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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Existence of an isometry such that $f(A)=A'$, $f(B)=B'$ and $f(C)=C'$
Let $A,B,C,A',B',C'\in \mathbb S^2_r$. If we have that:
$d(A,B) = d(A', B')$
$d(A,C) = d(A', C')$
$d(C,B) = d(C', B')$
Does that mean that there exists an isometry $f: \mathbb S^2_r \to \mathbb S^...
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find the point two lines intersect in 3d with a mixture of cartesian and spherical coordinate system knowns
I have line 1:
originating from origin (0,0,0) and magnitude (length) of 7
I have line 2:
originating from a shifted position (0, 0, 0.4) and the polar and azimuth are also known (see picture)
I ...
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Rectangle and ellipse earth coordinats from minor, major axis and orientation
I am trying to calculate area geographic coordinates on earth of different forms from given parameters:
Latitude and Longitude in decimal degree of the center point
Major and Minor axis in feet
Major ...
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Fake spheres and Liebmann’s theorem
In classical differential geometry, Liebmann’s theorem states that a compact and connected surface in $\mathbb{R}^{3}$ with constant Gauss curvature is a (standard) sphere; see e.g. Do Carmo's ...
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Declination angle
Currently I'm studying spherical geometry. I have come across the declination angle $\delta$ of the sun and there is an approximation to calculate $\delta$:
$$\delta \approx -23.45° \cdot \cos\left(\...
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The vertices of a tetrahedron lie on a sphere
I am struggling a bit with the following (elementary) question:
How to prove that every regular tetrahedron admits a circumsphere, i.e. there exist a sphere on which all four vertices lie.
I would ...
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How can I calculate the sun's angle relative to a window's normal?
The following is given:
The sun's azimuth $0 \le a \lt 360$ in degrees, that is the horizontal angle. North would be $a=0$, south would be $a=180$.
The sun's elevation $-90 \le e \le 90$ relative to ...
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Solve smallest circle problem in polar coordinates - that is, smallest cone
I'm working on a problem relevant to astronomy. I'm looking at a few dozen stars at the same time, and I want to identify a centroid direction. Not the average, such that most stars are as close as ...
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Do $4$ balanced points on a sphere form a tetrahedron or lie on a plane?
Let $\mathbb{S}^2$ be the unit sphere in $\mathbb{R}^3$, and let $x_1,x_2,x_3,x_4 \in \mathbb{S}^2$.
Suppose that $\sum_i x_i=0$, where we sum the vectors in $\mathbb{R}^3$.
Question: Does one of the ...
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Divergence Theorem for Centroid
JE Brock found the centroid of a spherical triangle $T=\triangle ABC$ with area $[T]$ to be
$$g=\frac{1}{2[T]}\left(\frac{A\times B}{|A\times B|}c+\frac{B\times C}{|B\times C|}a+\frac{C\times A}{|C\...
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How can I define a metric on a smooth manifold without looking to the space it is embedded in?
Consider the sphere, we denote $\psi^{-1}$ as the inverse coordinate chart $\mathbb{R}^2\rightarrow \mathbb{S}^2$, by:
$$(\theta,\phi)\mapsto(\cos\theta\sin\phi,\sin\theta\sin\phi,\cos\phi)$$
How can ...
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How can I calculate my position, if I have 3 points coordinates and distance? [closed]
How can I calculate my position, if I have 3 points coordinates and distance from every coordinate to my position. All coordinates by longitude and latitude.This is example
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Expressing a 'zero-sum' ratio as a point in space? ( Eg. $1:-9:8$ )
I have a collection of ratios (they are all the same degree) where the sum of their parts equate to $0$; and I need a way to represent these ratios as points in space (to perform k-means clustering on ...
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Calculating Diameter of Metric Space Built from Spherical Polygons
$\DeclareMathOperator{diam}{diam}$
Suppose we have a convex spherical polygon $P$ and suppose that we've figured out the two vertices which are the greatest distance apart, say $u$ and $v$. Consider ...
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Spherical right triangles identities
We have a spherical triangles with angles $\alpha,\beta,\gamma$ that have opposite sides of length $a,b,c$. So our triangle is right angled as we have that $\gamma = \frac{\pi}{2}$. I have to prove ...
