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

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37 views

Project a line onto a sphere to calculate parameterized spherical coordinates

I have a line segment and I want to find the arc that it projects to on a sphere. I know there are two arcs; I'm interested in the one that's closest to the line (or intersects it). The easy way to ...
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1answer
58 views

Relationship between regular tessellations in general space

Using the notation for regular tessellations $\{p,q\}$ denoting the tessellation consisting of p-gons , q of which meet at each vertex [e.g. $\{3,6\}$ in $\mathbb{R}^{2}$ is the equilateral triangle ...
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32 views

Triangles in spherical/elliptical geometry

In Euclidean geometry the sum of internal angles of a triangle is always $180^\circ$. In non-Eudlidean geometries it may be either less than this (in hyperbolic geometry) or more (spherical/elliptical ...
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30 views

Non commutative flows on the 2-sphere.

The title says everything, really. I'm looking for some flows on $S^2$ such that They do not commute. They are of some interest, or they are peculiar in some ways. Thanks
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3answers
47 views

Trying to create a inverse quare algorithm for expanding sphere

So, in a piece of software I am writing (this isn't homework), I want to have a sphere expand relative to time. I want it to expand quickly from start with the expansion slowing over time. I.e, the ...
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2answers
78 views

Find a spherical triangle with angle sum $5π/3$

Find a spherical triangle with angle sum $5π/3$ I am unsure how to answer this question and would like to be shown how I go about answering?
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1answer
60 views

Determine a point lie in bisect area between 2 circles on sphere

Given 2 points A,B,O on sphere of radius $R$. Point O is in middle of AB. E and F are deviation from O by geodesic distance $d$ (angle between EF and AB is $90^o$). Consider 2 circles $C_1,C_2$ on ...
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79 views

How can I efficiently check a point lie in 4 circles on sphere?

Given coordinate of 2 points $A,B$ (Cartisian or longitude-latitude coordinate) on sphere of radius $R_1$. Point $O$ is middle of $AB$, 2 points $E$ and $F$ is derivation from $O$ by a distance $d$. ...
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40 views

Locate a point on sphere with equal distance

Given 3 points A (lat1, lon1), B(lat2, lon2), O(lat3,lon3) on earth with geometric location longitude and latitude and a distance d, where O is middle point of A and B. Let GCD denote the great circle ...
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29 views

Find a point on extended great circle with given distance

Given 2 points on earth with longitude and latitude coordinate A(lat1, lon1), B(lat2, lon2), and a distance d. Find coordinate (in longitude and latitude) of 2 points C, and D on extended of great ...
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76 views

Area of a plane triangle as limit of a spherical triangle

We know that the area of an spherical triangle (in a unit sphere) is given by $A(\triangle) = \alpha + \beta + \gamma - \pi$, where $\alpha$, $\beta$, and $\gamma$ are the interior angles of the ...
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Colour of bear in Earth's surface [closed]

A bear stands in one point of the Earth's surface. Walking one kilometer south , then walking one kilometer east and immediately after one kilometer north and reaches the point from which started.Find ...
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168 views

Find all Equiangular Platonic triangles

A spherical triangle A is called equiangular if its 3 angles are equal. A is called Platonic if copies of A tile the unit sphere. I need to find all such triangles. Don't we have an infinite amount of ...
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20 views

Sample from distribution taking spherical statespace

I have a probability distribution over a 2-sphere, with density function $f(\phi)$, a function of polar angle only. Is there an efficient way to sample from this distribution?
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29 views

How to find height of two objects stacked at an angle

Consider the following situation: If there are two balls of diameter 50mm and 60mm stacked inside a tube with internal diameter of 60mm. If the smaller ball is stacked on the big ball, it is easy to ...
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32 views

Project locus of points on sphere using Mercator projection

Given a the locus of points on a sphere that are the same great-circle distance from some point, what is the shape described when that locus is projected onto a 2D plane using the Mercator projection? ...
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74 views

Area of the surface on a sphere surrounded by three externally tangent circles?

