Questions tagged [spherical-geometry]

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

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How to generate random points on a sphere?

How do I generate $1000$ points $\left(x, y, z\right)$ and make sure they land on a sphere whose center is $\left(0, 0, 0\right)$ and its diameter is $20$ ?. Simply, how do I manipulate a point's coordinates so that the point lies on the sphere's…
Filip
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What is the metric tensor on the n-sphere (hypersphere)?

I am considering the unit sphere (but an extension to one of radius $r$ would be appreciated) centered at the origin. Any coordinate system will do, though the standard angular one (with 1 radial and $n-1$ angular coordinates) would be preferable.…
34
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Are the points moving around a sphere in this manner always equidistant?

I recently encountered this gif: Pretend that there are visible circles constructed along the paths of the smaller black and white "discs", tracing how their individual centers move as they revolve around the center of the whole design. These…
Zxyrra
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Spherical cap area is $\pi r^2$. But why?

(If you're surprised by the title — $r$ is not what you (perhaps) think it is : ) Let $x$ be a point on a sphere $S$ and let $U$ be some sphere with center $x$ that intersects $S$. Claim¹. The spherical cap cut out from $S$ and the circle cut out…
Grigory M
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Deriving the Surface Area of a Spherical Triangle

A triangle on a sphere is composed of points $A$, $B$ and $C$. The $\alpha$, $\beta$ and $\gamma$ denote the angles at the corresponding points of the triangle: The Girard's theorem states that the surface area of any spherical triangle: $$ A = R^2…
ezpresso
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Distance between two points in spherical coordinates

I want to find the distance between two points in spherical coordinates, so I want to express $||x-x'||$ where $x=(r,\theta, \phi)$ and $x' = (r', \theta',\phi')$ by the respective components. Is this possible? I just know that this is…
user66906
17
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1 answer

Step forward, turn left, step forward, turn left ... where do you end up?

Take $1$ step forward, turn $90$ degrees to the left, take $1$ step forward, turn $90$ degrees to the left ... and keep going, alternating a step forward and a $90$-degree turn to the left. Where do you end up walking? It's very easy to see that you…
16
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Great arc distance between two points on a unit sphere

Suppose I have two points on a unit sphere whose spherical coordinates are $(\theta_1, \varphi_1)$ and $(\theta_2, \varphi_2)$. What is the great arc distance between these two points? I found something from Wiki here but it is written in terms of…
JACKY Li
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Is an equilateral triangle the same as an equiangular triangle, in any geometry?

I have heard of both equilateral triangles and equiangular triangles. (For example, this sporcle quiz lists both.) Are these always equivalent, regardless of geometry? I know they are the same in Euclidean geometry (a triangle that is equilateral…
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What's the name of a parabola mapped onto a sphere?

It seems that an 'arc' is a line-segment mapped onto the surface of a sphere (although I don't know if that name still holds if the segment wraps around the sphere more than once, i.e., if the angle subtended by the arc is $>2\pi$). Is there a name…
JCooper
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Relation between area of a triangle on a sphere and plane

We know area of a plane triangle $\Delta=\sqrt{s(s-a)(s-b)(s-c)}$ where $s=\frac{a+b+c}{2}$. I was just thinking: let we have a triangle with arc length $a,b,c$ on a sphere of radius $r$, do we have any similar kind of formula for that spherical…
Marso
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Locus of points such that facing Mecca is the same as facing east

We came to think of this problem: Ali is a good Muslim who happens to travel a lot. On one occasion when Ali is praying, properly oriented towards Mecca, he notices that he is also facing exactly east. Where can Ali be? The geographical…
Jeppe Stig Nielsen
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Mapping The Unit Disc To The Hemisphere?

Question: Can a disc drawn in the Euclidean plane be mapped to the surface of a hemisphere in Euclidean space ? If $U$ is the unit disc drawn in the Euclidean plane is there a map, $\pi$, which sends the points of $U$ to the surface of a…
13
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Law of sines: uniform proof of Euclidean, spherical & hyperbolic cases

There is a unified formulation of law of sines which is true in all 3 constant curvature geometries (Euclidean, spherical, hyperbolic): $$ \frac{l(a)}{\sin\alpha}= \frac{l(b)}{\sin\beta}= \frac{l(c)}{\sin\gamma}, $$ where $l(r)$ is the circumference…
Grigory M
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Are there exotic symplectic structures on $ S^2 $?

Besides the obvoius symplectic structure on $ S^2$ given by the area element in the standard embedding $ S^2 \to \Bbb R^3$, are there any other closed 2-forms on $ S^2$ which produce nonisomorphic symplectic structures on $ S^2$? If yes, is there a…
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