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Find the unknown angle
Suppose I have $4$ unit vectors in $3$D and I know all the $^4C_2=6$ angles between them. These angles provide the complete description of this group of vectors. Now, I want to add anther unit vector ...
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Worded spherical triangle problem
We consider a sphere with a radius of 4000 metres. We start at point A and travel on a spherical line segment to point B, turn 60$^{\circ}$ to our left then travel on a spherical line segment to point ...
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Spherical triangle problem: find the normal vector of the third edge
The following could be a problem that has already been answered to, I've looked at similar questions but I can't figure out the answer. Consider the sphere $S^2$ embedded in $\mathbb{R}^3$ whose ...
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On the nature of mosaic specified by Schlafli symbol $\{p,q\}$
I was reading a paper on hyperbolic pascal triangle and the author stated that for Schlafli symbol $\{p,q\}$, if $(p-2)\;(q-2)=4$, it determines the Euclidean mosaic. For $(p-2)\;(q-2) <4$ a sphere ...
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How do I change a 3D equation into a Spherical coordinates
I know how to change 2D Cartesian equations into polar equations, however I'm having some difficulty with a 3D equation and converting that into a Spherical coordinate system.
I am trying to take the ...
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Derivation of area of spherical polygon and pyramid
A certain reference shows the following formulas for spherical polygons and pyramids.
I am interested to know whether these formulas are valid and if there are any derivations on how it occurred. I ...
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Area of intersection between a spherical cap and a spherical segment
I am trying to calculate the area of the intersection between a spherical segment and a spherical cap as a function of the angle of the spherical segment with respect to the sphere's equator. The ...
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No projection of the sphere preserves straightness and area
I have been studying map projections (i.e. a homeomorphic embedding of a neighbourhood of a sphere into a plane or cylinder). Lambert projection preserves area, stereographic and Mercator projection ...
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how would I go about finding the distance between two points on earth
I want to try and find the distance between two points on earth, as much by hand as possible. Could anyone give me ideas on how I would go about doing this, (explanations as deep as possible please!)
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Are the Cartan structure equations invariant under gauge transformations?
(I'm a trying to build a specific gauge connection for a physical theory, but it turns out I need to use a kind of spherical basis for the Lie algebra, and it's confusing me. I'm trained in general ...
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In Spherical Mercator Tile coordinates, how to go one zoom-level deeper but span the same area?
For an arbitrary tile region defined by (z1, x1, y1) in Standard Web Mercator Tile format (Spherical Mercator) , how can we increase ...
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given a set of points, $T$, on the surface of $\mathbb{S}^n$, does a $t \in T$ lie in every possible hemisphere of $\mathbb{S}^n$?
Consider the $n$-dimensional hypersphere, $\mathbb{S}^n$. Given a set of points, $T$, on the surface of $\mathbb{S}^n$, we wish to determine whether at least one $t \in T$ lies in (including on the ...
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Vector Reflection in Spherical Coordinates Proof
Say I have a vector a = (1, $\theta_1, \phi_1$), and another vector b = (1, $\theta_2, \phi_2$). I want to come up with a formula for reflecting a through b in spherical coordinates which in Cartesian ...
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Intersection of circle and geodesic segment on sphere
I am trying to find an efficient way of computing the intersection point(s) of a circle and line segment on a spherical surface.
Say you have a sphere of radius R. On the surface of this sphere are
a ...
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Why are only greater circles considered lines in spherical geometry? [closed]
I was looking for a simple explanation why only greater circles are considered as straight lines in spherical geometry (in the context of invalidating Euclid's fifth postulate) and not any circles of ...
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Is spherical geometry "infinite" in the same sense that a Euclidean plane is?
This seems like a pretty straightforward question (assuming I worded it well), but I've never been able to find an answer anywhere. So, in Euclidean geometry, a plane extends infinitely in all ...
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Geodesic deviation in sphere and hyperbolic plane
Consider $H^2$ to be the hyperbolic $2$-space or radius $1$ (for instance, the upper half plane model) with its hyperbolic metric (coming from the corresponding Riemannian metric). Now, consider two ...
