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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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Relationship between a circle inscribed in a square and a sphere inscribed in a cylinder

The ratio of the area of a circle to the area of the square it is inscribed in is equal to ${\pi\over 4}$ and the ratio of the volume of a sphere to the volume of the cylinder it is inscribed in is $2\...
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1answer
23 views

Covering the 2-sphere with 6 hemispheres

While reading chapter 2 of Wald's General Relativity titled "Manifolds", I stumbled upon the fact that the 2-sphere $S^{2}$ cannot be mapped into $\mathbb{R}^{2}$ in a continuous 1-1 manner. Wald then ...
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1answer
25 views

Distance between a point and low-dimensional sphere

Is there a way to analytically calculate the distance between an arbitrary point $\mathbf{x}\in\mathbb{R}^n$ and a low-dimensional sphere embedded in $\mathbb{R}^n$, say one aligned with the axis ? ...
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Sphere immersions in 4-manifolds

I'm having trouble understanding the homomorphisms $\pi_1g(S^2)\longrightarrow \pi_1 M^4$ where $g: S^2\longrightarrow M^4$ can be a framed or unframed immersion. Any reference on this?
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2answers
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triple integral spherical coordinate

I have a problem converting this question into a spherical form. $∫∫∫ z/√(x^2+y^2+z^2)dxdydz$ where R is the interior of a sphere $x^2+y^2+z^2 = 2z$ the limits of integration I found are: 0≤r≤2cosθ ...
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1answer
35 views

Triple Integral in spherical coordinate

$\displaystyle\iiint_R (x^2+y^2+z^2)^{-2}\,dx\,dy\,dz$ where $R$ is in the region in the first octant outside the sphere $x^2+y^2+z^2 = 1$; Hi guys, I don't quite get which region is this, is it that ...
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1answer
16 views

Find sphere intersection point

I have a sphere which is originated at: $$ \begin{matrix} 0 & 0 & 0 \end{matrix} $$ Its radius r is 150. I have a line which goes from: $$ \begin{matrix} 0 & 0 & ...
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4 Spheres all touching each other??

If there are 4 spheres all touching each other and 3 of them have diameters 4, 6 and 12 what is the diameter of the fourth one? I imagine it like 3 balls on a flat table touching each other and then ...
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1answer
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How to calculate the geometric center of the surface-area of a part of a Sphere?

I have a regular sphere, $V=\pi r^3\frac4 3$ and $A=4\pi r^2$. Now I want to seperate it into four slices, with equal amounts of surface area - not counting the sliced area. Or in other words, I want ...
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43 views

Surface area of a sphere segment. [duplicate]

I got this problem and solution in a paper but cannot find how they have solved it. Consider I have a sphere that is equally divided into two different patch(P, S). The sphere can rotate and translate(...
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1answer
39 views

Area of touching part of Sphere to the wall.

I believe that it has a very simple explanation but one question stuck in my mind. What is the area between sphere and wall when it touches to it. If it is zero, why it is not occurring in real life?...
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Calculate circumsphere of truncated icosahedron with algebraically profiled faces

I have a truncated icosahedron with theoretical side lengths of $a$ units (if it were a perfect, flat shape). However, each face is deformed, as when viewed from the side instead of being a flat line ...
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0answers
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Is possible an eversion of a sphere $S^3$ in space $R^4$? [duplicate]

I read and find only a sphere $S^2$ eversion in space $R^3$, but I want to know the process to have a sphere $S^3$ eversion in space $R^4$ Over 50 years ago Smale proved that a sphere ($S^2$ in ...
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2answers
26 views

Check if 3D point is inside sphere

I know there is a pretty simply way to check if a 2D point is inside a circle, I wanted to know how to do there same but in the 3rd dimension. The variables are the point's X, Y, Z and the sphere's 3D ...
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1answer
39 views

Calculating the annulus of a sphere with a differential change in theta

Consider the following object: I want to calculate the area of the annulus. The annulus is within the region of $$ \theta $$ and $$ \theta + d\theta $$ The answer of the area of the annulus is ...
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1answer
44 views

How far is it from the equator to the north pole if you keep going northeast? [closed]

You may simplify Earth as a sphere with 40 Mm circumference. I thought about it when I read the Mercator projection has straight lines for constant bearing courses. You'd spiral around the north pole ...
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1answer
23 views

Calculate the radius of a circle or sphere given a section?

This is probably a basic Math101 problem for most of you, but I'm not a mathematician so I could use some help with it. In the diagram below, how would I solve for R (the radius of the circle or ...
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2answers
35 views

Bijection between spherical and planar triangle surfaces

I subdivide a unit sphere, centered at origin, onto 20 spherical triangles. For the sake of argument let's take one such triangle $Ts$, in $\mathbb{R}^3$, that has vertices $Normalize(-1,0,g), ...
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1answer
60 views

What is $d$**k**?

