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

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Find third geographic coordinate in triangle using spherical earth model

I'm trying to solve triangulation problems using geographic coordinates from a GPS. all calculations must use the spherical earth model (great circle distance). Given the points and lengths: Point $A$...
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274 views

Apollonius Pursuit Problem

Most references in a brief search under 'Apollonius' concern tangent circles. The problem I am interested in is the Apollonius pursuit problem. In the plane, the question concerns the point at which ...
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1answer
134 views

Real numbers mapped onto a sphere

We can compare real values if they were greater, lesser but we cannot do same for complex numbers. What if we map real values(within some small range) onto a sphere and declare each one of them as ...
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1answer
400 views

spherical triangle inequality for multiple edges

Consider a spherical triangle, with side lengths $0 \leq a,b,c \leq \pi$. In texts on spherical geometry we see proofs that $a \leq b + c$ but this inequality is unnecessarily weak, in that some sets ...
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1answer
754 views

Moving points along a curve on sphere.

I have two points on a unit sphere. I also have their coordinates. ...
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2answers
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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1answer
3k views

Angle between GPS coordinates

I realize GPS Coordinates are spherical coordinates. However I know the earth is more of an ellipsoid. I need to compute with a fairly high degree of accuracy the pitch and yaw between two objects ...
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1answer
286 views

dotting gradient in spherical coordinates with displacement vector

The gradient in spherical coordinates is given by: $\nabla f = \left(\frac{\partial f}{\partial r}, \frac{1}{r} \frac{\partial f}{\partial \theta}, \frac{1}{r \sin \theta} \frac{\partial f}{\partial \...
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1answer
73 views

Cardinality of Mappings on a 2-Sphere

I am wondering about the number of mappings from a point on a sphere to a neighboring point, and a not so neighboring point. If I take a 2-sphere, and place it on some $x,y,z$-axis and fix those so ...
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1answer
921 views

Coordinates of interception point Y with XY being the shortest distance of X to AB on sphere

How would one calculate the interception point $Y$ with $\overleftrightarrow{XY}$ being the shortest distance of $X$ to $\overleftrightarrow{AB}$? This answer to the question How to find the ...
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47 views

Internal angle of a vertex of degree $d$ in $\mathbb{E}^2$ and $\mathbb{S}^2$

I am currently working on determining the maximum number of times the minimum spherical distance can occur among $n$ points in $\mathbb{S}^2$, and I have the following question. In $\mathbb{E}^2$, ...
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1answer
797 views

Area of a Spherical Triangle from Side Lengths

I am currently working on a proof involving finding bounds for the f-vector of a simplicial $3$-complex given an $n$-element point set in $\mathbb{E}^3$, and (for a reason I won't explain) am needing ...
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Why does every direction at the north pole point south?

Why does every direction at the north pole point south? Why doesn't this happen at any other point on (face of the) earth? Is this due to convention used by humans or is there a geometrical ...
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1answer
505 views

Integrate linear function over spherical triangle

In the plane I can integrate a linear function over a triangle as the average of the corner values times the area of the triangle. Then if I subdivide the triangle into 4 triangles by introducing a ...
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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
294 views

radius around point

I have a question, I am trying to calculate a radius between two latitude and longitude points on the Google map. I understand the coding part, but I do not understand the mathematical side to it. I ...
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1answer
209 views

Travelling Salesman Problem on the unit sphere

Let vertices be defined on the unit sphere by its longitudes $\lambda_i$ and latitudes $\phi_i$. The distance between $i$ and $j$ is defined as follows: $\arccos(\sin(\lambda_i) \sin(\lambda_j) \cos(\...
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1answer
203 views

Why do derivatives of certain equations relating to circles yield other similar equations? [duplicate]

Possible Duplicate: Why is the derivative of a circle's area its perimeter (and similarly for spheres)? We all know that the volume of a sphere is: $V = \frac{4}{3}\pi r^{3}$ and its ...
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692 views