Let us draw three circles of radius $\dfrac23$ on a sphere of radius $1$, all of which are mutually tangent (externally). How do I calculate the area of the surface surrounded by the three circles?
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82 views

Find intersection points of two circles on sphere (earth)

Problem Assume that we have two devices that can measure distance to the target. Devices installed on earth at coordinats A and B. We measure distances Da, Db in meters. How to find points of ...
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67 views

Calculating the percent of area “covered” by a vector pointing on a sphere

The question is inspired by rotational dynamics and how much of the sky could a camera "cover" when it rotates in a specific way. Let's say that we have solved the equations for rotational motion of a ...
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14 views

Projection of great circles in flatprojected sphere

I am trying to construct a map using fractal fault algorithms. What I do is find a random great circle in a sphere and change the height of half of the sphere. Doing this ten thousands of time a map ...
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75 views

Placement of protons and neutrons in the nucleus

So, I'm creating a program that would represent a given atom (also different isotopes) in 3d view. I'd need some kind of formula to calculate the position of protons and neutrons to form a nucleus. ...
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3answers
74 views

How to know if an arc segment intercepts a circle in a sphere?

This problem is bugging for quite some time now, and I actually need to program the solution in my game. I have one point on a radius 1 sphere, and a rotation matrix. When I rotate the point with the ...
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1answer
25 views

Projection from triangle to spherical triangle

Consider a triangle, $T$, in $\mathbb{R}^3$ with vertices $(0,0,1), (0,1,0)$, and $(1,0,0)$. Let $S$ denote the sphere centered at the origin with radius 1 and let $S_1$ denote the portion of the ...
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2answers
144 views

How to find a plane that is tangent to 3 spheres?

So there are spheres with radius of 1 centered at (1,2,0), (4,5,0) and (1,3,2). How can one find a plane that is tangent to all 3 spheres? Visually, it looks like as if the spheres are sitting on a ...
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1answer
106 views

Determining North-South Line Via Watch Method: Theory & Reason

I recently read that if you're in the northern hemisphere and have an analog watch, then you can point the hour hand at the sun and know that a south line lies between (bisection) the hour hand and ...
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1answer
73 views

Frechet mean of $k$ elements in the n-dimentional sphere.

The Frechet mean of $k$ elements $x_1, \ldots, x_k \in S = \{ x \in \mathbb{R}^n,\, \| x \| = 1 \}$ is defined as the $\text{arg}\underset{\|x\|=1}{\text{min}} \sum_{i=1}^n d^2(x_i,x)$. Where $d(x_i,x)...
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27 views

Area of a sphere bounded by hyperplanes

Say we have a sphere in d-dimensional space, and k hyperplanes (d-1 dimensional) all passing through the origin. Is there a way to calculate (or approximate) the area of the surface of the sphere ...
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1answer
133 views

Plotting a skew ellipse

I tried to make a geometers sketchpad toolkit for spherical geometry that is more exact (and easier to understand) than the present available one. but then I soon realised my background is not good ...
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2answers
89 views

Computing a double integral over a surface S, where S is the unit sphere,

$$ \int \int_S (x^2+y^2)d\sigma$$ Where S is the unit sphere centered at (0,0,0), and $\sigma$ is surface area. I arrived at the correct answer of $\large \frac{8\pi}{3}$, but I took an (educated?) ...
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1answer
76 views

Is this algorithm for 3D spherical interpolation correct?

I am attempting to write a spherical interpolation algorithm for for the application of smooth 3D animation in a game. The scripting language that the game engine uses is Lua. It is often easier for ...
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2answers
627 views

Why is the polar triangle useful in spherical geometry?

We can solve many problems in spherical geometry by using the polar triangle. I am looking for an intuition why (and when) this is easier than working in the original triangle.
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89 views

mean square displacement on the 3-sphere

I would like to compute the mean square displacement (MSD) for a particle moving on the surface of a 3-sphere of radius R. I see that I could eventually use the polar coordinates and get a polar ...
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87 views

Orthogonal transformation of a triangle on $S^2$

Let $v_1, v_2, v_3$ and $w_1, w_2, w_3$ denote the vertices of two spherical triangles $\bigtriangleup_1, \bigtriangleup_2$ with the property, that $\|v_i - v_j \| = \|w_i - w_j \|$, e.g. their sides ...
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1answer
111 views

