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

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Intersection of perpendicular bisectors of a spherical triangle

I have 3 points on a unit sphere identified by their XYZ coordinates. They form a spherical triangle. If I'm not mistaken, perpendicular bisectors of a spherical triangle intersect in a single point, ...
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59 views

Implicit partial derivative of a spherical cap

Consider a spherical cap, for which the base radius is $a$ and the height is $h$. Then, the surface area and volume is (these equations can be found on Wolfram Mathworld) $A(a,h) = \pi(a^2 +h^2)$, ...
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49 views

How to reason about two points on the unit sphere.

I've recently been thinking about various problems involving two points on the surface of a unit sphere. Let's specify them with a pair of unit 3-vectors ${\bf \hat a}$ an ${\bf \hat b}$. Is there ...
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66 views

How to find Latitudes and Longitudes of projections of the vertices of a rectangular plane below earth's surface?

I want to find out the latitudes and longitudes of projections of the vertices of a rectangular plane inside the earth's surface. I know dimensions of rectangle, angles of orientation and latitude and ...
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64 views

Ratio of the Volume of n-spherical cap to the volume of n-sphere

Assume an n-dimensional sphere with radius $R$ and volume $V^{(n)}_s$. Also assume a corresponding n-spherical cap with height $h$ and volume $V^{(n)}_c$. what is the ratio of two volumes? ...
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85 views

Integrating a discrete 3D surface, in spherical coordinates

I have an matrix which contains height information for a sheet suspended in air. Like a checkerboard, each value in the matrix represents a sampled height. Here's the hard parts: the data in the ...
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43 views

Find the angles defining an hyperspherical cap

For the hyperspherical cap of dimension $n+1$ find all the angle $\phi_1, \phi_2, \ldots, \phi_n$ which defines the cap? I mean, I know a cap is usually define by its height $h$ and its base $a$. ...
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79 views

Is it true that the formulas of the volume and surface area of an n-dimensional sphere are best expressed in terms of $\eta = \frac{\pi}{2}$?

Someone told me that the formulas of the volume and surface area of an n-dimensional sphere get simplified a lot if we express them in terms of $\eta = \frac{\pi}{2}$ instead of $\pi$. . In terms of ...
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32 views

Convert Degrees of Latitude to Feet

I need to check this formula I have to compute the distance of a point with latitude $lat$ from the equator: $$ \mathrm{feet} = \mathrm{lat} * 10000 \times 3280 / 90 $$ Example: A point at ...
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15 views

How to show difference amount from scattering.

In our discussion, we need to show that the amount of photons at one wavelength is less than photons at a nearby wavelength. Photons (with a nearby wavelength) are created on the sun's photosphere ...
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53 views

Zero-distortion map projection

Is it possible to take a limit of map projections (from a sphere to a plane) with ever-smaller distortion factors to get some kind of dendritic limit projection that has zero distortion everywhere? My ...
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2answers
199 views

Curve on a basketball

The sewing pattern on a basketball is composed of two great circles and a single curve that intersects each great circle twice. Does this curve have a name? Are there any parametric descriptions of ...
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99 views

A basic question on surface area of spherical cap of sphere

Consider a sphere of radius 1. Now chop a spherical cap with latitude line $\phi$ at the bottom of the cap is removed from top (say $0<\phi<\frac{\pi}{2}$). I want to know the surface area of ...
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1answer
60 views

Do spherical triangles with the same base and altitude have the same height?

If two spherical triangles have the same base $\theta$ and the same altitude $\phi$, do they have the same area. Initially I believed they would have by the same logic flat triangles do. However I'm ...
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74 views

How many unit vectors can tile an n-sphere with a given angle?

Given a unit radius $n$-sphere, and a constant $c = cos(\theta)$, $0 \le \theta \le \pi$, what is the size of the largest possible set of unit vectors $U = \{u_1, u_2, ..., u_n\}$ such that $u_i \cdot ...
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36 views

Antipodal map and parallel transport on $S^3$

I'm reading Thurston and Levy's "Three-dimensional geometry and topology" where they have an informal explanation of what life in $S^3$ would look like. For the following detail that I am concerned ...
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53 views

Intersecting arcs on a sphere

I'm working through this paper and I'm hung up on Proposition $3.1$. To strip away the context of the problem and present it in another light: suppose there are two intersecting arcs $ab$ and $cd$ on ...
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Equation for calculating azimuth between two points

