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

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2answers
12k views

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 ...
0
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1answer
488 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 ...
5
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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
292 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 ...
1
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1answer
206 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) ...
4
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1answer
201 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 ...
1
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0answers
662 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 ...
0
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1answer
470 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 ...
2
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2answers
95 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 \ \ \ ...
0
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2answers
114 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 ...
6
votes
2answers
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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0answers
423 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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0answers
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 ...
0
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2answers
218 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 ...
3
votes
1answer
208 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 ...
1
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0answers
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 ...
0
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2answers
688 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 ...
1
vote
1answer
301 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 ...
8
votes
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 ...
3
votes
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 ...
2
votes
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 ...
6
votes
2answers
327 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?
2
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0answers
606 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 ...
0
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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 ...
2
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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 $A,B,C\in \mathbb{S}^2$, ...
3
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1answer
208 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 ...
3
votes
1answer
258 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 ...
3
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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 ...
3
votes
5answers
589 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 ...
2
votes
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.
1
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0answers
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 ...
3
votes
2answers
250 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 ...
4
votes
2answers
549 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 ...
3
votes
0answers
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 ...
0
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1answer
196 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 ...
2
votes
2answers
542 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. ...
0
votes
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 ...
0
votes
1answer
465 views

I have equations for getting x,y,z given latitude, longitude, and altitude. How do I reverse them?

I am using equations that look like the following to get x, y, and z given latitude, longitude, and altitude. ...
1
vote
1answer
851 views

How to calculate the number of latitude/longitude coordinates in a specific area at a given precision?

I am looking into ways of grouping large sets of latitude/longitude coordinates and am wondering if there is an easy/standard way to roughly calculate the number of different coordinates in a given ...
4
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3answers
784 views

Article or book about the history of spherical geometry?

I teach a course on non-Euclidean geometry to high schoolers. I'm looking for an article or book that gives a thorough and interesting history of spherical geometry and trigonometry. I'm looking for ...
0
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1answer
781 views

Vertices of intersection between N spheres

Just wanted to know what is the best algorithm (in terms of speed and accuracy) to determine the intersection of N spheres (in 3D). With intersection I mean the following; in 2D and in the case of two ...
2
votes
1answer
586 views

Trying to pick a random point on sphere end up picking from a lune

I was inspired by this question to play around a little bit (its a weekend). I was pretty confident of my derivation and thought it might be nice to supplement it with a pretty picture. However, ...
3
votes
1answer
832 views

Arc length of a great circle which is the hypotenuse of an isoceles right triangle on the sphere

I am doing a problem which requires me to find the arclength of the hypotenuse of an isosceles right triangle. (The book calls it a 2 Dimensional Sphere but I hope that is a typo) I start at the ...
2
votes
1answer
603 views

quaternion representation of the rotation of a sphere into plane displacement

I do have a sphere of known radius which does have a coordinate frame rigidly attached to it. Let's call the coordinate frame attached to the sphere XYZs. The sphere can be rotated and displaced ...
5
votes
1answer
703 views

approximating geodesic distances on the sphere by euclidean distances of a transformed sphere

Is there a way to find a function $F:\mathbb S^2 \rightarrow \mathbb R^3$ of class $C^1$, minimizing $$\int_{\mathbb S^2\times\mathbb S^2}(d(F(x),F(y))−\delta(x,y))^2 dx dy$$ , where $d$ stands for ...
6
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3answers
1k views

What is the area of the portion of 1/8 of an sphere cut off by two parallel planes?

So the problem that I'm trying to solve is as follows: Assume 1/8 of a sphere with radius $r$ whose center is at the origin (for example the 1/8 which is in $R^{+}$). Now two parallel planes are ...
3
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2answers
270 views

What is the average rotation angle needed to change the color of a sphere?

A sphere is painted in black and white. We are looking in the direction of the center of the sphere and see, in the direction of our vision, a point with a given color. When the sphere is rotated, at ...
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2answers
220 views

Calculating probabilities on a spherical map

A black and white colored sphere is given. We are looking at a random starting point on the sphere below us, which has a certain color. A random rotation can change the color of the spot below us. ...
2
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1answer
1k views

Formula for the coordinate of the midpoint in spherical coordinate system

Please let me know the formula for the coordinate of the midpoint of 2 points in spherical coordinate system . If possible , I want the answer includes the exact formula as , midpoint = point1 + ( ...