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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40 views

How to prove spherical harmonic addition theorem

I have been trying to prove that $$ P_\ell(\cos\gamma)=\frac{4\pi}{2\ell+1}\sum_{m=-\ell}^\ell Y_{\ell m}^{*}(\theta', \phi')Y_{\ell m}(\theta, \phi) $$ for $\cos\gamma=\cos\theta\cos\theta'+\sin\...
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How to check if other points are on the geodesic between 2 points on a sphere?

Given any 2 points $p_1, p_2 \neq p_1$ on a sphere, how can we check if $p_3$ is on the geodesic from $p_1$ to $p_2$? I think of checking the $||\text{arc } p_1 p_3|| + ||\text{arc } p_3 p_2|| = ||\...
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What happens with the convex hull of $6$ random points on a sphere?

Given a collection of points on the sphere, we can consider their spherical convex hull: add all points on the shortest path between two points in the set, repeat until the resulting set does not ...
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24 views

Checking that a point is in a spherical polygon

I am trying to find an algorithm to check if a point $p$ on the surface of a sphere lies within a spherical polygon with vertices at $v_1, v_2, \dots, v_J$. Any polygons I am considering will be "...
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2answers
36 views

How to generate a random vector, guaranteed to be within the hemisphere with respect to another vector?

Given a normalized vector N, how can one generate a random direction vector that is guaranteed to be in the hemisphere with respect to ...
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31 views

Are there known “native” sphere filling curves? [closed]

I am researching some applications of space filling curves in geospatial computation. I'm familiar with the concept of space-filling curves as they apply to Euclidean spaces. Currently major ...
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1answer
55 views

Distance between a point inside a circle and circumference on a line along a point outside of circle

Point $A$ is located outside of a circle centered at $C$ and with radius $r$. Point $B$ is given point inside the circle. How to calculate $d$, the length of line segment between $B$ and circumference ...
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Equation of a Sphere: question on squaring consecutive sphere radius

The equation for a sphere produces the following graphs and I have questions on them. I consecutively squared the radius of a sphere and graphed at an offset of the radius. As shown below, the ...
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1answer
41 views

Deriving the spherical law of sines using Clairaut's relation (without the law of cosines)

I'm studying geodesics in my differential geometry course. We derived and proved Clairaut's relation (see also Shifrin P73.), which states that for a surface of revolution, the geodesics satisfy the ...
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2D orthographic projection of a spherical lune (geometry of a crescent moon's terminator)

I am interested in drawing a crescent moon in a vector drawing program. Our moon is a sphere illuminated by the sun in a certain direction, and viewed from Earth in a rotated direction. Our view of ...
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2answers
106 views

The distribution of areas of a random triangle on the sphere - what are the second, third, etc. moments?

Suppose that we choose three points independently and uniformly at random on the surface of a unit sphere as the vertices of a triangle, and consider the area of this triangle. Call this random ...
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55 views

Projection of a Great Circle on another

Consider a great circle between $[lat_1, lon_1]$ and $[ lat_2, lon_2]$, on a perfectly spherical earth. Consider a second one : between $[lat_1, lon_1 +b]$ and $[ lat_2, lon_2+b]$. For a very small ...
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2answers
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An area of a convex polyhedral surface homeomorphic to a sphere vs an area of its shadow

Consider a convex polyhedral surface $P$ in $\mathbb{R}^3$. If $\pi_v : \mathbb{R}^3\rightarrow \{ x\in \mathbb{R}^3| x\perp v\}$ where $v\in \mathbb{S}^2$ is an orthogonal projection, then $$ \frac{...
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1answer
124 views

How can we calculate $\int_{r\in \mathbb{S}^2}\ {\rm Area}\ \Delta p_0q_0r\ d{\rm Area}_r$

Define a function $f : \mathbb{S}^2\times \mathbb{S}^2\times \mathbb{S}^2\rightarrow \mathbb{R}$ by $f(p,q,r)$ to be a area of the geodesic triangle $pqr$ in the unit sphere $\mathbb{S}^2$. (Here the ...
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1answer
33 views

Simple question about Power/Voronoi diagrams on the sphere

I am reading "Laguerre Voronoi Diagram on the Sphere" by Kokichi Sugihara (I have provided a link to the paper at the bottom of this post). In this paper the author defines a notion of ...
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Best fitting geodesic on a sphere

Given a list of points (latitudes and longitudes) on a sphere, I want to find the geodesic which minimizes the sum of squared deviations from the geodesic to the points. Is there a nice and efficient ...
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1answer
30 views

Surface Area of Cap below, Top Part of Sphere

How do I find the Surface Area of an oval cap below (Top of sphere)? Oval Cape Internet is stating: 2πRh However, a simple 8 radius circle is ...
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Calculate the rotation angle for a point on a small circle (tilted from XY Plane) to a known point on XY Plane

There are two great circles which are unit circles: Great Circle 1 (GC1): On the X-Y Plane Great Circle 2 (GC2): Tilted 45 degrees about the Y axis Here are some references and constraints: 3D ...
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1answer
43 views

How to calculate the volume of a rectangular section of a sphere?

