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

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Converting a geodesic into a set of Euler angles

Two non antipodal points on a sphere have a geodetic which is a segment of a great circle on that sphere. I'm trying to calculate the Euler angles that would rotate the "equator" great circle of the ...
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8 views

Relation between angle and curvature in sphere

Consider a canonical 2-dimensional sphere of radius $\frac{1}{\sqrt{k}}$ where $k>0$ is a sectional curvature Consider a geodesic triangle $ABC$ Prove that $$ \frac{d}{dk}\angle CAB > 0 $$ ...
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34 views

Is integration with respect to spherical measure equivalent to manifold integration over sphere?

Let $S$ be an $n$--sphere $\mathcal{S}^n(0,R)\subset\mathbb{R}^{n+1}$, $\Theta\subset\mathbb{R}^n$ an open subset and let $\phi:\Theta\subset\mathbb{R}^n\longrightarrow S\subset\mathbb{R}^{n+1}$ be a ...
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66 views

Can one define 'geodesic' solely in terms of the betweenness relations among the points on that geodesic?

In the Euclidean plane (though I assume the following result can be generalized to any Euclidean n-space), Tarski showed that one can define what it is to be a straight line solely in terms of the ...
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44 views

Area of a circle on sphere

On a (flat) Euclidean plane, the area of a circle with a radius $r$ can be described by the function $A(r) = \pi r^2.$ But how can one describe the area of the same circle on a spherical manifold? ...
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19 views

Hypersurfaces of a hemisphere

Let $S_+^{n+1}$ be the open hemisphere of the standard euclidean sphere centered at the north pole and let $M^n$ be a compact, connected and oriented hypersurface of $S_+^{n+1}$. Is it true that if $M$...
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30 views

Euler's Formula: $V-E+F=2$ by using spheric triangles

I just have a question to a proof found here: https://nrich.maths.org/1384 At one point it says: As eight copies of $\triangle$ will fill the sphere without overlapping. Why this? Why can I "...
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12 views

Looking for a particular parameterization of $S^n$

Say we have take vectors $(x_1,..,x_d) \in S^{d-1}$ and we look at vectors $(a_1,..,a_d) \in (\mathbb{Z^+ \cup \{0\}})^d$ such that $\sum_{i=1}^da_i =k$ for some positive integer $k$. Is there any ...
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22 views

sum of the angles of a spherical triangle

What is the sum of the angles of a spherical triangle formed on the surface of a sphere of radius R? The triangle is formed by the intersections of the arcs of great circles. Let A be the area of the ...
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13 views

Existence of spherical triangles and their uniqueness upto rigid motion

On a 2-dimensional sphere of radius $\frac{1}{\sqrt{k}}$, call it $S_k$, where $k > 0$, we have the metric $d$ that is the great circle distance between any two points. How do I prove the following?...
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477 views

Formula for the fourth side of a spherical quadrilateral

Given two sides $a,b$ of a spherical triangle and the angle $C$ between them, the spherical law of cosines gives an elegant formula for the missing edge length $c$: $$\cos c = \cos a \cos b+ \sin a \...
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46 views

Consider the function : $L_{ij}=x_{i}\frac{\partial}{\partial x_j}-x_{j}\frac{\partial}{\partial x_i}$

Let $\Omega$ be a smooth bounded domain of $\mathbb{S}^n$, the unit $n$ sphere centered at the origin of $\mathbb{R}^{n+1}$, and consider the functions $$L_{ij}u=x_{i}\frac{\partial u}{\partial x_j}-...
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31 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
47 views

Equation of a great circle passing through two points

I've searched everywhere for something to help me with this problem, but I can't find anything. What I want to calculate is the midpoint between two locations (latitude and longitude) on a sphere. The ...
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1answer
58 views

Sum of angles of a triangle on a sphere

What is the minimum and maximum of sum of angles of a spherical triangle? Let us remove a constraint from spherical triangles: sides are not necessarily circular arcs. Then what will be the minimum ...
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1answer
37 views

Normal vector field to this hypersurface

Let $M^n$ be a hypersurface of the unit sphere $S^{n+1} \subset \mathbb{R}^{n+2}$ contained in the open upper hemisphere $S^{n+1}_+$, and let $N : M \to \mathbb{R}^{n+2}$ be a unit normal vector field ...
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15 views

Change directly between spherical coordinate systems, without intermediate Cartesian coordinate system.

Is there a practical way to change from one spherical coordinate system to another spherical coordinate system without changing to an intermediate Cartesian coordinate system? The Stack Exchange-...
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2answers
34 views

Derivation of Spherical Law of Cosines

I am trying to get a derivation of the spherical law of cosines. The Wikipedia page [https://en.wikipedia.org/wiki/Spherical_law_of_cosines ] contains a proof that I don't understand because there are ...
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1answer
22 views

How to Find Equation of Line Given Latitude, Longitude, Heading

I need to find the equation of a line given X and Y coordinates (latitude and longitude) and a heading in degrees. I can assume that 0 degrees is North. So for example, I might have that the point ...
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77 views

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
23 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
51 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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17 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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61 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
55 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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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
38 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
105 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
37 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
103 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
31 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
84 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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31 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
121 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),\ \theta\in[0,\...
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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
48 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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48 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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30 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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26 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 ($\alpha+\...
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1answer
49 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
55 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
29 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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38 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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27 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
39 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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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 \frac{\...
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1answer
53 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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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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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
36 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 ...