How to project a spherical map onto a sphere / cube

I have this panorama, an spherical map from google streetview, and want to map this on a sphere/cube. Below are some examples and illustrations, i am going to implement it in c++ and are not sure ...
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1answer
476 views

Find Double of Distance Between 2 Quaternions

I want to find the geometric equivalent of vector addition and subtraction in 3d for quaternions. In 3d difference between 2 points(a and b) gives the vector from one point to another. (b-a) gives the ...
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2answers
96 views

angles between $n$ points on an $n-2$ dimensional sphere

$n$ points are placed on an $n-2$-sphere so that the smallest angle from the centre between any pair of the points is maximised. What is this smallest angle? $n=1 \ \ \ \cos^{-1}{1}\\ n=2 \ \ \ \...
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2answers
117 views

How many spherical quadrangles exist with a given ordered sequence of inner angles.

Well, I think the title already explains my question. Given a sphere and an ordered sequence of inner angles ($\alpha$, $\beta$, $\gamma$, $\delta$) how many spherical quadrangles do there exist that ...
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3k views

Deriving the Surface Area of a Spherical Triangle

A triangle on a sphere is composed of points $A$, $B$ and $C$. The $\alpha$, $\beta$ and $\gamma$ denote the angles at the corresponding points of the triangle: The Girard's theorem states that the ...
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435 views

Isomorphic triangles?

From my previous post I have learnt that spherical triangles can have different interior angle sums. Is this enough to argue that the triangles are not isomorphic? I am not sure how isomorphism works ...
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101 views

Definition of stereoprojection and Möbius maps

@WillieWong has kindly pointed out that there are 2 definitions of stereographic projection. One with the unit sphere placed on top of the plane, the other where the plane is at the equator of the ...
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2answers
220 views

The globe, spherical disks and spherical straight lines

Does a spherical triangle with 2 equal sides necessarily have 2 base angles of size $\pi/2$? The reason I think this is that if we have a triangle $ABC$ and $AB=AC$ (in spherical distance), we could ...
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1answer
209 views

Sine rule and equal angles

Is it true that if a triangle on a unit sphere has 2 sides with equal length then their opposit angles must be equal? I think it is true. I think we can use the spherical sine law. Call the sides with ...
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146 views

Stereographic projections and cross-ratios

Would anybody shed some light on question 2.11 in Wilson's Curved Spaces? The numbers $p,q\in \hat{\mathbb{C}}$ are stereographic projections of points $P,Q$ on the unit sphere. The spherical ...
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2answers
702 views

Line-circle intersection in spherical geometry?

How does one calculate the intersections between a "line" (a Great Circle) and a circle in spherical geometry? i.e. given start point (as lat,lon), heading, circle centre (as lat, lon) and circle ...
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1answer
305 views

How to scale a polyhedron contained a 3-sphere?

In the 3-sphere simulator I am building, the viewpoint is contained in the space of a 3-sphere (the surface of a 4-D hypersphere), and the user is able to navigate through it. There are some ...
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2answers
3k views

Is an equilateral triangle the same as an equiangular triangle, in any geometry?

I have heard of both equilateral triangles and equiangular triangles. (For example, this sporcle quiz lists both.) Are these always equivalent, regardless of geometry? I know they are the same in ...
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2answers
1k views

Lat/Long grid points covered by projecting rectangle onto sphere

Before my question proper, a little background: I'm wanting to optimise some computer rendering by eliminating the drawing of things that aren't visible given the current view. Suppose we have a ...
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1answer
4k views

Transforming from one spherical coordinate system to another

I have a set of points on the surface of a sphere specified in one coordinate system (specifically, the equatorial coordinate system), and for each point I need to work on all its neighbouring points ...
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330 views

Elementary arguments concerning the stereographic projection

How does one give a proof that is short; and strictly within the bounds of secondary-school geometry that the stereographic projection is conformal; and maps circles to circles?
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642 views

projection of a sphere onto a plane

Consider you have a sphere centered at the origin.The sphere has a diameter of $\frac{1}{2} \sqrt{\frac{3}{2}}$. This means that the inscribed cube has an edge of 1. Take any point from the plane (1,...
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1answer
178 views