Does anybody know an equation or approximation for calculating the azimuth as a function of latitudes and longitudes of both the points. For example I have Princeton, NJ is at 40.3571° N, 74.6702° ...
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87 views

Sphere Calculations: Determining the remaining surface area of the sphere when the sphere is cut by a vertical and horizontal plane

Sphere Calculations: Determining the remaining surface area of the sphere when the sphere is cut by a vertical and horizontal plane and the centre of the sphere is not on the vertical and horizontal ...
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229 views

how to find distance in longitude and latitute when center and radius is given? [closed]

How to find distance in longitude and latitute when center and radius is given? for example: ...
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2answers
164 views

Number of reflection symmetries of a basketball

Excerpt from John Horton Conway, The Symmetries of Things, pg. 12. Basketballs have two planes of reflective symmetry, as do tennis balls. I read this sentence and it immediately struck me as ...
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1answer
144 views

c.s.a of a hemisphere

Here is a question which I would like to understand. I want to know How to prove that c.s.a of a hemisphere is $2\pi r^2$ ? I'm a 10th CLASS average student,so please keep it simple. Thank you.... ...
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Spherical coordinate transformation

We have the following picture ($r=1$. $AB$ is the prime meridian): We can find the coordinates of $C$ using: $$ x = \sin(b)\cos(a)$$ $$ y=\sin(b)\sin(a)$$ $$ z = \cos(b) $$ I understand this ...
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40 views

creating a mono-monostatic body from a basketball and acrylic tube

I'm looking for a formula to calculate if it is possible to create a mono-monostatic body out of a miniature rubber basketball and an acrylic tube of a variable length. I have PUR casting resin that I ...
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1answer
54 views

Trouble with understanding a spherical coordinate system.

We have a sphere with $r=1$, and we want the coordinates of $C$. $A$ is the north pole, and $AB$ is our prime meridian. See picture: I'm familiar with an $(x,y,z)$ coordinate system, but not so ...
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How I cut my orange - spherical volume integral

I cut my orange in six eatable pieces, following some rules. My orange is a perfect sphere, and there is a cylindrical volume down through my orange, that is not eatable. In the diagram, the orange ...
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1answer
38 views

What are the umbilic hypersurfaces in a sphere?

It is a well-known result that all umbilic hypersurfaces (complete and connected, say) of $\mathbb{R}^n$ are spheres or planes. But what can we say about umbilic hypersurfaces of a constant curvature ...
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53 views

Intersection of Two Parabolas on a Sphere

I'm trying to implement the algorithm in this paper which describes an implementation of Fortune's algorithm on a sphere, and I'm getting hung up on the math explaining how to calculate the ...
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2answers
36 views

Implicit Representation of Surfaces - Basic Quesion

I am reading on implicit representations of surfaces and cant quite come around the following example. Take $F : \mathbb{R^3} \rightarrow \mathbb{R}$, where $F(x,y,z)=x^2+y^2+z^2$. Now we want to ...
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1answer
39 views

Curving points to a sphere

I think math.stackexchange is the right place to post this, but if not, feel free to tell me. I have a series of points to be plotted on a sphere (Each one has a latitude and longitude value). These ...
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1answer
42 views

Closed Operations in Spherical Space

Euclidean vector arithmetic (addition/subtraction) is not closed under a spherical space. For example: $\mathbb{S}^2=\{v \in{\mathbb{R}^3}|\|v\|=1\}$ We have $(1, 0, 0)\in\mathbb{S}^2$ and $(0, ...
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Computing distance from line to point in geodetic environment

Supposing to be in a cartesian plan and that I have the following point: $$A(x_{1},y_{1}), B(x_{2},y_{2}), C(x_{3},y_{3}), D(x_{4},y_{4})$$ $$P(x_{0},y_{0})$$ Now immagine two lines, the fist one ...
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How to construct three mutually orthogonal circles in stereographic projection?

I'm new to spherical geometry and I enjoy doing ruler-and-compass constructions, so I'm trying to teach myself to do them in stereographic projection. I'm finding it challenging, to put it mildly. ...
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Different ways for calculating distance between two geodetic points give me different results

I'm trying to calculate the distance between two geodetic points in two different ways. The points are: A:(41.466138, 15.547839) B:(41.467216, 15.547025) The ...
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92 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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31 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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22 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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42 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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77 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 ...
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1answer
94 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
60 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
114 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
128 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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1answer
226 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 ...
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1answer
60 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
67 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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38 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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126 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
66 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
1k 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 $(x,y,z)$ ...