I may not be using the correct terminology; please correct me if so. There is an existing question How to calculate the area covered by any spherical rectangle? which is similar to mine, but I am ...
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How do I find a plane that is tangent to two given spheres and passes through a given point?

My problem is the following: Given two spheres: $$(x-6)^2+(y-1)^2+(z+1)^2=1$$ and $$x^2+(y-5)^2+(z-4)^2=9$$ find a plane that is tangent to both of those spheres and passes through the point $$(5;2;0)$...
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1answer
49 views

Find convex hull or simplex on hypersphere that enclose all vectors

Assume that we have x1,x2,...xn vectors in d-dimensional (d>=64) hypersphere, How can we find a convex hull or simplex for all of them that encloses all the vectors? can we describe that shape with ...
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What are “vectors on the unit-sphere”?

I currently read and try to understand this paper and stumbled upon the following sentence: Given a distribution of vectors $x$ on the $d$-dimensional unit sphere, we consider a set... and later: ...
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Free action of a finite subgroup of $SO(4)$ on $\mathbb{S}^3$

Let $\Gamma$ be a finite subgroup of $SO(4)$ acting freely on $\mathbb{S}^3$. Just to check: is it true that the covering $\pi : \mathbb{S}^3 \to \mathbb{S}^3/\Gamma$ is regular? I think so. This ...
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1answer
159 views

Expected projected length of radial vectors of n-sphere

Situation In $n$-dimensional Euclidean space rests a unit $(n-1)$-dimensional sphere that is orthographically projected onto a $(n-1)$-dimensional plane. The topological definition of a sphere is used,...
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1answer
49 views

Show that isometries of $\mathbb{S^2}$ are determined by the image of three points $A,B,C$ not in a “line”.

Given information: If points $P,Q \in \mathbb{S^2}$ have the same distace from three points $A,B,C \in \mathbb{S^2}$ not in a "line", then $P=Q$. Deduce from the given information that an ...
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1answer
43 views

Showing that if $P,Q \in \mathbb{S^2}$ are equidistant from $A,B,C \in\mathbb{S^2}$, then $P=Q$

We are given that the equidistant set of two points on $\mathbb{S^2}$ is a "line" (great circle) on $\mathbb{S^2}$. Using this fact I must show that: If $P,Q \in \mathbb{S^2}$ are ...
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self-polar spherical triangles

I want to show that in $S^2$ all self-polar triangles are congruent. I know that if a triangle has angles $A,B,C$ and opposite sides $a,b,c$, then for its polar triangle we have : $A'=π-a$, $Β'=π-b$, $...
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Change of variable of surfaces integrals

Let $\mathbb{S}^{2}$ be the unit sphere, $\Delta_{s}$ the Laplace-Beltrami operator, $U_h$ the finite element space in $\mathbb{S}^{2}$ and $V_{h}$ the space of constant function associated to $U_{h}$....
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1answer
31 views

Spherical Geometry Distance Between 2 Points

I try to calculate distance between 2 points on Earth. I have an a car and this car goes with speed 100 km/h and I know the start point latirude and longitude and car goes 5 hours. I want to calculate ...
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1answer
63 views

Is every geodesic-preserving map of the sphere an isometry?

Let $\mathbb{S}^n$ be the $n$-dimensional unit sphere, equipped with the standard round Riemannian metric. Let $f:\mathbb{S}^n \to \mathbb{S}^n$ be a diffeomorphism and suppose that for every (...
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Time derivatives of great-circle or euclidean distance?

I have a function $f: \mathbb{R} \to S^2$ working on the unit sphere so that $f(t) = (x(t), y(t), z(t))$ (but $x(t), y(t), z(t)$ may be difficult to find). If I want to find/approximate the time ...
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Longitude, Latitude Question. Converting distance in miles to longitude and latitude

Let's say I have some points on a map. Point one is at a longitude of -48.6 and a latitude of 38.8. Point 2 on the map is at a longitude of 52.1 and a latitude of 30.3. A line between these two points ...
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1answer
50 views

If all lines on a sphere converge, what happens to latitude lines?