Earlier today, I saw $d$k $f(k)= $ ... in what I believe was a physics integration problem involving spheres. I cannot find the original problem. My question is what $d$k means; it is not part of an ...
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1answer
37 views

Integer grid points intersection with sphere

I'm wondering how many intersections does a centered sphere with radius $r$ ($r$ is an integer) have with an integer grid? For sure the 6 intersections with the axises, e.g. $(x,y,z)=(r,0,0)$. ...
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3answers
36 views

Tangent Line From a Point on a Sphere and $y$-axis

Let's say I have a sphere, $$100 = x^2+y^2 +z^2 $$ This indicates that the center of our sphere is at $$(0, 0, 0)$$ and we have a radius of $$radius = 10$$ I'm under the assumption that $$P = (1, ...
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1answer
65 views

Volume and surface area of sphere, cone, cylinder etc

Why isn't the volume of a sphere: $\pi$$^\text{2}$$r^\text{3}$, instead it is $\frac{4}{3}$$\pi$$r^\text{3}$? Like wise the surface area is 4$\pi$$r^\text{2}$and not 2$\pi$$^\text{2}$$r^\text{2}$. ...
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1answer
24 views

Finding weight in Megagrams (Mg) given circumference and density

Glaciers often deposit large rocks called erratics. The granite rock has a circumference of 9.5 m. Assuming it conforms to the shape of a sphere, what would be its weight in Megagrams (Mg), where 1 Mg ...
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Spheres and Squares

We usually study the geography of Earth using globe(spherical),maps(rectangular) now the problem is as follows .... We aim to find out the shape and area of a square of side R/4 when it is projected ...
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47 views

Need help to solve a question of Mensuration

The radius of base and slant height of a conical vessel is $3$ cm and $6$ cm respectively. Find the volume of sufficient water in the vessel such that when a sphere of radius $1$ cm is placed into it, ...
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1answer
20 views

spherical polar geometry change in elevation angle

how to calculate change in elevation angle if you know coordinates of two point on surface of sphere. let us say assume that a point move on the surface of sphere from [x1 y1 z1 ] = [0.1 0.1 0.9899] ...
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1answer
67 views

How can i see a CW-Structure of $S^n$ compatible with the antipodal map?

I dont understand this notion of compatibility. I saw two possibles CW-structures to $S^n$. Please, anyone can explain me this notion?
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1answer
119 views

Maximizing $x^4+y^4+z^4 + p(xy+xz+yz)$ on a sphere

What values of $x,y,z$ maximize $f(x,y,z,p) = x^4+y^4+z^4 + p(xy+xz+yz)$ with constant $p \geq 0$ with the constraint $x^2+y^2+z^2=1$? Some preliminary studies in Mathematica showed that the behavior ...
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1answer
36 views

$N$ points as far apart as possible in a sphere volumetrically?

Given a radius $R$ and points $N$, I want to distribute points in a sphere volumetrically so that they are as far apart as possible. I know that for $N = 1$, I can place it anywhere. For $N = 2$, ...
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1answer
22 views

Sphere inscribed in a cone

If a cone of height h and radius r has a sphere inscribed in it such that it touches the base and the curved surface area, how can I find the radius of the sphere? (Is this in the level of a 9 grader?)...
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1answer
98 views

Newtons law of cooling applied to spherical region

I'm having trouble solving the following problem: Formulate a mathmatical model for a stationary (steady) temperature distribution inside the spherical volume $$ R^2\leq x^2+y^2+z^2\leq (2R)^2, $$ ...
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Formalization of an Intuitive Proof of the Surface Area of a Sphere

Consider the following proof that the surface area of a sphere is $4 \pi r^2$. First let's try to squash an arbitrary sphere into its net. If we cut the sphere in half then open it up, we get two ...
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1answer
45 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 $2x^2 + z^2 = 1$ ?
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2answers
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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 \...
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1answer
52 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 ...
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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) ...
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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 ...
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1answer
38 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 ...
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0answers
43 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(...
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3answers
54 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)...
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0answers
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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 ...
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0answers
32 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+...
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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 $V$?
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1answer
47 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 : $V=\frac{4\pi R^3}{3}\frac{\Omega}{4\pi}=\frac{R^3 \Omega}{3}$ ?
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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, ...
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1answer
33 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 ...
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1answer
59 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 ...
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3answers
25 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 ...
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23 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 ...
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2answers
101 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 ...