Geodesic segments on the unit sphere

Let $\mathbb{S}^2$ be the unit sphere and $d$ be the geodesic distance. For any three points $A,B,C\in \mathbb{S}^2$ and $0<\lambda<1$, let $A_{\lambda}$ and $B_{\lambda}$ be points on the ...
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1answer
136 views

Ratio of geodesic segments on the sphere

Let $\mathbb{S}^2$ be the unit sphere. Let $0<\lambda<1$ be fixed. What is the smallest number $0<\mu<1$ (depending on $\lambda$) such that for any three points $A,B,C\in \mathbb{S}^2$, ...
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1answer
217 views

A property of Hilbert sphere

Let $X$ be (Edit: a closed convex subset of ) the unit sphere $Y=\{x\in \ell^2: \|x\|=1\}$ in $\ell^2$ with the great circle (geodesic) metric. (Edit: Suppose the diameter of $X$ is less than $\pi/2$.)...
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1answer
259 views

Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry?

In an Euclidean plane, we know that the area of a triangle is determined by the length of base and the height, then is there a similar thing do happen in Spherical and hyperbolic spaces? In Euclidean ...
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1answer
5k views

Maximum sum of angles in triangle in sphere

Recently my differential geometry lecturer demonstrated that the sum of the interior angles of a triangle in a sphere is not necessarily never $180^\circ$. This is one way to prove that the earth is ...
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1answer
136 views

Every point on the unit sphere has distance at most $d$ to some point in the set $S$, what is the lower bound for $|S|$?

Someone I know said "I wish no matter where I am, there is always a place near me so I can visit". I started to wonder what is the minimum number of places required if he give me what he consider as "...
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5answers
591 views

How do you parameterize a sphere so that there are “6 faces”?

I'm trying to parameterize a sphere so it has 6 faces of equal area, like this: But this is the closest I can get (simply jumping $\frac{\pi}{2}$ in $\phi$ azimuth angle for each "slice"). I can'...
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4answers
2k views

Latitude and longitude of points on a line

How could you get the latitude and longitude of four points (equal distance apart) on a line from $(27,-82)$ to $(28,-81)$? The four points should split the line into 5 parts.
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118 views

Analytical calculation of the resulting surface between a sphere and a spherical cap from another sphere

Let's say I have one spherical cap, resulting from cutting a sphere centered at origin and with radius R1 with a plane, whose normal goes into the direction of the <...
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2answers
264 views

How do I apply a digital filter to points on a sphere

Given a set of points on a sphere, how can I implement a higher order low pass filter on them? At the moment, I am just multiplying the vectors from the input and output set by their weights and ...
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2answers
561 views

Coordinates and distance in higher dimensional spherical and hyperbolic space

For n-dimensional spherical space, it seems to me the representation of points is easiest and most manipulable as unit vectors, with distance being the vector dot product (which is the cosine of the ...
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193 views

How to calculate rotated global location coordinates (Long, Lat)?

Given the current global location coordinate system: -180 <-> 180 Longitude -90 <-> 90 Latitude A rotation of the globe 90 degrees counter-clockwise around the Y-axis would bring the north ...
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1answer
198 views

Numerical software to solve partial differential equations in spherical coordinates?

Which numerical libraries / math software can allow me to solve partial differential equations in spherical coordinates? (my system consists of N degrees of freedom, each degree lives in $\mathbb{C}^{...
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2answers
562 views

How to use a Rhumb Line?

I am new to working with coordinate data and figured out the equation I am looking for is the Rhumb Line. I went to go research it and found a lot of equations and I still have no idea where to start. ...
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1answer
56 views

Spheres and converting formulas

If the volume V of a sphere with radius r is V=(4/3)πr^3. If the surface area is s=4πr^2, how can I express the volume as a function of the surface area S? My first thought was to set them equal to ...