geometry as on the surface of a sphere, where "lines" are great circles and any pair of lines must intersect
Questions tagged [spherical-geometry]
889 questions
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Proving a formula for spherical triangles
I'm trying to prove the following formula.
Given a spherical triangle with side lengths $a,b,c$ and interior angles $ \alpha,\beta,\gamma $ prove the following formula
$$2\sin \frac{A}{2} =\frac{\sqrt{\sin(s) \sin(s-a) \sin(s-b)…
Polymorph
- 1,225
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Existence of regular spherical n-gons
I'm having problems with the solution of the following exercise.
Let n $ \geq 3$.
a)What is the interior angle of a regular Euclidean n-gon? Given the side length, what is its area?
b) Show that for any $\alpha_n$ $\lt$ $\beta$ $\lt$ $\pi$ there…
Polymorph
- 1,225
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A simple question about angles on spherical-geometry
In the figure's sphere does:
$\widehat{AB} = \widehat{A'B'}$ or Not?
I mean the angles represented by arcs. (not the lengths).
In a book it said that they're not equal and they are:
$\widehat{AB} = \widehat{A'B'} * \cos{\widehat{AA'}}$.
Is it…
titansarus
- 749
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Does the R^n analog of a sphere, have the minimum surface area for a given volume?
The property is true for $n =3$. But is it also true for $n \gt 3$?
Is the object in $R^n$ with minimal surface area for a given volume, the higher dimensional analog of a sphere?
Lelouch
- 724
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The biggest square on a sphere
I wonder whether presented in wikipedia square (with angles $120^{\circ}$) is the biggest (taking into account lengths of sides) possible square on a sphere?
If not what is the biggest?
If so how it can be proved?
One can assume here definition…
Widawensen
- 8,172
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Spherical trigonometry has no identified invariant
For plane trig we have Law of Sines
$$ \frac{ a}{\sin A} =\frac{ b }{\sin B} =\frac{ c }{\sin C}= 2 R $$
where $2R$ is the diameter of the circum-circle.There is a clear understanding of the relationship of lengths and angles of a plane…
Narasimham
- 40,495
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Distance between points on a face
Given a 2D picture of a face, how is it possible to measure the distance between two different points on the surface of the face?
Thanks
Joel
Joel
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How can I find the center of gravity of a hollow spherical cap?
I am looking to find the center of gravity for a hollow spherical cap. Could I use that point as the point at which the entire mass of the spherical cap is for newtonian gravity problems?
Joe
- 534
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1 answer
Find Double of Distance Between 2 Quaternions
I want to find the geometric equivalent of vector addition and subtraction in 3d for quaternions. In 3d difference between 2 points(a and b) gives the vector from one point to another. (b-a) gives the vector from b to a and when I add this to b I…
Alp
- 113
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Reflections On Sphere Surface / Getting Great Circle from two 3D points
I'm trying to calculate the reflection of a point across another point, both of which are on the surface of a sphere. I believe I could do this by getting the formula for the great circle of the sphere that contains those two points, reflecting the…
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How to test whether spherical caps intersect?
I have a unit sphere, on the surface of which are defined spherical caps. I typically characterize the caps by the unit vector $n$ from the center of the sphere to the top of the cap, and the angle $\theta$.
My question is: given a pair of spherical…
F'x
- 1,450
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2 answers
Non congruent (spherical) quadrilateral with same angles
I want to construct two quadrilaterals on the unit sphere with same interior angles $\alpha_1,…,\alpha_4$ and the same perimeter, but which are not congruent to each other. Is that possible? How can one construct a counterexample or how can one…
Braten
- 1,955
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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, just like on a plane. What is the easiest way to…
Alexey
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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 the unit sphere of equal length $d$, then one of…
Cameron Williams
- 29,432
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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 axis
I would really appreciate some input…
Shane
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