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 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 ...
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38 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
53 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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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 ...
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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 ...
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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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3 circles on a sphere what is the radius of the sphere

On a sphere with radius R there are 3 circles: Circle $C_1$ with radius $r_1$ and circumference $c_1$ Circle $C_2$ with radius $r_2$ and circumference $c_2$ Circle $C_3$ with radius $r_3$ and ...
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33 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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18 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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21 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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45 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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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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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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75 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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37 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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48 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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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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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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80 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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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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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 ...
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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 ...
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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),\ ...
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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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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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25 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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29 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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44 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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52 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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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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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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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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37 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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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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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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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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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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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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1answer
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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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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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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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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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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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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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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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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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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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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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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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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 ...