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

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1answer
131 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
125 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
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1answer
215 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 ...
2
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1answer
3k 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
127 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
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5answers
428 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 ...
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4answers
1k 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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0answers
95 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
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2answers
147 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 ...
3
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2answers
407 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 ...
2
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0answers
178 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
179 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
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2answers
294 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
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1answer
52 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
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1answer
346 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. ...
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1answer
564 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
555 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 ...
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1answer
577 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 ...
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1answer
389 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
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1answer
580 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
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1answer
422 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 ...
4
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1answer
525 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 ...
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3answers
745 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
241 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
210 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. ...
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1answer
757 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 + ( ...
2
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0answers
379 views

Geographic coordinates to pitch+yaw+pitch

I'm creating a very basic simulation which involves air travel across the world and am trying to correctly position and orient my aircraft in a rendered 3D representation. I am representing the Earth ...
11
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1answer
597 views

What's the name of a parabola mapped onto a sphere?

It seems that an 'arc' is a line-segment mapped onto the surface of a sphere (although I don't know if that name still holds if the segment wraps around the sphere more than once, i.e., if the angle ...
6
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2answers
4k views

Proof that the angle sum of a triangle is always greater than 180 degrees in elliptic geometry

I've scoured the internet and have found many proofs showing that in Euclidean geometry, the angle sum of a triangle is always 180 degrees. I've also found many proofs showing that in hyperbolic ...
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2answers
198 views

Simplest form for locus of latitudes/longitudes equidistant from two given latitudes/longitudes?

Given two latitudes/longitudes (th1,ph1 and th2,ph2), I want to find a simple formula for the locus of th3,ph3 that are equidistant from th1,ph1 and th2,ph2. Mathematica happily spits out an answer ...
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1answer
741 views

Using the spherical law of cosines

Compute angular length c of the great-circle route between these two cities: Daytona Beach (location A): $29^\circ12'\ N, 81^\circ1' \ W$. Sidi Ifni (location B): $29^\circ23' \ N. 10^\circ10' \ ...
9
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1answer
427 views

Navigating though the surface of a hypersphere in a computer game

People in StackOverflow seems not so into this theme, so I thought I could have better luck in here. I had the idea of an spaceship game where the world is confined in the surface of an 4-D ...
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4answers
1k views

How to find the distance between a point and line joining two points on a sphere?

How do I calculate the distance between the line joining the two points on a spherical surface and another point on same surface? I have illustrated my problem in the image below. In the above ...
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4answers
390 views

Distance between points on a face

Given a 2D picture of a face, how is it possible to measure the distance between two different points on the surface of the face? Thanks Joel
3
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1answer
394 views

How to construct the midpoint in spherical geometry?

I am looking for the the method of constructing the midpoint of two points in spherical geometry. The only tools allowed for the construction are a pair of spherical compasses and a spherical ruler. ...
0
votes
1answer
247 views

How to find the intersection between the great circle and a hyperplane?

Let $s = (\frac{1}{\sqrt{d}}, \ldots, \frac{1}{\sqrt{d}})$ and $u \in \mathbb{R}^d$ be two distinct unit norm vectors in the first orthant. Consider moving along the great circle defined by $s$ and ...
1
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1answer
141 views

How to test whether spherical caps intersect?

I have a unit sphere, on the surface of which are defined spherical caps. I typically characterize the caps by the unit vector $n$ from the center of the sphere to the top of the cap, and the angle ...
3
votes
3answers
4k views

Area of a spherical triangle

A triangle on a sphere is comprised of points A, B and C. How to determine its area? I know the formula: A = E * R^2, where R is radius of sphere, and E is the excess angle of (a + b + c - pi), but ...
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1answer
394 views

Spherical geometry: Arbitrary point between two points

If A and B are two points on the earth, how could I find any arbitrary point between them along the shortest distance side of their great circle path? Points are in radians longitude = 0 to 2pi ...
3
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2answers
3k views

Napier's Rules applied to spherical distance calculations

I was in the middle of writing the same old geographic distance calculation using the Haversine formula when it occurred to me: shouldn't there be simpler way to do this? Haversine is of course ...
6
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2answers
878 views

How do I map a spherical triangle to a plane triangle?

My goal here is to make my own custom "polyhedral map" of Earth. If you print out something from the "Map Fold-outs" page, you will have something almost exactly like what I'm trying to make. I have ...
6
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1answer
913 views

How to calculate a heading on the earths surface?

Given an initial position and a subsequent position, each given by latitude and longitude in the WGS-84 system. How do you determine the heading in degrees clockwise from true north of movement?