres is a. I would like to ...
8
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1answer
67 views

Glass marble on a plane

There is a glass marble standing still on an absolutely smooth planar surface. Assuming ideal conditions, the marble touches the surface on a single point O (O is a point on the plane). We take away ...
0
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1answer
53 views

Finding a set of points that minimize distance from any point in a sphere

How can I generate a set of $N$ points that minimize $\sum_{x \in S^3} \min_{n \in N} \|x-n\| $, the distance between all points in the unit sphere and the closest point in the set? Intuitively, this ...
0
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1answer
68 views

Integration over Sphere

I have a question to a pretty basic integration problem. I was pretty sure about my solution but my tutor had a different one such that I am confused now. The integral is the following: $$\int_{S_r(...
0
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1answer
107 views

Find volume of region that lies under sphere x^2+y^2+z^2=4, above the plane z=0 using polar coordinates! [closed]

As you can see, I have done all the working from the image attatched. What I really need help with are the limits I am unsure of, which I have displayed in ? marks.
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0answers
100 views

Figuring out tangent and secant length on a circular plane intersection, where did I go wrong?

Here is a picture of my situation for tangent length (pardon the fact that the line in the picture isn't actually tangent visually, just pretend it is) Here is a picture of the secant line I'm ...
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0answers
62 views

Is there any shape with a flat base and an opening at the top that has a smaller surface area to volume ratio than a cylinder?

I am trying to optimize the shape of a food container with an opening at the top (meaning lowest surface area to volume ratio) that can be eaten from but I have no idea what shape can replace a ...
0
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1answer
541 views

Find the equation of a cone tangent to a sphere

The exercise is the following. Determine $(i)$ the equation for the cone $C$ with vertex $V=(1, 1, 1)$, tangent to the sphere $S$ with equation $x^2+y^2+z^2=1$; $(ii)$ a system of equations which ...
0
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1answer
221 views

Pack three largest sphere in a cube with given length.

I'd like to ask about sphere packing problem. The question is: Pack three largest and identical sphere in a cube with a given length 1. Find the diameter of the sphere. And can you kindly also draw ...
1
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1answer
36 views

Sphere reflection property (geometric proof).

I need some help with this exercise I found in chapter 3 of the book "The Geometry of Discrete Groups" by Beardon. Prove (analitically and geometrically) that for all non-zero $x,y \in \mathbb{R} ^n $...
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2answers
237 views

Volume of region inside sphere and cone

Let $R$ consist of the points lying inside of the sphere $$ x^2 + y^2 + z^2 = 3^2 $$ and inside the cone $$ z = \cot(\alpha)\sqrt{x^2 +y^2} $$ where $\alpha$ is $\arccos(\frac15)$ Find the volume of $...
0
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1answer
21 views

Show that $\bar mf=f\bar m$ for any isometry $f$ of $S^2$ [duplicate]

Let $\bar m$ be the isometry sending a point $P$ to its opposite point $-P$, show that $\bar m$ commutes with any isometry of $S^2$ I know I can use the fact that isometries preserve distances, but ...
0
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1answer
112 views

Show that antipodal points remain antipodal under any isometry of $S^2$

"The antipodal map of $S^2$ is the isometry $\bar m$ sending each point $P$ to its opposite point $-P$" (from textbook) Show that antipodal points remain antipodal under any isometry of $S^2$. In ...
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0answers
49 views

Goldberg polyhedron with a hexagon diameter

We say that a goldberg polyhedron $\operatorname{GP}(m,n)$ has a hexagon diameter if, in spherical form, there is a diameter of the sphere that only intersects hexagons, and goes through the center of ...
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2answers
33 views

Find the sum of the volumes

Let $d$ be the distance between the centers of two spheres which are in contact with each other. Let $A$ be the sum of the surface areas of the two spheres. Find the sum of the volumes of the two ...
1
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1answer
187 views

A right circular cone has a base radius of $6cm$ and a height of $8cm$. A cube-shaped box is inside the cone so that one face of the box …

A right circular cone has a base radius of $6$ cm and a height of $8$ cm. A cube-shaped box is inside the cone so that one face of the box is contained in the base of the cone and the four upper ...
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1answer
45 views

Compute the integral $\oint (ax+by+cz)^{2018} \,d\operatorname{vol} (S)$

Compute the integral $$\oint (ax+by+cz)^{2018} \,d\operatorname{vol} (S)$$ where $S = \{x^2 +y^2 +z^2 = 1\}$ (Unit sphere on $\mathbb{R^3}$) and $a,b,c\in \mathbb{R}$. I thought to use Divergence ...
0
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2answers
89 views

Sphere in terms of manifolds

Can we view a sphere as a two-dimensional manifold? Is this the reason why we call a sphere $S^2$?
2
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2answers
59 views

Is there a nice way to calculate $\int (1-x_3)^2$ over the hemisphere?

Let $M=\{ (x_1,x_2,x_3) \in \mathbb{S}^2 \, | \, x_3 \ge 0 \}$ be the closed upper-hemisphere in $\mathbb{R}^3$. Is there a nice way to calculate $\int_M(1-x_3)^2d\sigma$, where $d\sigma$ is the ...
2
votes
1answer
301 views

The hopf fibration is a submersion.

Let $F:\mathbb{S}^{3}\to \mathbb{S}^{2}$, where $F(x,y,u,v)=(2.(xu+yv),2.(xv-yu),,u^{2}+v^{2}-x^{2}-y^{2})$. A submersion has a rank equal to dimension of codomain, then the work is prove that $rank F=...
8
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2answers
140 views

The sphere is not uniformly close to being isometric to Euclidean space

$\newcommand{\SO}[1]{\text{SO}(#1)}$ $\newcommand{\Hom}[1]{\text{Hom}(#1)}$ $\newcommand{\R}{\mathbb{R}}$ Let $U \subseteq \mathbb{S}^n$ be an open subset of the round sphere. There exist an $\...
0
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0answers
130 views

Does $ S^n $ admit a group structure?

I know that some $ S^n $ s have group structures, whereas others do not. For example, $ S^0 = \{\pm 1\} \cong \mathbb{Z}/2\mathbb{Z}$, $ S^1 = \mathbb{R}/\mathbb{Z} $, $ S^3 = \{a+bi+cj+dk:a^2+...