I mean, on an image like this, how do latitude lines not converge? Everywhere i read says that there can be no parallel lines on a sphere. Is this related to the different definition of 'line' in ...
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Finding the minimum time for one object to intercept another object on an unit sphere

I am trying to find the minimum time, $t$, for one object A, at point $A$, to intercept another object C, at point $C$ on a unit sphere. The point of interception is $B$. A has a constant velocity of ...
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1answer
58 views

Bijective projection from a unit disk to a unit sphere

Is there any proper method that makes conformal one-to-one mapping (preserving the angles) from a unit disk to a unit sphere? I know with simple stereographic projection you can project a unit disk to ...
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1answer
32 views

How many points do I need on a sphere to force the minimum distance to be small?

Let $\mathbb{S}^2$ denote the unit sphere in $\mathbb{R}^3$ with its standard metric, and let $S\subset\mathbb{S}^2$ be some finite set of points. Given $\varepsilon>0$, how big does $S$ need to be ...
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1answer
166 views

How to calculate the arc lengths of a great circle inclined with the equator at $\phi°$ broken into $12$ arcs by longitudes $30°$ apart?

A great circle lies at $\phi°$ inclination to the equator. Longitudes $30°$ apart are drawn which divides the equator in $12$ equal arcs of size (radius of earth$*30$). The corresponding arcs on the ...
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1answer
24 views

Finding intersections of a spherical spiral with a geodesic segment

Long story short: Can we analytically solve for $φ$ in this equation? $\sin(\varphi) \cdot (A \cdot \cos(k \cdot \varphi) + B \cdot \sin(k \cdot \varphi)) + C \cdot \cos(\varphi)=0$ Given a point on ...
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1answer
156 views

Parallel lines in the geometry of a sphere

In Euclidean geometry, there is exactly one line through a point that is perpendicular to a line that does not go through the point. In spherical geometry, how do perpendicular lines exist? I can see ...
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1answer
48 views

Converting cartesian coordinates into latitude and longitude coordinates

I'm trying to convert a position vector on a unit sphere into the latitude and longitude coordinates but I'm not sure how to do it. I know that the formula for converting the latitude and longitude ...
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1answer
73 views

Angle between two position vectors on a sphere [closed]

I'm trying to find a general formula for the dot product of two position vectors of two points on a unit sphere given their latitude and longitude coordinates but I'm not sure how to find the angle ...
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0answers
27 views

Closure condition of spherical polygons

Given an open spherical polygon of geodesic edge length $L_i$ ($i=0,..,n$) and anterior spherical angles $A_i$ ($i=0,..,n-1$). My question is what are sufficient and nessecary conditions on the ...
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57 views

DIY 360 Cam: Fast Fisheye to Equirectangular Image Transform

I'm working on a DIY 360 camera project in which I stitch together the images from 180 fisheye cameras. The images from the 180 fisheye lenses are 2D circles. I wrote a python program that ...
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0answers
17 views

Decompose a product into angular harmonics

The function $f$ of two 3-component unit vectors $\hat r_1$ and $\hat r_2$ only depends on the angle from one vector to the other, so we can write $f(\hat r_1, \hat r_2)=g(\theta)$, where $\theta$ is ...
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76 views

What functions map points from unit sphere to unit sphere?

I am looking for examples of $f:\mathbb{S}^{n-1}\to\mathbb{S}^{n-1}$. The obvious one is: $f(\mathbf{x}) = \mathbf{Wx}$ where $\mathbf{W}$ is orthogonal and $\mathbf{x}\in\mathbb{S}^{n-1}$. Are there ...
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1answer
75 views

Determining the lune angles of a spherical quadrilateral

Suppose that we have a convex spherical quadrilateral, and we know its internal angles $\alpha,\beta,\gamma$ and $\delta$. Pairs of opposite sides of the quadrilateral are pieces of distinct great ...
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1answer
49 views

Points within an ellipse on the globe

I'm interested in finding the equation that will tell me if a given geographical coordinate (lat1, lon1) is within an ellipse centered on a another coordinate (lat2, lon2) with a given semi-major axis ...
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31 views

Intrinsic curvature without coordinates

Can curvature (of a surface) be derived without coordinates and completely intrinsically? For flat space two meeting lines would diverge from each other at a constant rate as you go along them. The ...
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1answer
67 views

What is the “most obtuse” triangle that can fit on a sphere?

Often, when someone introduces the idea of non-euclidean geometry they give the examples of spherical and hyperbolic geometry. To help visualize these concepts, they'll usually compare the sum of the ...
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167 views

How does spherical geometry contradict Euclid's parallel postulate?

Euclid's parallel postulate says: If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended ...

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