Azimuth angle limit in Spherical co-ordinate system

In spherical co-ordinate system $(r, \theta, \phi)$, $\theta$ can range from $0$ to $2\pi$, but $\phi$ only varies from $0$ to $\pi$. Why is that?
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76 views

Direction of rotation to transform from Point a to b on a unit sphere

If I have two points $a$, $b$, on a unit sphere, I believe I can determine the angle between them, expressed as vectors, as follows: $$\theta = \arccos\left(\frac{a\cdot b}{\|a\| \|b\|}\right)$$ ...
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96 views

Stereographic projection and lengths

We know the Stereographic Projection doesn't preserve areas except for the points on the plane such that $x^2+y^2=1$, because that's where $dA=dxdy$. I was wondering what would happen with lengths: ...
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1answer
58 views

constructing a spherical triangle using only the laws of sines and cosines

I have a spherical triangle with corners $A,B,C$, angles $\alpha, \beta, \gamma$ and sides $a,b,c$ (which are opposite to the corresponding corners/angles). I am given $a,b$ (with $a>b$) and $\...
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1answer
62 views

Spherical Geometry and Playfair's Axiom.

Recently I came across a variant of the Parallel Postulate known as Playfair's Axiom: In Euclidean (planar) geometry there is at most one line that can be drawn parallel to another given one ...
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149 views

Instruct geometer moths so you can learn about their true geometry.

I had a space-ship wreck in an unknown world of some kind of moths. I could observe geometer moths working. Everything looked strange. The moths claimed that they were using only straight edges and ...
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1answer
195 views

Tangent points on circle that placed on Earth surface

I need help about spherical geometry problem that I need to use it for my project. I try to calculate $T_1 $ and $T_2$ coordinates on $B$ centered small circle on the sphere and $AT_1$ and $AT_2$ ...
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1answer
39 views

How to find point on sphere from pitch and heading

I have a sphere of radius R and I would like to draw some vector positions on it given pitch and heading. I have a heading between 0 and 360 (0 being +x direction), and a pitch between -90 and 90 (90 ...
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153 views

Finding the northernmost latitude in a great circle that passes through two points on the sphere

I'm trying to solve the following problem from Smart's Text-Book on Spherical Astronomy (exercise 5 on p.23 of the 6th ed.): $A$ and $B$ are two places on the earth's surface with the same ...
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2answers
153 views

Finding the shortest distance to the north of the sphere

I found these problems in Alan F Beardon's Algebra and Geometry: Verify that any point with latitude α is a spherical distance R(π/2−α) from the north pole. Suppose that an aircraft flies on the ...
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1answer
1k views

How to calculate a solid angle (in Steradians) given only Horizontal Beam angle and Vertical Beam angle data.

I would like to convert a rectangular beam shape given in Horizontal and Vertical beam angle, into solid angle representing the surface area in steradians of projected light. For example a light ...
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137 views

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?
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106 views

Can the intersection of two balls be described?

Suppose, two spheres intersect. Subtracting the equations of the speheres, a linear equation appears which indicates the plane conataining all points belonging to the intersection of the spheres. But ...
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346 views

Circle from three points on surface of sphere

I need to compute the circle on the surface of a sphere given three points on that very surface. It is very easy to do that in Euclidean geometry, but the sphere has no x and y, but just two angles ...
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141 views

Construct the great circle (geodesic) in spherical or Riemanian geometry

Given: a circle $C$ with centre $M$ two points $P_1$ and $P_2$ inside circle $C$, so that $M$ is not on the line $P_1P_2$. Cunstruct an other circle $O$ so that: $P_1$ and $P_2$ are on ...
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Four circles touching one another on a spherical surface

The diagram above shows four identical circles, each having a flat radius $r$ (i.e. flat area $\pi r^2$), touching one another at six different points (i.e. each of four identical circles touches rest ...
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1answer
59 views

How to parametrize circles on a sphere by the distortion of the equator?

I guess am having a very silly problem right now. Considering a unit sphere $S^2$ and, for example, a curve, in spherical coordinates, $c(t)=(1, \frac{\pi}{2},t)$ that goes around the equator how can ...