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Lines through a point on sphere intersecting at antipodal points [closed]
Consider a line through a point p on a sphere, now take another line passing through that point. It is found that the second line always intersects the first line in antipodal points. How do I write a ...
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Probability a random spherical triangle has area $> \pi$
From Michigan State University's Herzog contest:
Problem 6, 1981
Three points are taken at random on a unit sphere. What is the probability that the area of the spherical triangle exceeds the area ...
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Spherical Pythagorean Theorem
In class, we came across the following relation for a right triangle on the surface of the sphere. $cos(\frac{c}{R})=cos(\frac{a}{R})cos(\frac{b}{R})$ where R is the radius of the sphere. Here a, b, c ...
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Rotated coordinates on a unit sphere
Given three points on a unit sphere, their coordinates are $p_1 = [\phi_1, \theta_1]$, $p_2 = [\phi_2, \theta_2]$, and $p_3 = [\phi_3, \theta_3]$, where $\phi$ and $\theta$ are azimuthal angle and ...
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Project a point onto a great circle on a hypersphere
Given two points, $A$ and $B$ that define a great circle on a unit hypersphere, how do I find the point $D$ on that great circle that is closest to the point $C$. (Closest in the sense of having the ...
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Random point on hypersphere surface at a uniform distance of another point on the surface
Suppose I have an n-sphere of radius 1 centered in $(0,0,...,0)$, where each point on the surface represents a multinomial distribution:
Given coordinates of a point $S=(x_1, x_2,...,x_n)$ on the ...
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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 ...
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What is the volume of the region that is within a distance r outside of the surface of an n-dimensional hypersphere of radius R?
Suppose you have an n-dimensional ball of radius R living within a d-dimensional space. Imagine the region in d-dimensional space consisting of all points that are a distance r outside the surface of ...
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Time and date from shadow
Given longitude, latitude and the height of a vertical stick at sea level, is it possible to determine date and time from the direction and length of its shadow? If yes, what are the formulas?
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Calculate the area of the hemisphere cut by a plane
I have the following problem. There is a unit hemisphere cut by the plane passing through the diameter.
The angle $\gamma$ is given.
The plane cuts a half of the great circle. I need to find the area ...
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Unit hyperspheres in different spaces
Let $L\in\mathbb{R}^{n\times n}$ be a matrix. So a unit hypersphere in the original space does not become a unit hypersphere in the $L$-transformed space. But is this similar to a unit hypersphere in ...
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Are there positively-curved spaces of infinite extent?
Unbounded flat Euclidean spaces can be either infinite (e.g., an infinite plane) or finite--e.g., a flat torus, constructed by starting with a square and identifying opposite edges.
Meanwhile, the ...
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How do spheres curve in hyperbolic space?
The surface of a sphere embedded in Euclidean space has positive curvature and (eponymous) spherical geometry.
But what if I construct an $n$-sphere (e.g., a 2-sphere), defined as the set of points ...
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Center of circle given by three points on sphere
For given points $A,B,C$ on a sphere, I need to find the centerpoint $MS$ on the sphere of the sperical circle that has $A,B,C$ on its boundary. By that I mean $d_{greatcircle}(MS,A)=d_{greatcircle}(...
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Surface area of unit hypersphere and a cap in a transformed space
Let us imagine a unit hypersphere (for example 3-dim) and transform it with a matrix $L$ in an $L$-transformed space. Now we cut the new body with a hyperplane into two caps. Let us assume the smaler ...
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Equivalence of chordal and spherical metric on Riemann Sphere
The chordal metric between two points $z,w \in \hat{\mathbb{C}} (=\mathbb{C}\cup{\{\infty\}})$ is defined as ,
\begin{align*}
d(z,w)=\displaystyle\frac{2|z-w|}{\sqrt{(1+|z|^2)(1+|w|^2)}}
\end{...
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Distance in between two oriented planes
I am trying to formalize mathematically the current problem I am facing. Current issue is:
Working in the medical field, we receives MR images that have been
acquired along the same direction, when ...
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What is this notation related to spherical geometry?
This is from Indexing the Sphere with the Hierarchical Triangular Mesh, section 4.2, equation 4.4. I don't recognize the notation.
$$ \vec e ( \vartheta ) = \mathbf v _ 1 \cdot \frac { \sin ( \theta - ...
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