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

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Explaining Spin(3)

I’m going to discuss the action of Spin(3) on Euclidean vectors. This thing has several alternative names: “versors”/“rotation quaternions”, “quaternionic adjoint representation”, “quaternion action ...
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1answer
19 views

Point along great circle line (aka arc) closest to a target point on the ground

Given: an arc (aka a great circle line, not a straight-line) defined by two arbitrary end points (which I can express in lat/lon/altitude or earth-centered fixed (ECF) 3D 'cartesian' space). Think ...
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1answer
41 views

How to find a point equidistant between two points on a sphere.

So I have this problem involving astronomy, but because astronomy uses all sorts of fancy words I'm going to make it more simple by using an analogy of the earth. The process, mathematically would be ...
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15 views

Geometrical Significance of dimensionless Invariant of Sphere

What is the geometric significance of the constant $$SI = \dfrac{\sin a}{\sin A}=\dfrac{\sin b}{\sin B}=\dfrac{\sin c}{\sin C }$$ in the Law of Sines in spherical trigonometry? Have a hunch that ...
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52 views

Trajectory on a sphere

I've asked a question before concerning a parallel problem, and I read a wikipedia page on spherical caps (Nominal Animal), which gave me an idea to do the following: I have the Cartesian coordinates ...
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1answer
43 views

Arc intersection on a sphere

Background: my JavaScript library https://github.com/mistic100/Photo-Sphere-Viewer allows to create 2D polygons overlaying a spherical photo. Polygons are defined by a serie a longitude/latitude ...
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3answers
67 views

How to divide a spherical triangle into three equal-area spherical triangles?

The Centroid point (at intersection of medians) divides a planar triangle into three equal-area smaller triangles. In case of spherical triangle, the three geodesics joining the vertex to the midpoint ...
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1answer
34 views

Find the edge angle of a dodecahedron using spherical trigonometry?

How can I find the edge angle (the angle at the center of a polyhedron subtended by an edge of the polyhedron) of a dodecahedron (a polyhedron with 3 pentagonal faces meeting at each vertex)? I know ...
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2answers
74 views

Compute angle between two points in a sphere.

Exist an equation that provide the angle between two points in a sphere. What I'm looking is not easy to explain, anyway assume to have two points in a sphere than connect this tho points with a line ...
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1answer
36 views

Write the equation $4x^{2}+4z^{2}=5$ in spherical coordinates.

Write the equation $4x^{2}+4z^{2}=5$ in spherical coordinates. I used the facts that $$ \begin{align} x&=ρ\sin\theta\cos\phi\;,\\ z&=ρ\cos\phi\;, \end{align} $$ And ended up with: $ 4 (ρ^2 ...
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1answer
63 views

condition for cones to be reciprocal

Question : Show that the cone $$ax^2 + by^2 + cz^2 - cxy - ayz - bzx = 0$$ is the reciprocal of the cone $$(a^2 - bc)x^2 + (b^2 - ac)y^2 + (c^2 - ab)z^2 - 2(a^2 + bc)yz - 2(b^2 + ac)zx - 2(c^2 + ab)xy ...
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1answer
29 views

sphere arc intersection

Given: an arc defined by two end points (which I can express in lat/lon/altitude or earth-centered fixed (ECF) 3D 'cartesian' space) a sphere defined by a center (lat/lon/alt or ECF) and a radius ...
5
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1answer
82 views

The centre of the earth

I'm a real beginner here (first post and first foray into math since high school, trying to catch up), so I'm going to try my best to explain my problem in mathematical terms then follow up with an ...
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0answers
30 views

Area of non-spherical triangle on a sphere

This is a followup to the question Area of triangle on a sphere (not spherical triangle) Since it's now almost two years later, I'm making it a new question. The problem is to find the area of ...
5
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1answer
120 views

Solving Laplace's equation in a sphere with mixed boundary conditions on the surface.

Can anyone help point me to a solution method for this problem? Solve $C(\vec{x})$, where $\vec{x}=(r,\theta,\phi)$ on $\Omega=\{\vec{x}\in\mathbb{R}^3\ |\ r\in[0,R],\ \phi\in[0,2\pi),\ ...
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2answers
2k views

How to calculate the area of a circle ( given: origin, radius ) on a sphere ( Earth )?

