Questions tagged [spherical-geometry]
geometry as on the surface of a sphere, where "lines" are great circles and any pair of lines must intersect
928 questions
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Can you find a great circle with only a compass?
Thinking about the construction of temari balls got me thinking about how one might begin the marking portion of the process, particularly how to build a great circle in the first place. To make the ...
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How do I find the point of latitude on a meridian that is closest to a specified point?
I need to find the latitude of the point on a meridian that has the shortest great-circle distance to a general but specified point elsewhere on the globe.
To find out I might (in theory) do some ...
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Intrinsic formula for calcuating latitude and longitude for a fraction of a great circle.
This is not a homework problem; my two degrees are in engineering. My interest in solving this problem stems from a broader amateur research project. Let $\phi$ be latitude and $\lambda$ longitude. ...
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deriving simplified haversine formula from standard version
On the wikipedia page on haversines:
https://en.wikipedia.org/wiki/Haversine_formula
It gives the following formula for the distances between any two points given their latitudes and longitudes (I ...
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Is there a generalization of the term "spherical lune" for the region of a sphere bounded by arcs of two not-necessarily-great circles?
The definition of a spherical lune is "the shape formed by two great circles and bounded by two great semicircles which meet at their antipodes". However, I haven't been able to find a term ...
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Sphere average of the second symmetric sum of a matrix
Say given $B$ symmetric and tr $B=0$. Let $P_u$ be the projection onto the tangent plane attached to $u$, i.e., $ P_u=Id-u\otimes u$. Let $\sigma_2(M)$ be the second symmetric sum of eigenvalues of $...
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Maximum area coverage of the sphere by 5 caps
Place 5 congruent spherical caps (geodesic balls) on the unit sphere such that no four or more of the caps' centers are coplanar. Must there exist a maximal area configuration of such caps in which ...
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"How do I compute RX, RY, RZ (Euler angles) to aim a tool toward the center of a sphere at each point on the surface?"
I have looked over several other posts and this seems to be a question that has been asked before, but never really answered. Probably, due to lack of information/examples being posted, so I am hoping ...
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What is the average euclidean distance between two points on the surface of a hypersphere of dimension $n$?
Take the surface of the unit hypersphere of dimension $n$:
$$
S = \{x\in\mathbb{R}^n: \lVert x \rVert = 1\}
$$
Given two uniformly sampled points on its surface, what is the expected distance between ...
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Finding an $n$-vector point a certain distance along a great circle arc
I want to find a point a certain distance from a starting point $\vec a$ on the great circle arc $\vec c$, which is defined by two known points $\vec n_1$ and $\vec n_2$. I am using normal vectors on ...
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Is this scaled version of the spherical cosine rule always valid?
In Euclidean geometry, similar triangles satisfy proportional side and angle relationships. This led me to wonder whether a similar kind of inequality or structure exists for spherical triangles, even ...
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Cannot isolate error in my solution to a geometry problem
This is a problem I found in a math competition:
The surface area of a closed rectangular box, which is inscribed in a sphere, is 846 sq cm , and
the sum of the lengths of all its edges is 144 cm . ...
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What is the official name for a "spherical pyramid cap"?
I know the blue shape is a spherical pyramid, but what is the red shape called? It's the spherical pyramid minus the standard pyramid - I couldn't find anything with a quick internet search.
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volume of a spherical pyramid
In Strehlke’s 1842 paper introducing what came to be known as Bretschneider’s formula (seems to have come out the same year as Bretschneider’s work), he includes a spherical version where the points ...
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Why is it necessary for steradians to be measured with square degrees?
I'm doing research on spheres (specifically spherical caps, sectors, and segments) and learned that steradians are measured in square degrees. Looking at the cone whose apex is the vertex of the ...
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Finitely sampling the n-dimensional unit sphere, such that the angular distance is bounded
Consider the unit, $d$-dimensional sphere $U^d = \{\mathbf{x} \in \mathbb{R}^d \mid \|\mathbf{x}\| = 1\}$. I am looking for a finite family of vectors $\mathcal{F} \subset U^d$, such that,
$$
\forall \...
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How to calculate the distance between a spherical coordinate and a longitude line
How to Calculate Great-Circle Distance to a Longitude Line?
In a spherical coordinate system (e.g., Earth's latitude and longitude):
I have a point $(\phi, \lambda)$ (latitude and longitude, ...
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Are the usual spherical coordinates $(\theta,\phi)$ not a "true" local coordinate chart on the 2-sphere?
I've been self-teaching myself differential geometry and still somewhat new. As I understand it, a local coordinate chart on an $n$-dimensional, real, smooth manifold, $M$, is a pair $(M_{(q)},q)$ ...
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Overlap of two circles on a sphere
If we have, in the plane, two circles of radius $R$ which are separated by a distance $d$ then its easy to compute the overlapping area:
$$\frac{A}{R^2} = 2\arccos\left(\frac{d}{2R}\right) - 2\left(\...
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Sufficient and necessary conditions for a solution of a certain system of ode's involving the cross product to be defined on the sphere $\mathbb{S}^2$
Consider the system of equations on $p,v: I\to \mathbb{R}^3$
\begin{align*}
p'&= -a v+b \cos(ct+d) (p\times v),\\
v'&= ap+ b\sin (ct+d) (p\times v),
\end{align*}
for constants $a,b,c,d\in \...
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A system of ordinary differential equations on the sphere involving the cross product
Consider the following o.d.e. system on $p,v:I\to \mathbb{R}^3$ given by
\begin{align*}
p' &= (\cos (at+b))(p\times v)\\
v' &=(\sin (at+b)) (p\times v)
\end{align*}
with orthonormal initial ...
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Identifying points on the hypersphere with points on the trace of a real projective geodesic
Let $\boldsymbol u^{(1)},\dots,\boldsymbol u^{(T)} \in \mathbb{R}^d$ live in the intersection of $\mathbb{S}^{d-1}$ with some $2$-dimensional linear subspace $\mathscr{U}$: $$\{ \boldsymbol u^{(i)} \...
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What is the proper term for the hemisphere of an $n$-sphere?
The question is in the title. My intuition would tell me that it should either be called an $n$-hemisphere, a hemi-$n$-sphere, or a semi $n$-sphere. However, I am primarily curious if there has been ...
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How to prove that in a geometry with no parallel lines, all the lines must be finite in length?
In the Veritasium video on the fifth Euclidean postulate, there was a statement (link with a timecode) that mathematicians (before discovering spherical geometry) managed to prove that in a geometry ...
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Regular spherical quadrilateral tiling for a game board
Are there constructions for tiling a sphere with mostly regular spherical quadrilaterals, but with correction spherical polygons whose number and total area are minimized?
In other words, a corrected ...
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