I know that the Earth isn't a sphere, not even an ellipsoid, but for my measurements, its an acceptable approximation. Assuming I have a coordinate(lat,lon) and a distance( e.g.: 1000km ), what is the ...
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1answer
44 views

Constructing a spherical triangle of a given surface area

Given three points $p_1$, $p_2$ and $p_3$ on the unit sphere $S^2$, we can construct a spherical triangle. The angles associated with the points are $A$, $B$ and $C$. Keeping the points $p_1$ and ...
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1answer
47 views

The furthest point to this torus

Recall that the (geodesic) distance on the unit sphere $S^n$ is given by $$ d(p, q) = \arccos \langle p, q \rangle. $$ Let $f_r = f : \mathbb{R}^2 \to S^3$ be defined by $$f(\theta, \phi) = \left(r ...
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0answers
28 views

Showing there is no local isometry between spheres of different radii

I wish to show there is no is no local isometry between 2-dim spheres of different radii, without the use of curvature, as it is not in my knowledge yet. Could you provide directions? If such ...
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0answers
21 views

Describing Area of Spherical Cap as Sum of Spherical Triangles

I was wondering how one could express the area of a spherical cap in terms of a sum of triangles. The area of a triangle on a sphere is: $A = E R^2$ where E is the excess of the triangle ...
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20 views

Area of Spherical Zone

"Let $\mathcal S$={$\mathbf x \in \mathbb{R}^3 : ||\mathbf x||=1$} Prove that the area of the part of $\mathcal S$ that lies between the two parallel planes given, say, by $x_3=a$ and $x_3=b$, is the ...
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1answer
41 views

Proving that there are only five Platonic solids using spherical geometry

In 100 Great Problems of Elementary Mathematics by Dorrie, it is proved that there are only five possible tessellations of the sphere using congruent regular (spherical) polygons: $4$ regular ...
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1answer
49 views

Cartesian to Spherical coordinate conversion specific case when Φ is zero and θ is indeterminant

Following is the conversion for spherical to cartesian coordinate \begin{align} x &= r \cos\theta \sin\varphi \\ y &= r \sin\theta \sin\varphi \\ z &= r \cos\varphi \end{align} and we are ...
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1answer
28 views

Problem finding coordinates in a earth like coordination system

A picture with the problem Hey guys Given: two coordinates $A(a_1,a_2), M(m_1,m_2)$ , the distance between $B$ & $C$ is known as $w, d(B,C) = w$ d(B,M) = d(M,C) where d is the great-circle ...
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34 views

rotation of spherical surface in spherical coordinates

I need to plot a spherical surface in computer (like the surface of a lens). I know the normal vector (as an example, say $\ n=(1,2,3) $) of this surface and it originates from the centre of the ...
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0answers
24 views

congruency of triangles in hyperbolic and spherical geometry

In Euclidean geometry, we have the following congruencies of triangles: side-side-side, side-angle-side, angle-angle-side = angle-side-angle (because of the angle sum) and side-side-angle (only if the ...
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1answer
35 views

Rotations of sphere $\mathbb S^2$

In the picture bellow; How to prove that the result of rotation about $P$ through angle $\theta$, followed by rotation about $Q$ through angle $\varphi$ is rotation about $R$ through some angle? ــ ...
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1answer
64 views

Spherical circle - Area

I am looking at the following exercise: The spherical circle of centre $p \in S^2$ and radius $R$ is the set of points of $S^2$ that are a spherical distance $R$ from $p$. If $0 \leq R \leq ...
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1answer
45 views

Similar spherical triangles are congruent

I look at the following exercise of A. Pressley: Show that similar spherical triangles are congruent. I have no clue how to show it. Can you give an idea how I can do that? In the book ...
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6answers
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How to generate random points on a sphere

How do I generate $1000$ points $(x, y, z)$ and make sure they land on a sphere whose center is $(0, 0, 0)$ and its diameter is $20$? Simply, how do I manipulate a point's coordinates so that the ...
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25 views

Proof equivalence of equal dihedral angles and vertices on a sphere for regular polyhedra.

I know that the following theorem is true: Theorem: Provided that all faces of a polyhedron are regular poygons, the statement ``all the dihedral angles are congruent'' is equivalent to saying ...
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1answer
27 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
55 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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29 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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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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28 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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1answer
72 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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3answers
46 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 ...
2
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1answer
69 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 ...
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2answers
72 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?
3
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1answer
57 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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2answers
530 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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37 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 ...
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17 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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2answers
243 views

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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2answers
157 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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0answers
19 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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1answer
25 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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2answers
148 views

Find the intersection point of a great circle arc and latitude line

In spherical geometry, I need to know at what longitude λ a great circle arc φ1,λ1-φ2,λ2 has intersected a line of latitude φ. I have found the equivalent equation for solving latitude φ for an ...
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0answers
30 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? ...